calc.texi   [plain text]


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@c [title]
@settitle GNU Emacs Calc 2.1 Manual
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@c @expr{expr} will typeset an expression;
@c $x$ in TeX, @samp{x} otherwise.

@iftex
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@math{@pi{}}
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@math{@pi/\den\}
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@expr{pi}
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@expr{pi/\den\}
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@copying
This file documents Calc, the GNU Emacs calculator.

Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
2005, 2006, 2007 Free Software Foundation, Inc.

@quotation
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with the
Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
Texts as in (a) below.  A copy of the license is included in the section
entitled ``GNU Free Documentation License.''

(a) The FSF's Back-Cover Text is: ``You have freedom to copy and modify
this GNU Manual, like GNU software.  Copies published by the Free
Software Foundation raise funds for GNU development.''
@end quotation
@end copying

@dircategory Emacs
@direntry
* Calc: (calc).         Advanced desk calculator and mathematical tool.
@end direntry

@titlepage
@sp 6
@center @titlefont{Calc Manual}
@sp 4
@center GNU Emacs Calc Version 2.1
@c [volume]
@sp 5
@center Dave Gillespie
@center daveg@@synaptics.com
@page

@vskip 0pt plus 1filll
Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
   2005, 2006, 2007 Free Software Foundation, Inc.
@insertcopying
@end titlepage

@c [begin]
@ifinfo
@node Top, Getting Started, (dir), (dir)
@chapter The GNU Emacs Calculator

@noindent
@dfn{Calc} is an advanced desk calculator and mathematical tool
that runs as part of the GNU Emacs environment.

This manual is divided into three major parts: ``Getting Started,''
the ``Calc Tutorial,'' and the ``Calc Reference.''  The Tutorial
introduces all the major aspects of Calculator use in an easy,
hands-on way.  The remainder of the manual is a complete reference to
the features of the Calculator.

For help in the Emacs Info system (which you are using to read this
file), type @kbd{?}.  (You can also type @kbd{h} to run through a
longer Info tutorial.)

@end ifinfo
@menu
* Getting Started::       General description and overview.
* Interactive Tutorial::
* Tutorial::              A step-by-step introduction for beginners.

* Introduction::          Introduction to the Calc reference manual.
* Data Types::            Types of objects manipulated by Calc.
* Stack and Trail::       Manipulating the stack and trail buffers.
* Mode Settings::         Adjusting display format and other modes.
* Arithmetic::            Basic arithmetic functions.
* Scientific Functions::  Transcendentals and other scientific functions.
* Matrix Functions::      Operations on vectors and matrices.
* Algebra::               Manipulating expressions algebraically.
* Units::                 Operations on numbers with units.
* Store and Recall::      Storing and recalling variables.
* Graphics::              Commands for making graphs of data.
* Kill and Yank::         Moving data into and out of Calc.
* Keypad Mode::           Operating Calc from a keypad.
* Embedded Mode::         Working with formulas embedded in a file.
* Programming::           Calc as a programmable calculator.

* Copying::               How you can copy and share Calc.
* GNU Free Documentation License:: The license for this documentation.
* Customizing Calc::      Customizing Calc.
* Reporting Bugs::        How to report bugs and make suggestions.

* Summary::               Summary of Calc commands and functions.

* Key Index::             The standard Calc key sequences.
* Command Index::         The interactive Calc commands.
* Function Index::        Functions (in algebraic formulas).
* Concept Index::         General concepts.
* Variable Index::        Variables used by Calc (both user and internal).
* Lisp Function Index::   Internal Lisp math functions.
@end menu

@node Getting Started, Interactive Tutorial, Top, Top
@chapter Getting Started
@noindent
This chapter provides a general overview of Calc, the GNU Emacs
Calculator:  What it is, how to start it and how to exit from it,
and what are the various ways that it can be used.

@menu
* What is Calc::
* About This Manual::
* Notations Used in This Manual::
* Demonstration of Calc::
* Using Calc::
* History and Acknowledgements::
@end menu

@node What is Calc, About This Manual, Getting Started, Getting Started
@section What is Calc?

@noindent
@dfn{Calc} is an advanced calculator and mathematical tool that runs as
part of the GNU Emacs environment.  Very roughly based on the HP-28/48
series of calculators, its many features include:

@itemize @bullet
@item
Choice of algebraic or RPN (stack-based) entry of calculations.

@item
Arbitrary precision integers and floating-point numbers.

@item
Arithmetic on rational numbers, complex numbers (rectangular and polar),
error forms with standard deviations, open and closed intervals, vectors
and matrices, dates and times, infinities, sets, quantities with units,
and algebraic formulas.

@item
Mathematical operations such as logarithms and trigonometric functions.

@item
Programmer's features (bitwise operations, non-decimal numbers).

@item
Financial functions such as future value and internal rate of return.

@item
Number theoretical features such as prime factorization and arithmetic
modulo @var{m} for any @var{m}.

@item
Algebraic manipulation features, including symbolic calculus.

@item
Moving data to and from regular editing buffers.

@item
Embedded mode for manipulating Calc formulas and data directly
inside any editing buffer.

@item
Graphics using GNUPLOT, a versatile (and free) plotting program.

@item
Easy programming using keyboard macros, algebraic formulas,
algebraic rewrite rules, or extended Emacs Lisp.
@end itemize

Calc tries to include a little something for everyone; as a result it is
large and might be intimidating to the first-time user.  If you plan to
use Calc only as a traditional desk calculator, all you really need to
read is the ``Getting Started'' chapter of this manual and possibly the
first few sections of the tutorial.  As you become more comfortable with
the program you can learn its additional features.  Calc does not
have the scope and depth of a fully-functional symbolic math package,
but Calc has the advantages of convenience, portability, and freedom.

@node About This Manual, Notations Used in This Manual, What is Calc, Getting Started
@section About This Manual

@noindent
This document serves as a complete description of the GNU Emacs
Calculator.  It works both as an introduction for novices, and as
a reference for experienced users.  While it helps to have some
experience with GNU Emacs in order to get the most out of Calc,
this manual ought to be readable even if you don't know or use Emacs
regularly.

@ifinfo
The manual is divided into three major parts:@: the ``Getting
Started'' chapter you are reading now, the Calc tutorial (chapter 2),
and the Calc reference manual (the remaining chapters and appendices).
@end ifinfo
@iftex
The manual is divided into three major parts:@: the ``Getting
Started'' chapter you are reading now, the Calc tutorial (chapter 2),
and the Calc reference manual (the remaining chapters and appendices).
@c [when-split]
@c This manual has been printed in two volumes, the @dfn{Tutorial} and the
@c @dfn{Reference}.  Both volumes include a copy of the ``Getting Started''
@c chapter.
@end iftex

If you are in a hurry to use Calc, there is a brief ``demonstration''
below which illustrates the major features of Calc in just a couple of
pages.  If you don't have time to go through the full tutorial, this
will show you everything you need to know to begin.
@xref{Demonstration of Calc}.

The tutorial chapter walks you through the various parts of Calc
with lots of hands-on examples and explanations.  If you are new
to Calc and you have some time, try going through at least the
beginning of the tutorial.  The tutorial includes about 70 exercises
with answers.  These exercises give you some guided practice with
Calc, as well as pointing out some interesting and unusual ways
to use its features.

The reference section discusses Calc in complete depth.  You can read
the reference from start to finish if you want to learn every aspect
of Calc.  Or, you can look in the table of contents or the Concept
Index to find the parts of the manual that discuss the things you
need to know.

@cindex Marginal notes
Every Calc keyboard command is listed in the Calc Summary, and also
in the Key Index.  Algebraic functions, @kbd{M-x} commands, and
variables also have their own indices.  
@texline Each
@infoline In the printed manual, each
paragraph that is referenced in the Key or Function Index is marked
in the margin with its index entry.

@c [fix-ref Help Commands]
You can access this manual on-line at any time within Calc by
pressing the @kbd{h i} key sequence.  Outside of the Calc window,
you can press @kbd{C-x * i} to read the manual on-line.  Also, you
can jump directly to the Tutorial by pressing @kbd{h t} or @kbd{C-x * t},
or to the Summary by pressing @kbd{h s} or @kbd{C-x * s}.  Within Calc,
you can also go to the part of the manual describing any Calc key,
function, or variable using @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v},
respectively.  @xref{Help Commands}.

The Calc manual can be printed, but because the manual is so large, you
should only make a printed copy if you really need it.  To print the
manual, you will need the @TeX{} typesetting program (this is a free
program by Donald Knuth at Stanford University) as well as the
@file{texindex} program and @file{texinfo.tex} file, both of which can
be obtained from the FSF as part of the @code{texinfo} package.
To print the Calc manual in one huge tome, you will need the
source code to this manual, @file{calc.texi}, available as part of the
Emacs source.  Once you have this file, type @kbd{texi2dvi calc.texi}.
Alternatively, change to the @file{man} subdirectory of the Emacs
source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
get some ``overfull box'' warnings while @TeX{} runs.)
The result will be a device-independent output file called
@file{calc.dvi}, which you must print in whatever way is right
for your system.  On many systems, the command is

@example
lpr -d calc.dvi
@end example

@noindent
or

@example
dvips calc.dvi
@end example

@c Printed copies of this manual are also available from the Free Software
@c Foundation.

@node Notations Used in This Manual, Demonstration of Calc, About This Manual, Getting Started
@section Notations Used in This Manual

@noindent
This section describes the various notations that are used
throughout the Calc manual.

In keystroke sequences, uppercase letters mean you must hold down
the shift key while typing the letter.  Keys pressed with Control
held down are shown as @kbd{C-x}.  Keys pressed with Meta held down
are shown as @kbd{M-x}.  Other notations are @key{RET} for the
Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
@key{DEL} for the Delete key, and @key{LFD} for the Line-Feed key.
The @key{DEL} key is called Backspace on some keyboards, it is
whatever key you would use to correct a simple typing error when
regularly using Emacs.

(If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
If you don't have a Meta key, look for Alt or Extend Char.  You can
also press @key{ESC} or @kbd{C-[} first to get the same effect, so
that @kbd{M-x}, @kbd{@key{ESC} x}, and @kbd{C-[ x} are all equivalent.)

Sometimes the @key{RET} key is not shown when it is ``obvious''
that you must press @key{RET} to proceed.  For example, the @key{RET}
is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.

Commands are generally shown like this:  @kbd{p} (@code{calc-precision})
or @kbd{C-x * k} (@code{calc-keypad}).  This means that the command is
normally used by pressing the @kbd{p} key or @kbd{C-x * k} key sequence,
but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.

Commands that correspond to functions in algebraic notation
are written:  @kbd{C} (@code{calc-cos}) [@code{cos}].  This means
the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
the corresponding function in an algebraic-style formula would
be @samp{cos(@var{x})}.

A few commands don't have key equivalents:  @code{calc-sincos}
[@code{sincos}].

@node Demonstration of Calc, Using Calc, Notations Used in This Manual, Getting Started
@section A Demonstration of Calc

@noindent
@cindex Demonstration of Calc
This section will show some typical small problems being solved with
Calc.  The focus is more on demonstration than explanation, but
everything you see here will be covered more thoroughly in the
Tutorial.

To begin, start Emacs if necessary (usually the command @code{emacs}
does this), and type @kbd{C-x * c} to start the
Calculator.  (You can also use @kbd{M-x calc} if this doesn't work.
@xref{Starting Calc}, for various ways of starting the Calculator.)

Be sure to type all the sample input exactly, especially noting the
difference between lower-case and upper-case letters.  Remember,
@key{RET}, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
Delete, and Space keys.

@strong{RPN calculation.}  In RPN, you type the input number(s) first,
then the command to operate on the numbers.

@noindent
Type @kbd{2 @key{RET} 3 + Q} to compute 
@texline @math{\sqrt{2+3} = 2.2360679775}.
@infoline the square root of 2+3, which is 2.2360679775.

@noindent
Type @kbd{P 2 ^} to compute 
@texline @math{\pi^2 = 9.86960440109}.
@infoline the value of `pi' squared, 9.86960440109.

@noindent
Type @key{TAB} to exchange the order of these two results.

@noindent
Type @kbd{- I H S} to subtract these results and compute the Inverse
Hyperbolic sine of the difference, 2.72996136574.

@noindent
Type @key{DEL} to erase this result.

@strong{Algebraic calculation.}  You can also enter calculations using
conventional ``algebraic'' notation.  To enter an algebraic formula,
use the apostrophe key.

@noindent
Type @kbd{' sqrt(2+3) @key{RET}} to compute 
@texline @math{\sqrt{2+3}}.
@infoline the square root of 2+3.

@noindent
Type @kbd{' pi^2 @key{RET}} to enter 
@texline @math{\pi^2}.
@infoline `pi' squared.  
To evaluate this symbolic formula as a number, type @kbd{=}.

@noindent
Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
result from the most-recent and compute the Inverse Hyperbolic sine.

@strong{Keypad mode.}  If you are using the X window system, press
@w{@kbd{C-x * k}} to get Keypad mode.  (If you don't use X, skip to
the next section.)

@noindent
Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
``buttons'' using your left mouse button.

@noindent
Click on @key{PI}, @key{2}, and @tfn{y^x}.

@noindent
Click on @key{INV}, then @key{ENTER} to swap the two results.

@noindent
Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.

@noindent
Click on @key{<-} to erase the result, then click @key{OFF} to turn
the Keypad Calculator off.

@strong{Grabbing data.}  Type @kbd{C-x * x} if necessary to exit Calc.
Now select the following numbers as an Emacs region:  ``Mark'' the
front of the list by typing @kbd{C-@key{SPC}} or @kbd{C-@@} there,
then move to the other end of the list.  (Either get this list from
the on-line copy of this manual, accessed by @w{@kbd{C-x * i}}, or just
type these numbers into a scratch file.)  Now type @kbd{C-x * g} to
``grab'' these numbers into Calc.

@example
@group
1.23  1.97
1.6   2
1.19  1.08
@end group
@end example

@noindent
The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
Type @w{@kbd{V R +}} to compute the sum of these numbers.

@noindent
Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
the product of the numbers.

@noindent
You can also grab data as a rectangular matrix.  Place the cursor on
the upper-leftmost @samp{1} and set the mark, then move to just after
the lower-right @samp{8} and press @kbd{C-x * r}.

@noindent
Type @kbd{v t} to transpose this 
@texline @math{3\times2}
@infoline 3x2 
matrix into a 
@texline @math{2\times3}
@infoline 2x3
matrix.  Type @w{@kbd{v u}} to unpack the rows into two separate
vectors.  Now type @w{@kbd{V R + @key{TAB} V R +}} to compute the sums
of the two original columns. (There is also a special
grab-and-sum-columns command, @kbd{C-x * :}.)

@strong{Units conversion.}  Units are entered algebraically.
Type @w{@kbd{' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour.
Type @w{@kbd{u c km/hr @key{RET}}}.  Type @w{@kbd{u c m/s @key{RET}}}.

@strong{Date arithmetic.}  Type @kbd{t N} to get the current date and
time.  Type @kbd{90 +} to find the date 90 days from now.  Type
@kbd{' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see how
many weeks have passed since then.

@strong{Algebra.}  Algebraic entries can also include formulas
or equations involving variables.  Type @kbd{@w{' [x + y} = a, x y = 1] @key{RET}}
to enter a pair of equations involving three variables.
(Note the leading apostrophe in this example; also, note that the space
between @samp{x y} is required.)  Type @w{@kbd{a S x,y @key{RET}}} to solve
these equations for the variables @expr{x} and @expr{y}.

@noindent
Type @kbd{d B} to view the solutions in more readable notation.
Type @w{@kbd{d C}} to view them in C language notation, @kbd{d T}
to view them in the notation for the @TeX{} typesetting system,
and @kbd{d L} to view them in the notation for the La@TeX{} typesetting
system.  Type @kbd{d N} to return to normal notation.

@noindent
Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these formulas.
(That's a letter @kbd{l}, not a numeral @kbd{1}.)

@iftex
@strong{Help functions.}  You can read about any command in the on-line
manual.  Type @kbd{C-x * c} to return to Calc after each of these
commands: @kbd{h k t N} to read about the @kbd{t N} command,
@kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
@kbd{h s} to read the Calc summary.
@end iftex
@ifinfo
@strong{Help functions.}  You can read about any command in the on-line
manual.  Remember to type the letter @kbd{l}, then @kbd{C-x * c}, to
return here after each of these commands: @w{@kbd{h k t N}} to read
about the @w{@kbd{t N}} command, @kbd{h f sqrt @key{RET}} to read about the
@code{sqrt} function, and @kbd{h s} to read the Calc summary.
@end ifinfo

Press @key{DEL} repeatedly to remove any leftover results from the stack.
To exit from Calc, press @kbd{q} or @kbd{C-x * c} again.

@node Using Calc, History and Acknowledgements, Demonstration of Calc, Getting Started
@section Using Calc

@noindent
Calc has several user interfaces that are specialized for
different kinds of tasks.  As well as Calc's standard interface,
there are Quick mode, Keypad mode, and Embedded mode.

@menu
* Starting Calc::
* The Standard Interface::
* Quick Mode Overview::
* Keypad Mode Overview::
* Standalone Operation::
* Embedded Mode Overview::
* Other C-x * Commands::
@end menu

@node Starting Calc, The Standard Interface, Using Calc, Using Calc
@subsection Starting Calc

@noindent
On most systems, you can type @kbd{C-x *} to start the Calculator.
The key sequence @kbd{C-x *} is bound to the command @code{calc-dispatch}, 
which can be rebound if convenient (@pxref{Customizing Calc}).

When you press @kbd{C-x *}, Emacs waits for you to press a second key to
complete the command.  In this case, you will follow @kbd{C-x *} with a
letter (upper- or lower-case, it doesn't matter for @kbd{C-x *}) that says
which Calc interface you want to use.

To get Calc's standard interface, type @kbd{C-x * c}.  To get
Keypad mode, type @kbd{C-x * k}.  Type @kbd{C-x * ?} to get a brief
list of the available options, and type a second @kbd{?} to get
a complete list.

To ease typing, @kbd{C-x * *} also works to start Calc.  It starts the
same interface (either @kbd{C-x * c} or @w{@kbd{C-x * k}}) that you last
used, selecting the @kbd{C-x * c} interface by default.

If @kbd{C-x *} doesn't work for you, you can always type explicit
commands like @kbd{M-x calc} (for the standard user interface) or
@w{@kbd{M-x calc-keypad}} (for Keypad mode).  First type @kbd{M-x}
(that's Meta with the letter @kbd{x}), then, at the prompt,
type the full command (like @kbd{calc-keypad}) and press Return.

The same commands (like @kbd{C-x * c} or @kbd{C-x * *}) that start
the Calculator also turn it off if it is already on.

@node The Standard Interface, Quick Mode Overview, Starting Calc, Using Calc
@subsection The Standard Calc Interface

@noindent
@cindex Standard user interface
Calc's standard interface acts like a traditional RPN calculator,
operated by the normal Emacs keyboard.  When you type @kbd{C-x * c}
to start the Calculator, the Emacs screen splits into two windows
with the file you were editing on top and Calc on the bottom.

@smallexample
@group

...
--**-Emacs: myfile             (Fundamental)----All----------------------
--- Emacs Calculator Mode ---                   |Emacs Calculator Trail
2:  17.3                                        |    17.3
1:  -5                                          |    3
    .                                           |    2
                                                |    4
                                                |  * 8
                                                |  ->-5
                                                |
--%%-Calc: 12 Deg       (Calculator)----All----- --%%-Emacs: *Calc Trail*
@end group
@end smallexample

In this figure, the mode-line for @file{myfile} has moved up and the
``Calculator'' window has appeared below it.  As you can see, Calc
actually makes two windows side-by-side.  The lefthand one is
called the @dfn{stack window} and the righthand one is called the
@dfn{trail window.}  The stack holds the numbers involved in the
calculation you are currently performing.  The trail holds a complete
record of all calculations you have done.  In a desk calculator with
a printer, the trail corresponds to the paper tape that records what
you do.

In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
were first entered into the Calculator, then the 2 and 4 were
multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
(The @samp{>} symbol shows that this was the most recent calculation.)
The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.

Most Calculator commands deal explicitly with the stack only, but
there is a set of commands that allow you to search back through
the trail and retrieve any previous result.

Calc commands use the digits, letters, and punctuation keys.
Shifted (i.e., upper-case) letters are different from lowercase
letters.  Some letters are @dfn{prefix} keys that begin two-letter
commands.  For example, @kbd{e} means ``enter exponent'' and shifted
@kbd{E} means @expr{e^x}.  With the @kbd{d} (``display modes'') prefix
the letter ``e'' takes on very different meanings:  @kbd{d e} means
``engineering notation'' and @kbd{d E} means ``@dfn{eqn} language mode.''

There is nothing stopping you from switching out of the Calc
window and back into your editing window, say by using the Emacs
@w{@kbd{C-x o}} (@code{other-window}) command.  When the cursor is
inside a regular window, Emacs acts just like normal.  When the
cursor is in the Calc stack or trail windows, keys are interpreted
as Calc commands.

When you quit by pressing @kbd{C-x * c} a second time, the Calculator
windows go away but the actual Stack and Trail are not gone, just
hidden.  When you press @kbd{C-x * c} once again you will get the
same stack and trail contents you had when you last used the
Calculator.

The Calculator does not remember its state between Emacs sessions.
Thus if you quit Emacs and start it again, @kbd{C-x * c} will give you
a fresh stack and trail.  There is a command (@kbd{m m}) that lets
you save your favorite mode settings between sessions, though.
One of the things it saves is which user interface (standard or
Keypad) you last used; otherwise, a freshly started Emacs will
always treat @kbd{C-x * *} the same as @kbd{C-x * c}.

The @kbd{q} key is another equivalent way to turn the Calculator off.

If you type @kbd{C-x * b} first and then @kbd{C-x * c}, you get a
full-screen version of Calc (@code{full-calc}) in which the stack and
trail windows are still side-by-side but are now as tall as the whole
Emacs screen.  When you press @kbd{q} or @kbd{C-x * c} again to quit,
the file you were editing before reappears.  The @kbd{C-x * b} key
switches back and forth between ``big'' full-screen mode and the
normal partial-screen mode.

Finally, @kbd{C-x * o} (@code{calc-other-window}) is like @kbd{C-x * c}
except that the Calc window is not selected.  The buffer you were
editing before remains selected instead.  @kbd{C-x * o} is a handy
way to switch out of Calc momentarily to edit your file; type
@kbd{C-x * c} to switch back into Calc when you are done.

@node Quick Mode Overview, Keypad Mode Overview, The Standard Interface, Using Calc
@subsection Quick Mode (Overview)

@noindent
@dfn{Quick mode} is a quick way to use Calc when you don't need the
full complexity of the stack and trail.  To use it, type @kbd{C-x * q}
(@code{quick-calc}) in any regular editing buffer.

Quick mode is very simple:  It prompts you to type any formula in
standard algebraic notation (like @samp{4 - 2/3}) and then displays
the result at the bottom of the Emacs screen (@mathit{3.33333333333}
in this case).  You are then back in the same editing buffer you
were in before, ready to continue editing or to type @kbd{C-x * q}
again to do another quick calculation.  The result of the calculation
will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
at this point will yank the result into your editing buffer.

Calc mode settings affect Quick mode, too, though you will have to
go into regular Calc (with @kbd{C-x * c}) to change the mode settings.

@c [fix-ref Quick Calculator mode]
@xref{Quick Calculator}, for further information.

@node Keypad Mode Overview, Standalone Operation, Quick Mode Overview, Using Calc
@subsection Keypad Mode (Overview)

@noindent
@dfn{Keypad mode} is a mouse-based interface to the Calculator.
It is designed for use with terminals that support a mouse.  If you
don't have a mouse, you will have to operate Keypad mode with your
arrow keys (which is probably more trouble than it's worth).

Type @kbd{C-x * k} to turn Keypad mode on or off.  Once again you
get two new windows, this time on the righthand side of the screen
instead of at the bottom.  The upper window is the familiar Calc
Stack; the lower window is a picture of a typical calculator keypad.

@tex
\dimen0=\pagetotal%
\advance \dimen0 by 24\baselineskip%
\ifdim \dimen0>\pagegoal \vfill\eject \fi%
\medskip
@end tex
@smallexample
@group
|--- Emacs Calculator Mode ---
|2:  17.3
|1:  -5
|    .
|--%%-Calc: 12 Deg       (Calcul
|----+-----Calc 2.1------+----1
|FLR |CEIL|RND |TRNC|CLN2|FLT |
|----+----+----+----+----+----|
| LN |EXP |    |ABS |IDIV|MOD |
|----+----+----+----+----+----|
|SIN |COS |TAN |SQRT|y^x |1/x |
|----+----+----+----+----+----|
|  ENTER  |+/- |EEX |UNDO| <- |
|-----+---+-+--+--+-+---++----|
| INV |  7  |  8  |  9  |  /  |
|-----+-----+-----+-----+-----|
| HYP |  4  |  5  |  6  |  *  |
|-----+-----+-----+-----+-----|
|EXEC |  1  |  2  |  3  |  -  |
|-----+-----+-----+-----+-----|
| OFF |  0  |  .  | PI  |  +  |
|-----+-----+-----+-----+-----+
@end group
@end smallexample

Keypad mode is much easier for beginners to learn, because there
is no need to memorize lots of obscure key sequences.  But not all
commands in regular Calc are available on the Keypad.  You can
always switch the cursor into the Calc stack window to use
standard Calc commands if you need.  Serious Calc users, though,
often find they prefer the standard interface over Keypad mode.

To operate the Calculator, just click on the ``buttons'' of the
keypad using your left mouse button.  To enter the two numbers
shown here you would click @w{@kbd{1 7 .@: 3 ENTER 5 +/- ENTER}}; to
add them together you would then click @kbd{+} (to get 12.3 on
the stack).

If you click the right mouse button, the top three rows of the
keypad change to show other sets of commands, such as advanced
math functions, vector operations, and operations on binary
numbers.

Because Keypad mode doesn't use the regular keyboard, Calc leaves
the cursor in your original editing buffer.  You can type in
this buffer in the usual way while also clicking on the Calculator
keypad.  One advantage of Keypad mode is that you don't need an
explicit command to switch between editing and calculating.

If you press @kbd{C-x * b} first, you get a full-screen Keypad mode
(@code{full-calc-keypad}) with three windows:  The keypad in the lower
left, the stack in the lower right, and the trail on top.

@c [fix-ref Keypad Mode]
@xref{Keypad Mode}, for further information.

@node Standalone Operation, Embedded Mode Overview, Keypad Mode Overview, Using Calc
@subsection Standalone Operation

@noindent
@cindex Standalone Operation
If you are not in Emacs at the moment but you wish to use Calc,
you must start Emacs first.  If all you want is to run Calc, you
can give the commands:

@example
emacs -f full-calc
@end example

@noindent
or

@example
emacs -f full-calc-keypad
@end example

@noindent
which run a full-screen Calculator (as if by @kbd{C-x * b C-x * c}) or
a full-screen X-based Calculator (as if by @kbd{C-x * b C-x * k}).
In standalone operation, quitting the Calculator (by pressing
@kbd{q} or clicking on the keypad @key{EXIT} button) quits Emacs
itself.

@node Embedded Mode Overview, Other C-x * Commands, Standalone Operation, Using Calc
@subsection Embedded Mode (Overview)

@noindent
@dfn{Embedded mode} is a way to use Calc directly from inside an
editing buffer.  Suppose you have a formula written as part of a
document like this:

@smallexample
@group
The derivative of

                                   ln(ln(x))

is
@end group
@end smallexample

@noindent
and you wish to have Calc compute and format the derivative for
you and store this derivative in the buffer automatically.  To
do this with Embedded mode, first copy the formula down to where
you want the result to be:

@smallexample
@group
The derivative of

                                   ln(ln(x))

is

                                   ln(ln(x))
@end group
@end smallexample

Now, move the cursor onto this new formula and press @kbd{C-x * e}.
Calc will read the formula (using the surrounding blank lines to
tell how much text to read), then push this formula (invisibly)
onto the Calc stack.  The cursor will stay on the formula in the
editing buffer, but the buffer's mode line will change to look
like the Calc mode line (with mode indicators like @samp{12 Deg}
and so on).  Even though you are still in your editing buffer,
the keyboard now acts like the Calc keyboard, and any new result
you get is copied from the stack back into the buffer.  To take
the derivative, you would type @kbd{a d x @key{RET}}.

@smallexample
@group
The derivative of

                                   ln(ln(x))

is

1 / ln(x) x
@end group
@end smallexample

To make this look nicer, you might want to press @kbd{d =} to center
the formula, and even @kbd{d B} to use Big display mode.

@smallexample
@group
The derivative of

                                   ln(ln(x))

is
% [calc-mode: justify: center]
% [calc-mode: language: big]

                                       1
                                    -------
                                    ln(x) x
@end group
@end smallexample

Calc has added annotations to the file to help it remember the modes
that were used for this formula.  They are formatted like comments
in the @TeX{} typesetting language, just in case you are using @TeX{} or
La@TeX{}. (In this example @TeX{} is not being used, so you might want
to move these comments up to the top of the file or otherwise put them
out of the way.)

As an extra flourish, we can add an equation number using a
righthand label:  Type @kbd{d @} (1) @key{RET}}.

@smallexample
@group
% [calc-mode: justify: center]
% [calc-mode: language: big]
% [calc-mode: right-label: " (1)"]

                                       1
                                    -------                      (1)
                                    ln(x) x
@end group
@end smallexample

To leave Embedded mode, type @kbd{C-x * e} again.  The mode line
and keyboard will revert to the way they were before.

The related command @kbd{C-x * w} operates on a single word, which
generally means a single number, inside text.  It uses any
non-numeric characters rather than blank lines to delimit the
formula it reads.  Here's an example of its use:

@smallexample
A slope of one-third corresponds to an angle of 1 degrees.
@end smallexample

Place the cursor on the @samp{1}, then type @kbd{C-x * w} to enable
Embedded mode on that number.  Now type @kbd{3 /} (to get one-third),
and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
then @w{@kbd{C-x * w}} again to exit Embedded mode.

@smallexample
A slope of one-third corresponds to an angle of 18.4349488229 degrees.
@end smallexample

@c [fix-ref Embedded Mode]
@xref{Embedded Mode}, for full details.

@node Other C-x * Commands,  , Embedded Mode Overview, Using Calc
@subsection Other @kbd{C-x *} Commands

@noindent
Two more Calc-related commands are @kbd{C-x * g} and @kbd{C-x * r},
which ``grab'' data from a selected region of a buffer into the
Calculator.  The region is defined in the usual Emacs way, by
a ``mark'' placed at one end of the region, and the Emacs
cursor or ``point'' placed at the other.

The @kbd{C-x * g} command reads the region in the usual left-to-right,
top-to-bottom order.  The result is packaged into a Calc vector
of numbers and placed on the stack.  Calc (in its standard
user interface) is then started.  Type @kbd{v u} if you want
to unpack this vector into separate numbers on the stack.  Also,
@kbd{C-u C-x * g} interprets the region as a single number or
formula.

The @kbd{C-x * r} command reads a rectangle, with the point and
mark defining opposite corners of the rectangle.  The result
is a matrix of numbers on the Calculator stack.

Complementary to these is @kbd{C-x * y}, which ``yanks'' the
value at the top of the Calc stack back into an editing buffer.
If you type @w{@kbd{C-x * y}} while in such a buffer, the value is
yanked at the current position.  If you type @kbd{C-x * y} while
in the Calc buffer, Calc makes an educated guess as to which
editing buffer you want to use.  The Calc window does not have
to be visible in order to use this command, as long as there
is something on the Calc stack.

Here, for reference, is the complete list of @kbd{C-x *} commands.
The shift, control, and meta keys are ignored for the keystroke
following @kbd{C-x *}.

@noindent
Commands for turning Calc on and off:

@table @kbd
@item *
Turn Calc on or off, employing the same user interface as last time.

@item =, +, -, /, \, &, #
Alternatives for @kbd{*}.

@item C
Turn Calc on or off using its standard bottom-of-the-screen
interface.  If Calc is already turned on but the cursor is not
in the Calc window, move the cursor into the window.

@item O
Same as @kbd{C}, but don't select the new Calc window.  If
Calc is already turned on and the cursor is in the Calc window,
move it out of that window.

@item B
Control whether @kbd{C-x * c} and @kbd{C-x * k} use the full screen.

@item Q
Use Quick mode for a single short calculation.

@item K
Turn Calc Keypad mode on or off.

@item E
Turn Calc Embedded mode on or off at the current formula.

@item J
Turn Calc Embedded mode on or off, select the interesting part.

@item W
Turn Calc Embedded mode on or off at the current word (number).

@item Z
Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.

@item X
Quit Calc; turn off standard, Keypad, or Embedded mode if on.
(This is like @kbd{q} or @key{OFF} inside of Calc.)
@end table
@iftex
@sp 2
@end iftex

@noindent
Commands for moving data into and out of the Calculator:

@table @kbd
@item G
Grab the region into the Calculator as a vector.

@item R
Grab the rectangular region into the Calculator as a matrix.

@item :
Grab the rectangular region and compute the sums of its columns.

@item _
Grab the rectangular region and compute the sums of its rows.

@item Y
Yank a value from the Calculator into the current editing buffer.
@end table
@iftex
@sp 2
@end iftex

@noindent
Commands for use with Embedded mode:

@table @kbd
@item A
``Activate'' the current buffer.  Locate all formulas that
contain @samp{:=} or @samp{=>} symbols and record their locations
so that they can be updated automatically as variables are changed.

@item D
Duplicate the current formula immediately below and select
the duplicate.

@item F
Insert a new formula at the current point.

@item N
Move the cursor to the next active formula in the buffer.

@item P
Move the cursor to the previous active formula in the buffer.

@item U
Update (i.e., as if by the @kbd{=} key) the formula at the current point.

@item `
Edit (as if by @code{calc-edit}) the formula at the current point.
@end table
@iftex
@sp 2
@end iftex

@noindent
Miscellaneous commands:

@table @kbd
@item I
Run the Emacs Info system to read the Calc manual.
(This is the same as @kbd{h i} inside of Calc.)

@item T
Run the Emacs Info system to read the Calc Tutorial.

@item S
Run the Emacs Info system to read the Calc Summary.

@item L
Load Calc entirely into memory.  (Normally the various parts
are loaded only as they are needed.)

@item M
Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
and record them as the current keyboard macro.

@item 0
(This is the ``zero'' digit key.)  Reset the Calculator to
its initial state:  Empty stack, and initial mode settings.
@end table

@node History and Acknowledgements,  , Using Calc, Getting Started
@section History and Acknowledgements

@noindent
Calc was originally started as a two-week project to occupy a lull
in the author's schedule.  Basically, a friend asked if I remembered
the value of 
@texline @math{2^{32}}.
@infoline @expr{2^32}.  
I didn't offhand, but I said, ``that's easy, just call up an
@code{xcalc}.''  @code{Xcalc} duly reported that the answer to our
question was @samp{4.294967e+09}---with no way to see the full ten
digits even though we knew they were there in the program's memory!  I
was so annoyed, I vowed to write a calculator of my own, once and for
all.

I chose Emacs Lisp, a) because I had always been curious about it
and b) because, being only a text editor extension language after
all, Emacs Lisp would surely reach its limits long before the project
got too far out of hand.

To make a long story short, Emacs Lisp turned out to be a distressingly
solid implementation of Lisp, and the humble task of calculating
turned out to be more open-ended than one might have expected.

Emacs Lisp didn't have built-in floating point math (now it does), so
this had to be
simulated in software.  In fact, Emacs integers will only comfortably
fit six decimal digits or so---not enough for a decent calculator.  So
I had to write my own high-precision integer code as well, and once I had
this I figured that arbitrary-size integers were just as easy as large
integers.  Arbitrary floating-point precision was the logical next step.
Also, since the large integer arithmetic was there anyway it seemed only
fair to give the user direct access to it, which in turn made it practical
to support fractions as well as floats.  All these features inspired me
to look around for other data types that might be worth having.

Around this time, my friend Rick Koshi showed me his nifty new HP-28
calculator.  It allowed the user to manipulate formulas as well as
numerical quantities, and it could also operate on matrices.  I
decided that these would be good for Calc to have, too.  And once
things had gone this far, I figured I might as well take a look at
serious algebra systems for further ideas.  Since these systems did
far more than I could ever hope to implement, I decided to focus on
rewrite rules and other programming features so that users could
implement what they needed for themselves.

Rick complained that matrices were hard to read, so I put in code to
format them in a 2D style.  Once these routines were in place, Big mode
was obligatory.  Gee, what other language modes would be useful?

Scott Hemphill and Allen Knutson, two friends with a strong mathematical
bent, contributed ideas and algorithms for a number of Calc features
including modulo forms, primality testing, and float-to-fraction conversion.

Units were added at the eager insistence of Mass Sivilotti.  Later,
Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
expert assistance with the units table.  As far as I can remember, the
idea of using algebraic formulas and variables to represent units dates
back to an ancient article in Byte magazine about muMath, an early
algebra system for microcomputers.

Many people have contributed to Calc by reporting bugs and suggesting
features, large and small.  A few deserve special mention:  Tim Peters,
who helped develop the ideas that led to the selection commands, rewrite
rules, and many other algebra features; 
@texline Fran\c{c}ois
@infoline Francois
Pinard, who contributed an early prototype of the Calc Summary appendix
as well as providing valuable suggestions in many other areas of Calc;
Carl Witty, whose eagle eyes discovered many typographical and factual
errors in the Calc manual; Tim Kay, who drove the development of
Embedded mode; Ove Ewerlid, who made many suggestions relating to the
algebra commands and contributed some code for polynomial operations;
Randal Schwartz, who suggested the @code{calc-eval} function; Robert
J. Chassell, who suggested the Calc Tutorial and exercises; and Juha
Sarlin, who first worked out how to split Calc into quickly-loading
parts.  Bob Weiner helped immensely with the Lucid Emacs port.

@cindex Bibliography
@cindex Knuth, Art of Computer Programming
@cindex Numerical Recipes
@c Should these be expanded into more complete references?
Among the books used in the development of Calc were Knuth's @emph{Art
of Computer Programming} (especially volume II, @emph{Seminumerical
Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
@emph{CRC Standard Math Tables} (William H. Beyer, ed.); and
Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
Functions}.  Also, of course, Calc could not have been written without
the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
Dan LaLiberte.

Final thanks go to Richard Stallman, without whose fine implementations
of the Emacs editor, language, and environment, Calc would have been
finished in two weeks.

@c [tutorial]

@ifinfo
@c This node is accessed by the `C-x * t' command.
@node Interactive Tutorial, Tutorial, Getting Started, Top
@chapter Tutorial

@noindent
Some brief instructions on using the Emacs Info system for this tutorial:

Press the space bar and Delete keys to go forward and backward in a
section by screenfuls (or use the regular Emacs scrolling commands
for this).

Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
If the section has a @dfn{menu}, press a digit key like @kbd{1}
or @kbd{2} to go to a sub-section from the menu.  Press @kbd{u} to
go back up from a sub-section to the menu it is part of.

Exercises in the tutorial all have cross-references to the
appropriate page of the ``answers'' section.  Press @kbd{f}, then
the exercise number, to see the answer to an exercise.  After
you have followed a cross-reference, you can press the letter
@kbd{l} to return to where you were before.

You can press @kbd{?} at any time for a brief summary of Info commands.

Press @kbd{1} now to enter the first section of the Tutorial.

@menu
* Tutorial::
@end menu
@end ifinfo

@node Tutorial, Introduction, Interactive Tutorial, Top
@chapter Tutorial

@noindent
This chapter explains how to use Calc and its many features, in
a step-by-step, tutorial way.  You are encouraged to run Calc and
work along with the examples as you read (@pxref{Starting Calc}).
If you are already familiar with advanced calculators, you may wish
@c [not-split]
to skip on to the rest of this manual.
@c [when-split]
@c to skip on to volume II of this manual, the @dfn{Calc Reference}.

@c [fix-ref Embedded Mode]
This tutorial describes the standard user interface of Calc only.
The Quick mode and Keypad mode interfaces are fairly
self-explanatory.  @xref{Embedded Mode}, for a description of
the Embedded mode interface.

@ifinfo
The easiest way to read this tutorial on-line is to have two windows on
your Emacs screen, one with Calc and one with the Info system.  (If you
have a printed copy of the manual you can use that instead.)  Press
@kbd{C-x * c} to turn Calc on or to switch into the Calc window, and
press @kbd{C-x * i} to start the Info system or to switch into its window.
Or, you may prefer to use the tutorial in printed form.
@end ifinfo
@iftex
The easiest way to read this tutorial on-line is to have two windows on
your Emacs screen, one with Calc and one with the Info system.  (If you
have a printed copy of the manual you can use that instead.)  Press
@kbd{C-x * c} to turn Calc on or to switch into the Calc window, and
press @kbd{C-x * i} to start the Info system or to switch into its window.
@end iftex

This tutorial is designed to be done in sequence.  But the rest of this
manual does not assume you have gone through the tutorial.  The tutorial
does not cover everything in the Calculator, but it touches on most
general areas.

@ifinfo
You may wish to print out a copy of the Calc Summary and keep notes on
it as you learn Calc.  @xref{About This Manual}, to see how to make a
printed summary.  @xref{Summary}.
@end ifinfo
@iftex
The Calc Summary at the end of the reference manual includes some blank
space for your own use.  You may wish to keep notes there as you learn
Calc.
@end iftex

@menu
* Basic Tutorial::
* Arithmetic Tutorial::
* Vector/Matrix Tutorial::
* Types Tutorial::
* Algebra Tutorial::
* Programming Tutorial::

* Answers to Exercises::
@end menu

@node Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
@section Basic Tutorial

@noindent
In this section, we learn how RPN and algebraic-style calculations
work, how to undo and redo an operation done by mistake, and how
to control various modes of the Calculator.

@menu
* RPN Tutorial::            Basic operations with the stack.
* Algebraic Tutorial::      Algebraic entry; variables.
* Undo Tutorial::           If you make a mistake: Undo and the trail.
* Modes Tutorial::          Common mode-setting commands.
@end menu

@node RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
@subsection RPN Calculations and the Stack

@cindex RPN notation
@ifinfo
@noindent
Calc normally uses RPN notation.  You may be familiar with the RPN
system from Hewlett-Packard calculators, FORTH, or PostScript.
(Reverse Polish Notation, RPN, is named after the Polish mathematician
Jan Lukasiewicz.)
@end ifinfo
@tex
\noindent
Calc normally uses RPN notation.  You may be familiar with the RPN
system from Hewlett-Packard calculators, FORTH, or PostScript.
(Reverse Polish Notation, RPN, is named after the Polish mathematician
Jan \L ukasiewicz.)
@end tex

The central component of an RPN calculator is the @dfn{stack}.  A
calculator stack is like a stack of dishes.  New dishes (numbers) are
added at the top of the stack, and numbers are normally only removed
from the top of the stack.

@cindex Operators
@cindex Operands
In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
and the @expr{+} is the @dfn{operator}.  In an RPN calculator you always
enter the operands first, then the operator.  Each time you type a
number, Calc adds or @dfn{pushes} it onto the top of the Stack.
When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
number of operands from the stack and pushes back the result.

Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
@kbd{2 @key{RET} 3 @key{RET} +}.  (The @key{RET} key, Return, corresponds to
the @key{ENTER} key on traditional RPN calculators.)  Try this now if
you wish; type @kbd{C-x * c} to switch into the Calc window (you can type
@kbd{C-x * c} again or @kbd{C-x * o} to switch back to the Tutorial window).
The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
and pushes the result (5) back onto the stack.  Here's how the stack
will look at various points throughout the calculation:

@smallexample
@group
    .          1:  2          2:  2          1:  5              .
                   .          1:  3              .
                                  .

  C-x * c          2 @key{RET}          3 @key{RET}            +             @key{DEL}
@end group
@end smallexample

The @samp{.} symbol is a marker that represents the top of the stack.
Note that the ``top'' of the stack is really shown at the bottom of
the Stack window.  This may seem backwards, but it turns out to be
less distracting in regular use.

@cindex Stack levels
@cindex Levels of stack
The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
numbers}.  Old RPN calculators always had four stack levels called
@expr{x}, @expr{y}, @expr{z}, and @expr{t}.  Calc's stack can grow
as large as you like, so it uses numbers instead of letters.  Some
stack-manipulation commands accept a numeric argument that says
which stack level to work on.  Normal commands like @kbd{+} always
work on the top few levels of the stack.

@c [fix-ref Truncating the Stack]
The Stack buffer is just an Emacs buffer, and you can move around in
it using the regular Emacs motion commands.  But no matter where the
cursor is, even if you have scrolled the @samp{.} marker out of
view, most Calc commands always move the cursor back down to level 1
before doing anything.  It is possible to move the @samp{.} marker
upwards through the stack, temporarily ``hiding'' some numbers from
commands like @kbd{+}.  This is called @dfn{stack truncation} and
we will not cover it in this tutorial; @pxref{Truncating the Stack},
if you are interested.

You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
@key{RET} +}.  That's because if you type any operator name or
other non-numeric key when you are entering a number, the Calculator
automatically enters that number and then does the requested command.
Thus @kbd{2 @key{RET} 3 +} will work just as well.

Examples in this tutorial will often omit @key{RET} even when the
stack displays shown would only happen if you did press @key{RET}:

@smallexample
@group
1:  2          2:  2          1:  5
    .          1:  3              .
                   .

  2 @key{RET}            3              +
@end group
@end smallexample

@noindent
Here, after pressing @kbd{3} the stack would really show @samp{1:  2}
with @samp{Calc:@: 3} in the minibuffer.  In these situations, you can
press the optional @key{RET} to see the stack as the figure shows.

(@bullet{}) @strong{Exercise 1.}  (This tutorial will include exercises
at various points.  Try them if you wish.  Answers to all the exercises
are located at the end of the Tutorial chapter.  Each exercise will
include a cross-reference to its particular answer.  If you are
reading with the Emacs Info system, press @kbd{f} and the
exercise number to go to the answer, then the letter @kbd{l} to
return to where you were.)

@noindent
Here's the first exercise:  What will the keystrokes @kbd{1 @key{RET} 2
@key{RET} 3 @key{RET} 4 + * -} compute?  (@samp{*} is the symbol for
multiplication.)  Figure it out by hand, then try it with Calc to see
if you're right.  @xref{RPN Answer 1, 1}. (@bullet{})

(@bullet{}) @strong{Exercise 2.}  Compute 
@texline @math{(2\times4) + (7\times9.4) + {5\over4}}
@infoline @expr{2*4 + 7*9.5 + 5/4} 
using the stack.  @xref{RPN Answer 2, 2}. (@bullet{})

The @key{DEL} key is called Backspace on some keyboards.  It is
whatever key you would use to correct a simple typing error when
regularly using Emacs.  The @key{DEL} key pops and throws away the
top value on the stack.  (You can still get that value back from
the Trail if you should need it later on.)  There are many places
in this tutorial where we assume you have used @key{DEL} to erase the
results of the previous example at the beginning of a new example.
In the few places where it is really important to use @key{DEL} to
clear away old results, the text will remind you to do so.

(It won't hurt to let things accumulate on the stack, except that
whenever you give a display-mode-changing command Calc will have to
spend a long time reformatting such a large stack.)

Since the @kbd{-} key is also an operator (it subtracts the top two
stack elements), how does one enter a negative number?  Calc uses
the @kbd{_} (underscore) key to act like the minus sign in a number.
So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.

You can also press @kbd{n}, which means ``change sign.''  It changes
the number at the top of the stack (or the number being entered)
from positive to negative or vice-versa:  @kbd{5 n @key{RET}}.

@cindex Duplicating a stack entry
If you press @key{RET} when you're not entering a number, the effect
is to duplicate the top number on the stack.  Consider this calculation:

@smallexample
@group
1:  3          2:  3          1:  9          2:  9          1:  81
    .          1:  3              .          1:  9              .
                   .                             .

  3 @key{RET}           @key{RET}             *             @key{RET}             *
@end group
@end smallexample

@noindent
(Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
to raise 3 to the fourth power.)

The space-bar key (denoted @key{SPC} here) performs the same function
as @key{RET}; you could replace all three occurrences of @key{RET} in
the above example with @key{SPC} and the effect would be the same.

@cindex Exchanging stack entries
Another stack manipulation key is @key{TAB}.  This exchanges the top
two stack entries.  Suppose you have computed @kbd{2 @key{RET} 3 +}
to get 5, and then you realize what you really wanted to compute
was @expr{20 / (2+3)}.

@smallexample
@group
1:  5          2:  5          2:  20         1:  4
    .          1:  20         1:  5              .
                   .              .

 2 @key{RET} 3 +         20            @key{TAB}             /
@end group
@end smallexample

@noindent
Planning ahead, the calculation would have gone like this:

@smallexample
@group
1:  20         2:  20         3:  20         2:  20         1:  4
    .          1:  2          2:  2          1:  5              .
                   .          1:  3              .
                                  .

  20 @key{RET}         2 @key{RET}            3              +              /
@end group
@end smallexample

A related stack command is @kbd{M-@key{TAB}} (hold @key{META} and type
@key{TAB}).  It rotates the top three elements of the stack upward,
bringing the object in level 3 to the top.

@smallexample
@group
1:  10         2:  10         3:  10         3:  20         3:  30
    .          1:  20         2:  20         2:  30         2:  10
                   .          1:  30         1:  10         1:  20
                                  .              .              .

  10 @key{RET}         20 @key{RET}         30 @key{RET}         M-@key{TAB}          M-@key{TAB}
@end group
@end smallexample

(@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
on the stack.  Figure out how to add one to the number in level 2
without affecting the rest of the stack.  Also figure out how to add
one to the number in level 3.  @xref{RPN Answer 3, 3}. (@bullet{})

Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
arguments from the stack and push a result.  Operations like @kbd{n} and
@kbd{Q} (square root) pop a single number and push the result.  You can
think of them as simply operating on the top element of the stack.

@smallexample
@group
1:  3          1:  9          2:  9          1:  25         1:  5
    .              .          1:  16             .              .
                                  .

  3 @key{RET}          @key{RET} *        4 @key{RET} @key{RET} *        +              Q
@end group
@end smallexample

@noindent
(Note that capital @kbd{Q} means to hold down the Shift key while
typing @kbd{q}.  Remember, plain unshifted @kbd{q} is the Quit command.)

@cindex Pythagorean Theorem
Here we've used the Pythagorean Theorem to determine the hypotenuse of a
right triangle.  Calc actually has a built-in command for that called
@kbd{f h}, but let's suppose we can't remember the necessary keystrokes.
We can still enter it by its full name using @kbd{M-x} notation:

@smallexample
@group
1:  3          2:  3          1:  5
    .          1:  4              .
                   .

  3 @key{RET}          4 @key{RET}      M-x calc-hypot
@end group
@end smallexample

All Calculator commands begin with the word @samp{calc-}.  Since it
gets tiring to type this, Calc provides an @kbd{x} key which is just
like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
prefix for you:

@smallexample
@group
1:  3          2:  3          1:  5
    .          1:  4              .
                   .

  3 @key{RET}          4 @key{RET}         x hypot
@end group
@end smallexample

What happens if you take the square root of a negative number?

@smallexample
@group
1:  4          1:  -4         1:  (0, 2)
    .              .              .

  4 @key{RET}            n              Q
@end group
@end smallexample

@noindent
The notation @expr{(a, b)} represents a complex number.
Complex numbers are more traditionally written @expr{a + b i};
Calc can display in this format, too, but for now we'll stick to the
@expr{(a, b)} notation.

If you don't know how complex numbers work, you can safely ignore this
feature.  Complex numbers only arise from operations that would be
errors in a calculator that didn't have complex numbers.  (For example,
taking the square root or logarithm of a negative number produces a
complex result.)

Complex numbers are entered in the notation shown.  The @kbd{(} and
@kbd{,} and @kbd{)} keys manipulate ``incomplete complex numbers.''

@smallexample
@group
1:  ( ...      2:  ( ...      1:  (2, ...    1:  (2, ...    1:  (2, 3)
    .          1:  2              .              3              .
                   .                             .

    (              2              ,              3              )
@end group
@end smallexample

You can perform calculations while entering parts of incomplete objects.
However, an incomplete object cannot actually participate in a calculation:

@smallexample
@group
1:  ( ...      2:  ( ...      3:  ( ...      1:  ( ...      1:  ( ...
    .          1:  2          2:  2              5              5
                   .          1:  3              .              .
                                  .
                                                             (error)
    (             2 @key{RET}           3              +              +
@end group
@end smallexample

@noindent
Adding 5 to an incomplete object makes no sense, so the last command
produces an error message and leaves the stack the same.

Incomplete objects can't participate in arithmetic, but they can be
moved around by the regular stack commands.

@smallexample
@group
2:  2          3:  2          3:  3          1:  ( ...      1:  (2, 3)
1:  3          2:  3          2:  ( ...          2              .
    .          1:  ( ...      1:  2              3
                   .              .              .

2 @key{RET} 3 @key{RET}        (            M-@key{TAB}          M-@key{TAB}            )
@end group
@end smallexample

@noindent
Note that the @kbd{,} (comma) key did not have to be used here.
When you press @kbd{)} all the stack entries between the incomplete
entry and the top are collected, so there's never really a reason
to use the comma.  It's up to you.

(@bullet{}) @strong{Exercise 4.}  To enter the complex number @expr{(2, 3)},
your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}.  What happened?
(Joe thought of a clever way to correct his mistake in only two
keystrokes, but it didn't quite work.  Try it to find out why.)
@xref{RPN Answer 4, 4}. (@bullet{})

Vectors are entered the same way as complex numbers, but with square
brackets in place of parentheses.  We'll meet vectors again later in
the tutorial.

Any Emacs command can be given a @dfn{numeric prefix argument} by
typing a series of @key{META}-digits beforehand.  If @key{META} is
awkward for you, you can instead type @kbd{C-u} followed by the
necessary digits.  Numeric prefix arguments can be negative, as in
@kbd{M-- M-3 M-5} or @w{@kbd{C-u - 3 5}}.  Calc commands use numeric
prefix arguments in a variety of ways.  For example, a numeric prefix
on the @kbd{+} operator adds any number of stack entries at once:

@smallexample
@group
1:  10         2:  10         3:  10         3:  10         1:  60
    .          1:  20         2:  20         2:  20             .
                   .          1:  30         1:  30
                                  .              .

  10 @key{RET}         20 @key{RET}         30 @key{RET}         C-u 3            +
@end group
@end smallexample

For stack manipulation commands like @key{RET}, a positive numeric
prefix argument operates on the top @var{n} stack entries at once.  A
negative argument operates on the entry in level @var{n} only.  An
argument of zero operates on the entire stack.  In this example, we copy
the second-to-top element of the stack:

@smallexample
@group
1:  10         2:  10         3:  10         3:  10         4:  10
    .          1:  20         2:  20         2:  20         3:  20
                   .          1:  30         1:  30         2:  30
                                  .              .          1:  20
                                                                .

  10 @key{RET}         20 @key{RET}         30 @key{RET}         C-u -2          @key{RET}
@end group
@end smallexample

@cindex Clearing the stack
@cindex Emptying the stack
Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
(The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
entire stack.)

@node Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
@subsection Algebraic-Style Calculations

@noindent
If you are not used to RPN notation, you may prefer to operate the
Calculator in Algebraic mode, which is closer to the way
non-RPN calculators work.  In Algebraic mode, you enter formulas
in traditional @expr{2+3} notation.

@strong{Warning:} Note that @samp{/} has lower precedence than
@samp{*}, so that @samp{a/b*c} is interpreted as @samp{a/(b*c)}.  See
below for details.

You don't really need any special ``mode'' to enter algebraic formulas.
You can enter a formula at any time by pressing the apostrophe (@kbd{'})
key.  Answer the prompt with the desired formula, then press @key{RET}.
The formula is evaluated and the result is pushed onto the RPN stack.
If you don't want to think in RPN at all, you can enter your whole
computation as a formula, read the result from the stack, then press
@key{DEL} to delete it from the stack.

Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
The result should be the number 9.

Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
@samp{/}, and @samp{^}.  You can use parentheses to make the order
of evaluation clear.  In the absence of parentheses, @samp{^} is
evaluated first, then @samp{*}, then @samp{/}, then finally
@samp{+} and @samp{-}.  For example, the expression

@example
2 + 3*4*5 / 6*7^8 - 9
@end example

@noindent
is equivalent to

@example
2 + ((3*4*5) / (6*(7^8)) - 9
@end example

@noindent
or, in large mathematical notation,

@ifinfo
@example
@group
    3 * 4 * 5
2 + --------- - 9
          8
     6 * 7
@end group
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
\afterdisplay
@end tex

@noindent
The result of this expression will be the number @mathit{-6.99999826533}.

Calc's order of evaluation is the same as for most computer languages,
except that @samp{*} binds more strongly than @samp{/}, as the above
example shows.  As in normal mathematical notation, the @samp{*} symbol
can often be omitted:  @samp{2 a} is the same as @samp{2*a}.

Operators at the same level are evaluated from left to right, except
that @samp{^} is evaluated from right to left.  Thus, @samp{2-3-4} is
equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
to @samp{2^(3^4)} (a very large integer; try it!).

If you tire of typing the apostrophe all the time, there is
Algebraic mode, where Calc automatically senses
when you are about to type an algebraic expression.  To enter this
mode, press the two letters @w{@kbd{m a}}.  (An @samp{Alg} indicator
should appear in the Calc window's mode line.)

Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.

In Algebraic mode, when you press any key that would normally begin
entering a number (such as a digit, a decimal point, or the @kbd{_}
key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
an algebraic entry.

Functions which do not have operator symbols like @samp{+} and @samp{*}
must be entered in formulas using function-call notation.  For example,
the function name corresponding to the square-root key @kbd{Q} is
@code{sqrt}.  To compute a square root in a formula, you would use
the notation @samp{sqrt(@var{x})}.

Press the apostrophe, then type @kbd{sqrt(5*2) - 3}.  The result should
be @expr{0.16227766017}.

Note that if the formula begins with a function name, you need to use
the apostrophe even if you are in Algebraic mode.  If you type @kbd{arcsin}
out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
command, and the @kbd{csin} will be taken as the name of the rewrite
rule to use!

Some people prefer to enter complex numbers and vectors in algebraic
form because they find RPN entry with incomplete objects to be too
distracting, even though they otherwise use Calc as an RPN calculator.

Still in Algebraic mode, type:

@smallexample
@group
1:  (2, 3)     2:  (2, 3)     1:  (8, -1)    2:  (8, -1)    1:  (9, -1)
    .          1:  (1, -2)        .          1:  1              .
                   .                             .

 (2,3) @key{RET}      (1,-2) @key{RET}        *              1 @key{RET}          +
@end group
@end smallexample

Algebraic mode allows us to enter complex numbers without pressing
an apostrophe first, but it also means we need to press @key{RET}
after every entry, even for a simple number like @expr{1}.

(You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
though regular numeric keys still use RPN numeric entry.  There is also
Total Algebraic mode, started by typing @kbd{m t}, in which all
normal keys begin algebraic entry.  You must then use the @key{META} key
to type Calc commands:  @kbd{M-m t} to get back out of Total Algebraic
mode, @kbd{M-q} to quit, etc.)

If you're still in Algebraic mode, press @kbd{m a} again to turn it off.

Actual non-RPN calculators use a mixture of algebraic and RPN styles.
In general, operators of two numbers (like @kbd{+} and @kbd{*})
use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
use RPN form.  Also, a non-RPN calculator allows you to see the
intermediate results of a calculation as you go along.  You can
accomplish this in Calc by performing your calculation as a series
of algebraic entries, using the @kbd{$} sign to tie them together.
In an algebraic formula, @kbd{$} represents the number on the top
of the stack.  Here, we perform the calculation 
@texline @math{\sqrt{2\times4+1}},
@infoline @expr{sqrt(2*4+1)},
which on a traditional calculator would be done by pressing
@kbd{2 * 4 + 1 =} and then the square-root key.

@smallexample
@group
1:  8          1:  9          1:  3
    .              .              .

  ' 2*4 @key{RET}        $+1 @key{RET}        Q
@end group
@end smallexample

@noindent
Notice that we didn't need to press an apostrophe for the @kbd{$+1},
because the dollar sign always begins an algebraic entry.

(@bullet{}) @strong{Exercise 1.}  How could you get the same effect as
pressing @kbd{Q} but using an algebraic entry instead?  How about
if the @kbd{Q} key on your keyboard were broken?
@xref{Algebraic Answer 1, 1}. (@bullet{})

The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
entries.  For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.

Algebraic formulas can include @dfn{variables}.  To store in a
variable, press @kbd{s s}, then type the variable name, then press
@key{RET}.  (There are actually two flavors of store command:
@kbd{s s} stores a number in a variable but also leaves the number
on the stack, while @w{@kbd{s t}} removes a number from the stack and
stores it in the variable.)  A variable name should consist of one
or more letters or digits, beginning with a letter.

@smallexample
@group
1:  17             .          1:  a + a^2    1:  306
    .                             .              .

    17          s t a @key{RET}      ' a+a^2 @key{RET}       =
@end group
@end smallexample

@noindent
The @kbd{=} key @dfn{evaluates} a formula by replacing all its
variables by the values that were stored in them.

For RPN calculations, you can recall a variable's value on the
stack either by entering its name as a formula and pressing @kbd{=},
or by using the @kbd{s r} command.

@smallexample
@group
1:  17         2:  17         3:  17         2:  17         1:  306
    .          1:  17         2:  17         1:  289            .
                   .          1:  2              .
                                  .

  s r a @key{RET}     ' a @key{RET} =         2              ^              +
@end group
@end smallexample

If you press a single digit for a variable name (as in @kbd{s t 3}, you
get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
They are ``quick'' simply because you don't have to type the letter
@code{q} or the @key{RET} after their names.  In fact, you can type
simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
@kbd{t 3} and @w{@kbd{r 3}}.

Any variables in an algebraic formula for which you have not stored
values are left alone, even when you evaluate the formula.

@smallexample
@group
1:  2 a + 2 b     1:  34 + 2 b
    .                 .

 ' 2a+2b @key{RET}          =
@end group
@end smallexample

Calls to function names which are undefined in Calc are also left
alone, as are calls for which the value is undefined.

@smallexample
@group
1:  2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
    .

 ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
@end group
@end smallexample

@noindent
In this example, the first call to @code{log10} works, but the other
calls are not evaluated.  In the second call, the logarithm is
undefined for that value of the argument; in the third, the argument
is symbolic, and in the fourth, there are too many arguments.  In the
fifth case, there is no function called @code{foo}.  You will see a
``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
Press the @kbd{w} (``why'') key to see any other messages that may
have arisen from the last calculation.  In this case you will get
``logarithm of zero,'' then ``number expected: @code{x}''.  Calc
automatically displays the first message only if the message is
sufficiently important; for example, Calc considers ``wrong number
of arguments'' and ``logarithm of zero'' to be important enough to
report automatically, while a message like ``number expected: @code{x}''
will only show up if you explicitly press the @kbd{w} key.

(@bullet{}) @strong{Exercise 2.}  Joe entered the formula @samp{2 x y},
stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
@samp{10 y}.  He then tried the same for the formula @samp{2 x (1+y)},
expecting @samp{10 (1+y)}, but it didn't work.  Why not?
@xref{Algebraic Answer 2, 2}. (@bullet{})

(@bullet{}) @strong{Exercise 3.}  What result would you expect
@kbd{1 @key{RET} 0 /} to give?  What if you then type @kbd{0 *}?
@xref{Algebraic Answer 3, 3}. (@bullet{})

One interesting way to work with variables is to use the
@dfn{evaluates-to} (@samp{=>}) operator.  It works like this:
Enter a formula algebraically in the usual way, but follow
the formula with an @samp{=>} symbol.  (There is also an @kbd{s =}
command which builds an @samp{=>} formula using the stack.)  On
the stack, you will see two copies of the formula with an @samp{=>}
between them.  The lefthand formula is exactly like you typed it;
the righthand formula has been evaluated as if by typing @kbd{=}.

@smallexample
@group
2:  2 + 3 => 5                     2:  2 + 3 => 5
1:  2 a + 2 b => 34 + 2 b          1:  2 a + 2 b => 20 + 2 b
    .                                  .

' 2+3 => @key{RET}  ' 2a+2b @key{RET} s =          10 s t a @key{RET}
@end group
@end smallexample

@noindent
Notice that the instant we stored a new value in @code{a}, all
@samp{=>} operators already on the stack that referred to @expr{a}
were updated to use the new value.  With @samp{=>}, you can push a
set of formulas on the stack, then change the variables experimentally
to see the effects on the formulas' values.

You can also ``unstore'' a variable when you are through with it:

@smallexample
@group
2:  2 + 5 => 5
1:  2 a + 2 b => 2 a + 2 b
    .

    s u a @key{RET}
@end group
@end smallexample

We will encounter formulas involving variables and functions again
when we discuss the algebra and calculus features of the Calculator.

@node Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
@subsection Undo and Redo

@noindent
If you make a mistake, you can usually correct it by pressing shift-@kbd{U},
the ``undo'' command.  First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
and restart Calc (@kbd{C-x * * C-x * *}) to make sure things start off
with a clean slate.  Now:

@smallexample
@group
1:  2          2:  2          1:  8          2:  2          1:  6
    .          1:  3              .          1:  3              .
                   .                             .

   2 @key{RET}           3              ^              U              *
@end group
@end smallexample

You can undo any number of times.  Calc keeps a complete record of
all you have done since you last opened the Calc window.  After the
above example, you could type:

@smallexample
@group
1:  6          2:  2          1:  2              .              .
    .          1:  3              .
                   .
                                                             (error)
                   U              U              U              U
@end group
@end smallexample

You can also type @kbd{D} to ``redo'' a command that you have undone
mistakenly.

@smallexample
@group
    .          1:  2          2:  2          1:  6          1:  6
                   .          1:  3              .              .
                                  .
                                                             (error)
                   D              D              D              D
@end group
@end smallexample

@noindent
It was not possible to redo past the @expr{6}, since that was placed there
by something other than an undo command.

@cindex Time travel
You can think of undo and redo as a sort of ``time machine.''  Press
@kbd{U} to go backward in time, @kbd{D} to go forward.  If you go
backward and do something (like @kbd{*}) then, as any science fiction
reader knows, you have changed your future and you cannot go forward
again.  Thus, the inability to redo past the @expr{6} even though there
was an earlier undo command.

You can always recall an earlier result using the Trail.  We've ignored
the trail so far, but it has been faithfully recording everything we
did since we loaded the Calculator.  If the Trail is not displayed,
press @kbd{t d} now to turn it on.

Let's try grabbing an earlier result.  The @expr{8} we computed was
undone by a @kbd{U} command, and was lost even to Redo when we pressed
@kbd{*}, but it's still there in the trail.  There should be a little
@samp{>} arrow (the @dfn{trail pointer}) resting on the last trail
entry.  If there isn't, press @kbd{t ]} to reset the trail pointer.
Now, press @w{@kbd{t p}} to move the arrow onto the line containing
@expr{8}, and press @w{@kbd{t y}} to ``yank'' that number back onto the
stack.

If you press @kbd{t ]} again, you will see that even our Yank command
went into the trail.

Let's go further back in time.  Earlier in the tutorial we computed
a huge integer using the formula @samp{2^3^4}.  We don't remember
what it was, but the first digits were ``241''.  Press @kbd{t r}
(which stands for trail-search-reverse), then type @kbd{241}.
The trail cursor will jump back to the next previous occurrence of
the string ``241'' in the trail.  This is just a regular Emacs
incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
continue the search forwards or backwards as you like.

To finish the search, press @key{RET}.  This halts the incremental
search and leaves the trail pointer at the thing we found.  Now we
can type @kbd{t y} to yank that number onto the stack.  If we hadn't
remembered the ``241'', we could simply have searched for @kbd{2^3^4},
then pressed @kbd{@key{RET} t n} to halt and then move to the next item.

You may have noticed that all the trail-related commands begin with
the letter @kbd{t}.  (The store-and-recall commands, on the other hand,
all began with @kbd{s}.)  Calc has so many commands that there aren't
enough keys for all of them, so various commands are grouped into
two-letter sequences where the first letter is called the @dfn{prefix}
key.  If you type a prefix key by accident, you can press @kbd{C-g}
to cancel it.  (In fact, you can press @kbd{C-g} to cancel almost
anything in Emacs.)  To get help on a prefix key, press that key
followed by @kbd{?}.  Some prefixes have several lines of help,
so you need to press @kbd{?} repeatedly to see them all.  
You can also type @kbd{h h} to see all the help at once.

Try pressing @kbd{t ?} now.  You will see a line of the form,

@smallexample
trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank:  [MORE]  t-
@end smallexample

@noindent
The word ``trail'' indicates that the @kbd{t} prefix key contains
trail-related commands.  Each entry on the line shows one command,
with a single capital letter showing which letter you press to get
that command.  We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
@kbd{t y} so far.  The @samp{[MORE]} means you can press @kbd{?}
again to see more @kbd{t}-prefix commands.  Notice that the commands
are roughly divided (by semicolons) into related groups.

When you are in the help display for a prefix key, the prefix is
still active.  If you press another key, like @kbd{y} for example,
it will be interpreted as a @kbd{t y} command.  If all you wanted
was to look at the help messages, press @kbd{C-g} afterwards to cancel
the prefix.

One more way to correct an error is by editing the stack entries.
The actual Stack buffer is marked read-only and must not be edited
directly, but you can press @kbd{`} (the backquote or accent grave)
to edit a stack entry.

Try entering @samp{3.141439} now.  If this is supposed to represent
@cpi{}, it's got several errors.  Press @kbd{`} to edit this number.
Now use the normal Emacs cursor motion and editing keys to change
the second 4 to a 5, and to transpose the 3 and the 9.  When you
press @key{RET}, the number on the stack will be replaced by your
new number.  This works for formulas, vectors, and all other types
of values you can put on the stack.  The @kbd{`} key also works
during entry of a number or algebraic formula.

@node Modes Tutorial,  , Undo Tutorial, Basic Tutorial
@subsection Mode-Setting Commands

@noindent
Calc has many types of @dfn{modes} that affect the way it interprets
your commands or the way it displays data.  We have already seen one
mode, namely Algebraic mode.  There are many others, too; we'll
try some of the most common ones here.

Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
Notice the @samp{12} on the Calc window's mode line:

@smallexample
--%%-Calc: 12 Deg       (Calculator)----All------
@end smallexample

@noindent
Most of the symbols there are Emacs things you don't need to worry
about, but the @samp{12} and the @samp{Deg} are mode indicators.
The @samp{12} means that calculations should always be carried to
12 significant figures.  That is why, when we type @kbd{1 @key{RET} 7 /},
we get @expr{0.142857142857} with exactly 12 digits, not counting
leading and trailing zeros.

You can set the precision to anything you like by pressing @kbd{p},
then entering a suitable number.  Try pressing @kbd{p 30 @key{RET}},
then doing @kbd{1 @key{RET} 7 /} again:

@smallexample
@group
1:  0.142857142857
2:  0.142857142857142857142857142857
    .
@end group
@end smallexample

Although the precision can be set arbitrarily high, Calc always
has to have @emph{some} value for the current precision.  After
all, the true value @expr{1/7} is an infinitely repeating decimal;
Calc has to stop somewhere.

Of course, calculations are slower the more digits you request.
Press @w{@kbd{p 12}} now to set the precision back down to the default.

Calculations always use the current precision.  For example, even
though we have a 30-digit value for @expr{1/7} on the stack, if
we use it in a calculation in 12-digit mode it will be rounded
down to 12 digits before it is used.  Try it; press @key{RET} to
duplicate the number, then @w{@kbd{1 +}}.  Notice that the @key{RET}
key didn't round the number, because it doesn't do any calculation.
But the instant we pressed @kbd{+}, the number was rounded down.

@smallexample
@group
1:  0.142857142857
2:  0.142857142857142857142857142857
3:  1.14285714286
    .
@end group
@end smallexample

@noindent
In fact, since we added a digit on the left, we had to lose one
digit on the right from even the 12-digit value of @expr{1/7}.

How did we get more than 12 digits when we computed @samp{2^3^4}?  The
answer is that Calc makes a distinction between @dfn{integers} and
@dfn{floating-point} numbers, or @dfn{floats}.  An integer is a number
that does not contain a decimal point.  There is no such thing as an
``infinitely repeating fraction integer,'' so Calc doesn't have to limit
itself.  If you asked for @samp{2^10000} (don't try this!), you would
have to wait a long time but you would eventually get an exact answer.
If you ask for @samp{2.^10000}, you will quickly get an answer which is
correct only to 12 places.  The decimal point tells Calc that it should
use floating-point arithmetic to get the answer, not exact integer
arithmetic.

You can use the @kbd{F} (@code{calc-floor}) command to convert a
floating-point value to an integer, and @kbd{c f} (@code{calc-float})
to convert an integer to floating-point form.

Let's try entering that last calculation:

@smallexample
@group
1:  2.         2:  2.         1:  1.99506311689e3010
    .          1:  10000          .
                   .

  2.0 @key{RET}          10000 @key{RET}      ^
@end group
@end smallexample

@noindent
@cindex Scientific notation, entry of
Notice the letter @samp{e} in there.  It represents ``times ten to the
power of,'' and is used by Calc automatically whenever writing the
number out fully would introduce more extra zeros than you probably
want to see.  You can enter numbers in this notation, too.

@smallexample
@group
1:  2.         2:  2.         1:  1.99506311678e3010
    .          1:  10000.         .
                   .

  2.0 @key{RET}          1e4 @key{RET}        ^
@end group
@end smallexample

@cindex Round-off errors
@noindent
Hey, the answer is different!  Look closely at the middle columns
of the two examples.  In the first, the stack contained the
exact integer @expr{10000}, but in the second it contained
a floating-point value with a decimal point.  When you raise a
number to an integer power, Calc uses repeated squaring and
multiplication to get the answer.  When you use a floating-point
power, Calc uses logarithms and exponentials.  As you can see,
a slight error crept in during one of these methods.  Which
one should we trust?  Let's raise the precision a bit and find
out:

@smallexample
@group
    .          1:  2.         2:  2.         1:  1.995063116880828e3010
                   .          1:  10000.         .
                                  .

 p 16 @key{RET}        2. @key{RET}           1e4            ^    p 12 @key{RET}
@end group
@end smallexample

@noindent
@cindex Guard digits
Presumably, it doesn't matter whether we do this higher-precision
calculation using an integer or floating-point power, since we
have added enough ``guard digits'' to trust the first 12 digits
no matter what.  And the verdict is@dots{}  Integer powers were more
accurate; in fact, the result was only off by one unit in the
last place.

@cindex Guard digits
Calc does many of its internal calculations to a slightly higher
precision, but it doesn't always bump the precision up enough.
In each case, Calc added about two digits of precision during
its calculation and then rounded back down to 12 digits
afterward.  In one case, it was enough; in the other, it
wasn't.  If you really need @var{x} digits of precision, it
never hurts to do the calculation with a few extra guard digits.

What if we want guard digits but don't want to look at them?
We can set the @dfn{float format}.  Calc supports four major
formats for floating-point numbers, called @dfn{normal},
@dfn{fixed-point}, @dfn{scientific notation}, and @dfn{engineering
notation}.  You get them by pressing @w{@kbd{d n}}, @kbd{d f},
@kbd{d s}, and @kbd{d e}, respectively.  In each case, you can
supply a numeric prefix argument which says how many digits
should be displayed.  As an example, let's put a few numbers
onto the stack and try some different display modes.  First,
use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
numbers shown here:

@smallexample
@group
4:  12345      4:  12345      4:  12345      4:  12345      4:  12345
3:  12345.     3:  12300.     3:  1.2345e4   3:  1.23e4     3:  12345.000
2:  123.45     2:  123.       2:  1.2345e2   2:  1.23e2     2:  123.450
1:  12.345     1:  12.3       1:  1.2345e1   1:  1.23e1     1:  12.345
    .              .              .              .              .

   d n          M-3 d n          d s          M-3 d s        M-3 d f
@end group
@end smallexample

@noindent
Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
to three significant digits, but then when we typed @kbd{d s} all
five significant figures reappeared.  The float format does not
affect how numbers are stored, it only affects how they are
displayed.  Only the current precision governs the actual rounding
of numbers in the Calculator's memory.

Engineering notation, not shown here, is like scientific notation
except the exponent (the power-of-ten part) is always adjusted to be
a multiple of three (as in ``kilo,'' ``micro,'' etc.).  As a result
there will be one, two, or three digits before the decimal point.

Whenever you change a display-related mode, Calc redraws everything
in the stack.  This may be slow if there are many things on the stack,
so Calc allows you to type shift-@kbd{H} before any mode command to
prevent it from updating the stack.  Anything Calc displays after the
mode-changing command will appear in the new format.

@smallexample
@group
4:  12345      4:  12345      4:  12345      4:  12345      4:  12345
3:  12345.000  3:  12345.000  3:  12345.000  3:  1.2345e4   3:  12345.
2:  123.450    2:  123.450    2:  1.2345e1   2:  1.2345e1   2:  123.45
1:  12.345     1:  1.2345e1   1:  1.2345e2   1:  1.2345e2   1:  12.345
    .              .              .              .              .

    H d s          @key{DEL} U          @key{TAB}            d @key{SPC}          d n
@end group
@end smallexample

@noindent
Here the @kbd{H d s} command changes to scientific notation but without
updating the screen.  Deleting the top stack entry and undoing it back
causes it to show up in the new format; swapping the top two stack
entries reformats both entries.  The @kbd{d @key{SPC}} command refreshes the
whole stack.  The @kbd{d n} command changes back to the normal float
format; since it doesn't have an @kbd{H} prefix, it also updates all
the stack entries to be in @kbd{d n} format.

Notice that the integer @expr{12345} was not affected by any
of the float formats.  Integers are integers, and are always
displayed exactly.

@cindex Large numbers, readability
Large integers have their own problems.  Let's look back at
the result of @kbd{2^3^4}.

@example
2417851639229258349412352
@end example

@noindent
Quick---how many digits does this have?  Try typing @kbd{d g}:

@example
2,417,851,639,229,258,349,412,352
@end example

@noindent
Now how many digits does this have?  It's much easier to tell!
We can actually group digits into clumps of any size.  Some
people prefer @kbd{M-5 d g}:

@example
24178,51639,22925,83494,12352
@end example

Let's see what happens to floating-point numbers when they are grouped.
First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
to get ourselves into trouble.  Now, type @kbd{1e13 /}:

@example
24,17851,63922.9258349412352
@end example

@noindent
The integer part is grouped but the fractional part isn't.  Now try
@kbd{M-- M-5 d g} (that's meta-minus-sign, meta-five):

@example
24,17851,63922.92583,49412,352
@end example

If you find it hard to tell the decimal point from the commas, try
changing the grouping character to a space with @kbd{d , @key{SPC}}:

@example
24 17851 63922.92583 49412 352
@end example

Type @kbd{d , ,} to restore the normal grouping character, then
@kbd{d g} again to turn grouping off.  Also, press @kbd{p 12} to
restore the default precision.

Press @kbd{U} enough times to get the original big integer back.
(Notice that @kbd{U} does not undo each mode-setting command; if
you want to undo a mode-setting command, you have to do it yourself.)
Now, type @kbd{d r 16 @key{RET}}:

@example
16#200000000000000000000
@end example

@noindent
The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
Suddenly it looks pretty simple; this should be no surprise, since we
got this number by computing a power of two, and 16 is a power of 2.
In fact, we can use @w{@kbd{d r 2 @key{RET}}} to see it in actual binary
form:

@example
2#1000000000000000000000000000000000000000000000000000000 @dots{}
@end example

@noindent
We don't have enough space here to show all the zeros!  They won't
fit on a typical screen, either, so you will have to use horizontal
scrolling to see them all.  Press @kbd{<} and @kbd{>} to scroll the
stack window left and right by half its width.  Another way to view
something large is to press @kbd{`} (back-quote) to edit the top of
stack in a separate window.  (Press @kbd{C-c C-c} when you are done.)

You can enter non-decimal numbers using the @kbd{#} symbol, too.
Let's see what the hexadecimal number @samp{5FE} looks like in
binary.  Type @kbd{16#5FE} (the letters can be typed in upper or
lower case; they will always appear in upper case).  It will also
help to turn grouping on with @kbd{d g}:

@example
2#101,1111,1110
@end example

Notice that @kbd{d g} groups by fours by default if the display radix
is binary or hexadecimal, but by threes if it is decimal, octal, or any
other radix.

Now let's see that number in decimal; type @kbd{d r 10}:

@example
1,534
@end example

Numbers are not @emph{stored} with any particular radix attached.  They're
just numbers; they can be entered in any radix, and are always displayed
in whatever radix you've chosen with @kbd{d r}.  The current radix applies
to integers, fractions, and floats.

@cindex Roundoff errors, in non-decimal numbers
(@bullet{}) @strong{Exercise 1.}  Your friend Joe tried to enter one-third
as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12.  He got
@samp{3#0.0222222...} (with 25 2's) in the display.  When he multiplied
that by three, he got @samp{3#0.222222...} instead of the expected
@samp{3#1}.  Next, Joe entered @samp{3#0.2} and, to his great relief,
saw @samp{3#0.2} on the screen.  But when he typed @kbd{2 /}, he got
@samp{3#0.10000001} (some zeros omitted).  What's going on here?
@xref{Modes Answer 1, 1}. (@bullet{})

@cindex Scientific notation, in non-decimal numbers
(@bullet{}) @strong{Exercise 2.}  Scientific notation works in non-decimal
modes in the natural way (the exponent is a power of the radix instead of
a power of ten, although the exponent itself is always written in decimal).
Thus @samp{8#1.23e3 = 8#1230.0}.  Suppose we have the hexadecimal number
@samp{f.e8f} times 16 to the 15th power:  We write @samp{16#f.e8fe15}.
What is wrong with this picture?  What could we write instead that would
work better?  @xref{Modes Answer 2, 2}. (@bullet{})

The @kbd{m} prefix key has another set of modes, relating to the way
Calc interprets your inputs and does computations.  Whereas @kbd{d}-prefix
modes generally affect the way things look, @kbd{m}-prefix modes affect
the way they are actually computed.

The most popular @kbd{m}-prefix mode is the @dfn{angular mode}.  Notice
the @samp{Deg} indicator in the mode line.  This means that if you use
a command that interprets a number as an angle, it will assume the
angle is measured in degrees.  For example,

@smallexample
@group
1:  45         1:  0.707106781187   1:  0.500000000001    1:  0.5
    .              .                    .                     .

    45             S                    2 ^                   c 1
@end group
@end smallexample

@noindent
The shift-@kbd{S} command computes the sine of an angle.  The sine
of 45 degrees is 
@texline @math{\sqrt{2}/2};
@infoline @expr{sqrt(2)/2}; 
squaring this yields @expr{2/4 = 0.5}.  However, there has been a slight
roundoff error because the representation of 
@texline @math{\sqrt{2}/2}
@infoline @expr{sqrt(2)/2} 
wasn't exact.  The @kbd{c 1} command is a handy way to clean up numbers
in this case; it temporarily reduces the precision by one digit while it
re-rounds the number on the top of the stack.

@cindex Roundoff errors, examples
(@bullet{}) @strong{Exercise 3.}  Your friend Joe computed the sine
of 45 degrees as shown above, then, hoping to avoid an inexact
result, he increased the precision to 16 digits before squaring.
What happened?  @xref{Modes Answer 3, 3}. (@bullet{})

To do this calculation in radians, we would type @kbd{m r} first.
(The indicator changes to @samp{Rad}.)  45 degrees corresponds to
@cpiover{4} radians.  To get @cpi{}, press the @kbd{P} key.  (Once
again, this is a shifted capital @kbd{P}.  Remember, unshifted
@kbd{p} sets the precision.)

@smallexample
@group
1:  3.14159265359   1:  0.785398163398   1:  0.707106781187
    .                   .                .

    P                   4 /       m r    S
@end group
@end smallexample

Likewise, inverse trigonometric functions generate results in
either radians or degrees, depending on the current angular mode.

@smallexample
@group
1:  0.707106781187   1:  0.785398163398   1:  45.
    .                    .                    .

    .5 Q        m r      I S        m d       U I S
@end group
@end smallexample

@noindent
Here we compute the Inverse Sine of 
@texline @math{\sqrt{0.5}},
@infoline @expr{sqrt(0.5)}, 
first in radians, then in degrees.

Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
and vice-versa.

@smallexample
@group
1:  45         1:  0.785398163397     1:  45.
    .              .                      .

    45             c r                    c d
@end group
@end smallexample

Another interesting mode is @dfn{Fraction mode}.  Normally,
dividing two integers produces a floating-point result if the
quotient can't be expressed as an exact integer.  Fraction mode
causes integer division to produce a fraction, i.e., a rational
number, instead.

@smallexample
@group
2:  12         1:  1.33333333333    1:  4:3
1:  9              .                    .
    .

 12 @key{RET} 9          /          m f       U /      m f
@end group
@end smallexample

@noindent
In the first case, we get an approximate floating-point result.
In the second case, we get an exact fractional result (four-thirds).

You can enter a fraction at any time using @kbd{:} notation.
(Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
because @kbd{/} is already used to divide the top two stack
elements.)  Calculations involving fractions will always
produce exact fractional results; Fraction mode only says
what to do when dividing two integers.

@cindex Fractions vs. floats
@cindex Floats vs. fractions
(@bullet{}) @strong{Exercise 4.}  If fractional arithmetic is exact,
why would you ever use floating-point numbers instead?
@xref{Modes Answer 4, 4}. (@bullet{})

Typing @kbd{m f} doesn't change any existing values in the stack.
In the above example, we had to Undo the division and do it over
again when we changed to Fraction mode.  But if you use the
evaluates-to operator you can get commands like @kbd{m f} to
recompute for you.

@smallexample
@group
1:  12 / 9 => 1.33333333333    1:  12 / 9 => 1.333    1:  12 / 9 => 4:3
    .                              .                      .

   ' 12/9 => @key{RET}                   p 4 @key{RET}                m f
@end group
@end smallexample

@noindent
In this example, the righthand side of the @samp{=>} operator
on the stack is recomputed when we change the precision, then
again when we change to Fraction mode.  All @samp{=>} expressions
on the stack are recomputed every time you change any mode that
might affect their values.

@node Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial, Tutorial
@section Arithmetic Tutorial

@noindent
In this section, we explore the arithmetic and scientific functions
available in the Calculator.

The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
and @kbd{^}.  Each normally takes two numbers from the top of the stack
and pushes back a result.  The @kbd{n} and @kbd{&} keys perform
change-sign and reciprocal operations, respectively.

@smallexample
@group
1:  5          1:  0.2        1:  5.         1:  -5.        1:  5.
    .              .              .              .              .

    5              &              &              n              n
@end group
@end smallexample

@cindex Binary operators
You can apply a ``binary operator'' like @kbd{+} across any number of
stack entries by giving it a numeric prefix.  You can also apply it
pairwise to several stack elements along with the top one if you use
a negative prefix.

@smallexample
@group
3:  2          1:  9          3:  2          4:  2          3:  12
2:  3              .          2:  3          3:  3          2:  13
1:  4                         1:  4          2:  4          1:  14
    .                             .          1:  10             .
                                                 .

2 @key{RET} 3 @key{RET} 4     M-3 +           U              10          M-- M-3 +
@end group
@end smallexample

@cindex Unary operators
You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
stack entries with a numeric prefix, too.

@smallexample
@group
3:  2          3:  0.5                3:  0.5
2:  3          2:  0.333333333333     2:  3.
1:  4          1:  0.25               1:  4.
    .              .                      .

2 @key{RET} 3 @key{RET} 4      M-3 &                  M-2 &
@end group
@end smallexample

Notice that the results here are left in floating-point form.
We can convert them back to integers by pressing @kbd{F}, the
``floor'' function.  This function rounds down to the next lower
integer.  There is also @kbd{R}, which rounds to the nearest
integer.

@smallexample
@group
7:  2.         7:  2          7:  2
6:  2.4        6:  2          6:  2
5:  2.5        5:  2          5:  3
4:  2.6        4:  2          4:  3
3:  -2.        3:  -2         3:  -2
2:  -2.4       2:  -3         2:  -2
1:  -2.6       1:  -3         1:  -3
    .              .              .

                  M-7 F        U M-7 R
@end group
@end smallexample

Since dividing-and-flooring (i.e., ``integer quotient'') is such a
common operation, Calc provides a special command for that purpose, the
backslash @kbd{\}.  Another common arithmetic operator is @kbd{%}, which
computes the remainder that would arise from a @kbd{\} operation, i.e.,
the ``modulo'' of two numbers.  For example,

@smallexample
@group
2:  1234       1:  12         2:  1234       1:  34
1:  100            .          1:  100            .
    .                             .

1234 @key{RET} 100       \              U              %
@end group
@end smallexample

These commands actually work for any real numbers, not just integers.

@smallexample
@group
2:  3.1415     1:  3          2:  3.1415     1:  0.1415
1:  1              .          1:  1              .
    .                             .

3.1415 @key{RET} 1       \              U              %
@end group
@end smallexample

(@bullet{}) @strong{Exercise 1.}  The @kbd{\} command would appear to be a
frill, since you could always do the same thing with @kbd{/ F}.  Think
of a situation where this is not true---@kbd{/ F} would be inadequate.
Now think of a way you could get around the problem if Calc didn't
provide a @kbd{\} command.  @xref{Arithmetic Answer 1, 1}. (@bullet{})

We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
commands.  Other commands along those lines are @kbd{C} (cosine),
@kbd{T} (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
logarithm).  These can be modified by the @kbd{I} (inverse) and
@kbd{H} (hyperbolic) prefix keys.

Let's compute the sine and cosine of an angle, and verify the
identity 
@texline @math{\sin^2x + \cos^2x = 1}.
@infoline @expr{sin(x)^2 + cos(x)^2 = 1}.  
We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
With the angular mode set to degrees (type @w{@kbd{m d}}), do:

@smallexample
@group
2:  -64        2:  -64        2:  -0.89879   2:  -0.89879   1:  1.
1:  -64        1:  -0.89879   1:  -64        1:  0.43837        .
    .              .              .              .

 64 n @key{RET} @key{RET}      S              @key{TAB}            C              f h
@end group
@end smallexample

@noindent
(For brevity, we're showing only five digits of the results here.
You can of course do these calculations to any precision you like.)

Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
of squares, command.

Another identity is 
@texline @math{\displaystyle\tan x = {\sin x \over \cos x}}.
@infoline @expr{tan(x) = sin(x) / cos(x)}.
@smallexample
@group

2:  -0.89879   1:  -2.0503    1:  -64.
1:  0.43837        .              .
    .

    U              /              I T
@end group
@end smallexample

A physical interpretation of this calculation is that if you move
@expr{0.89879} units downward and @expr{0.43837} units to the right,
your direction of motion is @mathit{-64} degrees from horizontal.  Suppose
we move in the opposite direction, up and to the left:

@smallexample
@group
2:  -0.89879   2:  0.89879    1:  -2.0503    1:  -64.
1:  0.43837    1:  -0.43837       .              .
    .              .

    U U            M-2 n          /              I T
@end group
@end smallexample

@noindent
How can the angle be the same?  The answer is that the @kbd{/} operation
loses information about the signs of its inputs.  Because the quotient
is negative, we know exactly one of the inputs was negative, but we
can't tell which one.  There is an @kbd{f T} [@code{arctan2}] function which
computes the inverse tangent of the quotient of a pair of numbers.
Since you feed it the two original numbers, it has enough information
to give you a full 360-degree answer.

@smallexample
@group
2:  0.89879    1:  116.       3:  116.       2:  116.       1:  180.
1:  -0.43837       .          2:  -0.89879   1:  -64.           .
    .                         1:  0.43837        .
                                  .

    U U            f T         M-@key{RET} M-2 n       f T            -
@end group
@end smallexample

@noindent
The resulting angles differ by 180 degrees; in other words, they
point in opposite directions, just as we would expect.

The @key{META}-@key{RET} we used in the third step is the
``last-arguments'' command.  It is sort of like Undo, except that it
restores the arguments of the last command to the stack without removing
the command's result.  It is useful in situations like this one,
where we need to do several operations on the same inputs.  We could
have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
the top two stack elements right after the @kbd{U U}, then a pair of
@kbd{M-@key{TAB}} commands to cycle the 116 up around the duplicates.

A similar identity is supposed to hold for hyperbolic sines and cosines,
except that it is the @emph{difference}
@texline @math{\cosh^2x - \sinh^2x}
@infoline @expr{cosh(x)^2 - sinh(x)^2} 
that always equals one.  Let's try to verify this identity.

@smallexample
@group
2:  -64        2:  -64        2:  -64        2:  9.7192e54  2:  9.7192e54
1:  -64        1:  -3.1175e27 1:  9.7192e54  1:  -64        1:  9.7192e54
    .              .              .              .              .

 64 n @key{RET} @key{RET}      H C            2 ^            @key{TAB}            H S 2 ^
@end group
@end smallexample

@noindent
@cindex Roundoff errors, examples
Something's obviously wrong, because when we subtract these numbers
the answer will clearly be zero!  But if you think about it, if these
numbers @emph{did} differ by one, it would be in the 55th decimal
place.  The difference we seek has been lost entirely to roundoff
error.

We could verify this hypothesis by doing the actual calculation with,
say, 60 decimal places of precision.  This will be slow, but not
enormously so.  Try it if you wish; sure enough, the answer is
0.99999, reasonably close to 1.

Of course, a more reasonable way to verify the identity is to use
a more reasonable value for @expr{x}!

@cindex Common logarithm
Some Calculator commands use the Hyperbolic prefix for other purposes.
The logarithm and exponential functions, for example, work to the base
@expr{e} normally but use base-10 instead if you use the Hyperbolic
prefix.

@smallexample
@group
1:  1000       1:  6.9077     1:  1000       1:  3
    .              .              .              .

    1000           L              U              H L
@end group
@end smallexample

@noindent
First, we mistakenly compute a natural logarithm.  Then we undo
and compute a common logarithm instead.

The @kbd{B} key computes a general base-@var{b} logarithm for any
value of @var{b}.

@smallexample
@group
2:  1000       1:  3          1:  1000.      2:  1000.      1:  6.9077
1:  10             .              .          1:  2.71828        .
    .                                            .

 1000 @key{RET} 10       B              H E            H P            B
@end group
@end smallexample

@noindent
Here we first use @kbd{B} to compute the base-10 logarithm, then use
the ``hyperbolic'' exponential as a cheap hack to recover the number
1000, then use @kbd{B} again to compute the natural logarithm.  Note
that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
onto the stack.

You may have noticed that both times we took the base-10 logarithm
of 1000, we got an exact integer result.  Calc always tries to give
an exact rational result for calculations involving rational numbers
where possible.  But when we used @kbd{H E}, the result was a
floating-point number for no apparent reason.  In fact, if we had
computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
exact integer 1000.  But the @kbd{H E} command is rigged to generate
a floating-point result all of the time so that @kbd{1000 H E} will
not waste time computing a thousand-digit integer when all you
probably wanted was @samp{1e1000}.

(@bullet{}) @strong{Exercise 2.}  Find a pair of integer inputs to
the @kbd{B} command for which Calc could find an exact rational
result but doesn't.  @xref{Arithmetic Answer 2, 2}. (@bullet{})

The Calculator also has a set of functions relating to combinatorics
and statistics.  You may be familiar with the @dfn{factorial} function,
which computes the product of all the integers up to a given number.

@smallexample
@group
1:  100        1:  93326215443...    1:  100.       1:  9.3326e157
    .              .                     .              .

    100            !                     U c f          !
@end group
@end smallexample

@noindent
Recall, the @kbd{c f} command converts the integer or fraction at the
top of the stack to floating-point format.  If you take the factorial
of a floating-point number, you get a floating-point result
accurate to the current precision.  But if you give @kbd{!} an
exact integer, you get an exact integer result (158 digits long
in this case).

If you take the factorial of a non-integer, Calc uses a generalized
factorial function defined in terms of Euler's Gamma function
@texline @math{\Gamma(n)}
@infoline @expr{gamma(n)}
(which is itself available as the @kbd{f g} command).

@smallexample
@group
3:  4.         3:  24.               1:  5.5        1:  52.342777847
2:  4.5        2:  52.3427777847         .              .
1:  5.         1:  120.
    .              .

                   M-3 !              M-0 @key{DEL} 5.5       f g
@end group
@end smallexample

@noindent
Here we verify the identity 
@texline @math{n! = \Gamma(n+1)}.
@infoline @expr{@var{n}!@: = gamma(@var{n}+1)}.

The binomial coefficient @var{n}-choose-@var{m}
@texline or @math{\displaystyle {n \choose m}}
is defined by
@texline @math{\displaystyle {n! \over m! \, (n-m)!}}
@infoline @expr{n!@: / m!@: (n-m)!}
for all reals @expr{n} and @expr{m}.  The intermediate results in this
formula can become quite large even if the final result is small; the
@kbd{k c} command computes a binomial coefficient in a way that avoids
large intermediate values.

The @kbd{k} prefix key defines several common functions out of
combinatorics and number theory.  Here we compute the binomial
coefficient 30-choose-20, then determine its prime factorization.

@smallexample
@group
2:  30         1:  30045015   1:  [3, 3, 5, 7, 11, 13, 23, 29]
1:  20             .              .
    .

 30 @key{RET} 20         k c            k f
@end group
@end smallexample

@noindent
You can verify these prime factors by using @kbd{v u} to ``unpack''
this vector into 8 separate stack entries, then @kbd{M-8 *} to
multiply them back together.  The result is the original number,
30045015.

@cindex Hash tables
Suppose a program you are writing needs a hash table with at least
10000 entries.  It's best to use a prime number as the actual size
of a hash table.  Calc can compute the next prime number after 10000:

@smallexample
@group
1:  10000      1:  10007      1:  9973
    .              .              .

    10000          k n            I k n
@end group
@end smallexample

@noindent
Just for kicks we've also computed the next prime @emph{less} than
10000.

@c [fix-ref Financial Functions]
@xref{Financial Functions}, for a description of the Calculator
commands that deal with business and financial calculations (functions
like @code{pv}, @code{rate}, and @code{sln}).

@c [fix-ref Binary Number Functions]
@xref{Binary Functions}, to read about the commands for operating
on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).

@node Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial, Tutorial
@section Vector/Matrix Tutorial

@noindent
A @dfn{vector} is a list of numbers or other Calc data objects.
Calc provides a large set of commands that operate on vectors.  Some
are familiar operations from vector analysis.  Others simply treat
a vector as a list of objects.

@menu
* Vector Analysis Tutorial::
* Matrix Tutorial::
* List Tutorial::
@end menu

@node Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix Tutorial, Vector/Matrix Tutorial
@subsection Vector Analysis

@noindent
If you add two vectors, the result is a vector of the sums of the
elements, taken pairwise.

@smallexample
@group
1:  [1, 2, 3]     2:  [1, 2, 3]     1:  [8, 8, 3]
    .             1:  [7, 6, 0]         .
                      .

    [1,2,3]  s 1      [7 6 0]  s 2      +
@end group
@end smallexample

@noindent
Note that we can separate the vector elements with either commas or
spaces.  This is true whether we are using incomplete vectors or
algebraic entry.  The @kbd{s 1} and @kbd{s 2} commands save these
vectors so we can easily reuse them later.

If you multiply two vectors, the result is the sum of the products
of the elements taken pairwise.  This is called the @dfn{dot product}
of the vectors.

@smallexample
@group
2:  [1, 2, 3]     1:  19
1:  [7, 6, 0]         .
    .

    r 1 r 2           *
@end group
@end smallexample

@cindex Dot product
The dot product of two vectors is equal to the product of their
lengths times the cosine of the angle between them.  (Here the vector
is interpreted as a line from the origin @expr{(0,0,0)} to the
specified point in three-dimensional space.)  The @kbd{A}
(absolute value) command can be used to compute the length of a
vector.

@smallexample
@group
3:  19            3:  19          1:  0.550782    1:  56.579
2:  [1, 2, 3]     2:  3.741657        .               .
1:  [7, 6, 0]     1:  9.219544
    .                 .

    M-@key{RET}             M-2 A          * /             I C
@end group
@end smallexample

@noindent
First we recall the arguments to the dot product command, then
we compute the absolute values of the top two stack entries to
obtain the lengths of the vectors, then we divide the dot product
by the product of the lengths to get the cosine of the angle.
The inverse cosine finds that the angle between the vectors
is about 56 degrees.

@cindex Cross product
@cindex Perpendicular vectors
The @dfn{cross product} of two vectors is a vector whose length
is the product of the lengths of the inputs times the sine of the
angle between them, and whose direction is perpendicular to both
input vectors.  Unlike the dot product, the cross product is
defined only for three-dimensional vectors.  Let's double-check
our computation of the angle using the cross product.

@smallexample
@group
2:  [1, 2, 3]  3:  [-18, 21, -8]  1:  [-0.52, 0.61, -0.23]  1:  56.579
1:  [7, 6, 0]  2:  [1, 2, 3]          .                         .
    .          1:  [7, 6, 0]
                   .

    r 1 r 2        V C  s 3  M-@key{RET}    M-2 A * /                 A I S
@end group
@end smallexample

@noindent
First we recall the original vectors and compute their cross product,
which we also store for later reference.  Now we divide the vector
by the product of the lengths of the original vectors.  The length of
this vector should be the sine of the angle; sure enough, it is!

@c [fix-ref General Mode Commands]
Vector-related commands generally begin with the @kbd{v} prefix key.
Some are uppercase letters and some are lowercase.  To make it easier
to type these commands, the shift-@kbd{V} prefix key acts the same as
the @kbd{v} key.  (@xref{General Mode Commands}, for a way to make all
prefix keys have this property.)

If we take the dot product of two perpendicular vectors we expect
to get zero, since the cosine of 90 degrees is zero.  Let's check
that the cross product is indeed perpendicular to both inputs:

@smallexample
@group
2:  [1, 2, 3]      1:  0          2:  [7, 6, 0]      1:  0
1:  [-18, 21, -8]      .          1:  [-18, 21, -8]      .
    .                                 .

    r 1 r 3            *          @key{DEL} r 2 r 3            *
@end group
@end smallexample

@cindex Normalizing a vector
@cindex Unit vectors
(@bullet{}) @strong{Exercise 1.}  Given a vector on the top of the
stack, what keystrokes would you use to @dfn{normalize} the
vector, i.e., to reduce its length to one without changing its
direction?  @xref{Vector Answer 1, 1}. (@bullet{})

(@bullet{}) @strong{Exercise 2.}  Suppose a certain particle can be
at any of several positions along a ruler.  You have a list of
those positions in the form of a vector, and another list of the
probabilities for the particle to be at the corresponding positions.
Find the average position of the particle.
@xref{Vector Answer 2, 2}. (@bullet{})

@node Matrix Tutorial, List Tutorial, Vector Analysis Tutorial, Vector/Matrix Tutorial
@subsection Matrices

@noindent
A @dfn{matrix} is just a vector of vectors, all the same length.
This means you can enter a matrix using nested brackets.  You can
also use the semicolon character to enter a matrix.  We'll show
both methods here:

@smallexample
@group
1:  [ [ 1, 2, 3 ]             1:  [ [ 1, 2, 3 ]
      [ 4, 5, 6 ] ]                 [ 4, 5, 6 ] ]
    .                             .

  [[1 2 3] [4 5 6]]             ' [1 2 3; 4 5 6] @key{RET}
@end group
@end smallexample

@noindent
We'll be using this matrix again, so type @kbd{s 4} to save it now.

Note that semicolons work with incomplete vectors, but they work
better in algebraic entry.  That's why we use the apostrophe in
the second example.

When two matrices are multiplied, the lefthand matrix must have
the same number of columns as the righthand matrix has rows.
Row @expr{i}, column @expr{j} of the result is effectively the
dot product of row @expr{i} of the left matrix by column @expr{j}
of the right matrix.

If we try to duplicate this matrix and multiply it by itself,
the dimensions are wrong and the multiplication cannot take place:

@smallexample
@group
1:  [ [ 1, 2, 3 ]   * [ [ 1, 2, 3 ]
      [ 4, 5, 6 ] ]     [ 4, 5, 6 ] ]
    .

    @key{RET} *
@end group
@end smallexample

@noindent
Though rather hard to read, this is a formula which shows the product
of two matrices.  The @samp{*} function, having invalid arguments, has
been left in symbolic form.

We can multiply the matrices if we @dfn{transpose} one of them first.

@smallexample
@group
2:  [ [ 1, 2, 3 ]       1:  [ [ 14, 32 ]      1:  [ [ 17, 22, 27 ]
      [ 4, 5, 6 ] ]           [ 32, 77 ] ]          [ 22, 29, 36 ]
1:  [ [ 1, 4 ]              .                       [ 27, 36, 45 ] ]
      [ 2, 5 ]                                    .
      [ 3, 6 ] ]
    .

    U v t                   *                     U @key{TAB} *
@end group
@end smallexample

Matrix multiplication is not commutative; indeed, switching the
order of the operands can even change the dimensions of the result
matrix, as happened here!

If you multiply a plain vector by a matrix, it is treated as a
single row or column depending on which side of the matrix it is
on.  The result is a plain vector which should also be interpreted
as a row or column as appropriate.

@smallexample
@group
2:  [ [ 1, 2, 3 ]      1:  [14, 32]
      [ 4, 5, 6 ] ]        .
1:  [1, 2, 3]
    .

    r 4 r 1                *
@end group
@end smallexample

Multiplying in the other order wouldn't work because the number of
rows in the matrix is different from the number of elements in the
vector.

(@bullet{}) @strong{Exercise 1.}  Use @samp{*} to sum along the rows
of the above 
@texline @math{2\times3}
@infoline 2x3 
matrix to get @expr{[6, 15]}.  Now use @samp{*} to sum along the columns
to get @expr{[5, 7, 9]}. 
@xref{Matrix Answer 1, 1}. (@bullet{})

@cindex Identity matrix
An @dfn{identity matrix} is a square matrix with ones along the
diagonal and zeros elsewhere.  It has the property that multiplication
by an identity matrix, on the left or on the right, always produces
the original matrix.

@smallexample
@group
1:  [ [ 1, 2, 3 ]      2:  [ [ 1, 2, 3 ]      1:  [ [ 1, 2, 3 ]
      [ 4, 5, 6 ] ]          [ 4, 5, 6 ] ]          [ 4, 5, 6 ] ]
    .                  1:  [ [ 1, 0, 0 ]          .
                             [ 0, 1, 0 ]
                             [ 0, 0, 1 ] ]
                           .

    r 4                    v i 3 @key{RET}              *
@end group
@end smallexample

If a matrix is square, it is often possible to find its @dfn{inverse},
that is, a matrix which, when multiplied by the original matrix, yields
an identity matrix.  The @kbd{&} (reciprocal) key also computes the
inverse of a matrix.

@smallexample
@group
1:  [ [ 1, 2, 3 ]      1:  [ [   -2.4,     1.2,   -0.2 ]
      [ 4, 5, 6 ]            [    2.8,    -1.4,    0.4 ]
      [ 7, 6, 0 ] ]          [ -0.73333, 0.53333, -0.2 ] ]
    .                      .

    r 4 r 2 |  s 5         &
@end group
@end smallexample

@noindent
The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
matrices together.  Here we have used it to add a new row onto
our matrix to make it square.

We can multiply these two matrices in either order to get an identity.

@smallexample
@group
1:  [ [ 1., 0., 0. ]      1:  [ [ 1., 0., 0. ]
      [ 0., 1., 0. ]            [ 0., 1., 0. ]
      [ 0., 0., 1. ] ]          [ 0., 0., 1. ] ]
    .                         .

    M-@key{RET}  *                  U @key{TAB} *
@end group
@end smallexample

@cindex Systems of linear equations
@cindex Linear equations, systems of
Matrix inverses are related to systems of linear equations in algebra.
Suppose we had the following set of equations:

@ifinfo
@group
@example
    a + 2b + 3c = 6
   4a + 5b + 6c = 2
   7a + 6b      = 3
@end example
@end group
@end ifinfo
@tex
\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
   $\hfil#$&$\hfil{}#{}$&
   $\hfil#$&$\hfil{}#{}$&
   $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
  a&+&2b&+&3c&=6 \cr
 4a&+&5b&+&6c&=2 \cr
 7a&+&6b& &  &=3 \cr}
$$
\afterdisplayh
@end tex

@noindent
This can be cast into the matrix equation,

@ifinfo
@group
@example
   [ [ 1, 2, 3 ]     [ [ a ]     [ [ 6 ]
     [ 4, 5, 6 ]   *   [ b ]   =   [ 2 ]
     [ 7, 6, 0 ] ]     [ c ] ]     [ 3 ] ]
@end example
@end group
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
   \times
   \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
$$
\afterdisplay
@end tex

We can solve this system of equations by multiplying both sides by the
inverse of the matrix.  Calc can do this all in one step:

@smallexample
@group
2:  [6, 2, 3]          1:  [-12.6, 15.2, -3.93333]
1:  [ [ 1, 2, 3 ]          .
      [ 4, 5, 6 ]
      [ 7, 6, 0 ] ]
    .

    [6,2,3] r 5            /
@end group
@end smallexample

@noindent
The result is the @expr{[a, b, c]} vector that solves the equations.
(Dividing by a square matrix is equivalent to multiplying by its
inverse.)

Let's verify this solution:

@smallexample
@group
2:  [ [ 1, 2, 3 ]                1:  [6., 2., 3.]
      [ 4, 5, 6 ]                    .
      [ 7, 6, 0 ] ]
1:  [-12.6, 15.2, -3.93333]
    .

    r 5  @key{TAB}                         *
@end group
@end smallexample

@noindent
Note that we had to be careful about the order in which we multiplied
the matrix and vector.  If we multiplied in the other order, Calc would
assume the vector was a row vector in order to make the dimensions
come out right, and the answer would be incorrect.  If you
don't feel safe letting Calc take either interpretation of your
vectors, use explicit 
@texline @math{N\times1}
@infoline Nx1
or
@texline @math{1\times N}
@infoline 1xN
matrices instead.  In this case, you would enter the original column
vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.

(@bullet{}) @strong{Exercise 2.}  Algebraic entry allows you to make
vectors and matrices that include variables.  Solve the following
system of equations to get expressions for @expr{x} and @expr{y}
in terms of @expr{a} and @expr{b}.

@ifinfo
@group
@example
   x + a y = 6
   x + b y = 10
@end example
@end group
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \eqalign{ x &+ a y = 6 \cr
             x &+ b y = 10}
$$
\afterdisplay
@end tex

@noindent
@xref{Matrix Answer 2, 2}. (@bullet{})

@cindex Least-squares for over-determined systems
@cindex Over-determined systems of equations
(@bullet{}) @strong{Exercise 3.}  A system of equations is ``over-determined''
if it has more equations than variables.  It is often the case that
there are no values for the variables that will satisfy all the
equations at once, but it is still useful to find a set of values
which ``nearly'' satisfy all the equations.  In terms of matrix equations,
you can't solve @expr{A X = B} directly because the matrix @expr{A}
is not square for an over-determined system.  Matrix inversion works
only for square matrices.  One common trick is to multiply both sides
on the left by the transpose of @expr{A}:
@ifinfo
@samp{trn(A)*A*X = trn(A)*B}.
@end ifinfo
@tex
\turnoffactive
$A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
@end tex
Now 
@texline @math{A^T A}
@infoline @expr{trn(A)*A} 
is a square matrix so a solution is possible.  It turns out that the
@expr{X} vector you compute in this way will be a ``least-squares''
solution, which can be regarded as the ``closest'' solution to the set
of equations.  Use Calc to solve the following over-determined
system:

@ifinfo
@group
@example
    a + 2b + 3c = 6
   4a + 5b + 6c = 2
   7a + 6b      = 3
   2a + 4b + 6c = 11
@end example
@end group
@end ifinfo
@tex
\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
   $\hfil#$&$\hfil{}#{}$&
   $\hfil#$&$\hfil{}#{}$&
   $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
  a&+&2b&+&3c&=6 \cr
 4a&+&5b&+&6c&=2 \cr
 7a&+&6b& &  &=3 \cr
 2a&+&4b&+&6c&=11 \cr}
$$
\afterdisplayh
@end tex

@noindent
@xref{Matrix Answer 3, 3}. (@bullet{})

@node List Tutorial,  , Matrix Tutorial, Vector/Matrix Tutorial
@subsection Vectors as Lists

@noindent
@cindex Lists
Although Calc has a number of features for manipulating vectors and
matrices as mathematical objects, you can also treat vectors as
simple lists of values.  For example, we saw that the @kbd{k f}
command returns a vector which is a list of the prime factors of a
number.

You can pack and unpack stack entries into vectors:

@smallexample
@group
3:  10         1:  [10, 20, 30]     3:  10
2:  20             .                2:  20
1:  30                              1:  30
    .                                   .

                   M-3 v p              v u
@end group
@end smallexample

You can also build vectors out of consecutive integers, or out
of many copies of a given value:

@smallexample
@group
1:  [1, 2, 3, 4]    2:  [1, 2, 3, 4]    2:  [1, 2, 3, 4]
    .               1:  17              1:  [17, 17, 17, 17]
                        .                   .

    v x 4 @key{RET}           17                  v b 4 @key{RET}
@end group
@end smallexample

You can apply an operator to every element of a vector using the
@dfn{map} command.

@smallexample
@group
1:  [17, 34, 51, 68]   1:  [289, 1156, 2601, 4624]  1:  [17, 34, 51, 68]
    .                      .                            .

    V M *                  2 V M ^                      V M Q
@end group
@end smallexample

@noindent
In the first step, we multiply the vector of integers by the vector
of 17's elementwise.  In the second step, we raise each element to
the power two.  (The general rule is that both operands must be
vectors of the same length, or else one must be a vector and the
other a plain number.)  In the final step, we take the square root
of each element.

(@bullet{}) @strong{Exercise 1.}  Compute a vector of powers of two
from 
@texline @math{2^{-4}}
@infoline @expr{2^-4} 
to @expr{2^4}.  @xref{List Answer 1, 1}. (@bullet{})

You can also @dfn{reduce} a binary operator across a vector.
For example, reducing @samp{*} computes the product of all the
elements in the vector:

@smallexample
@group
1:  123123     1:  [3, 7, 11, 13, 41]      1:  123123
    .              .                           .

    123123         k f                         V R *
@end group
@end smallexample

@noindent
In this example, we decompose 123123 into its prime factors, then
multiply those factors together again to yield the original number.

We could compute a dot product ``by hand'' using mapping and
reduction:

@smallexample
@group
2:  [1, 2, 3]     1:  [7, 12, 0]     1:  19
1:  [7, 6, 0]         .                  .
    .

    r 1 r 2           V M *              V R +
@end group
@end smallexample

@noindent
Recalling two vectors from the previous section, we compute the
sum of pairwise products of the elements to get the same answer
for the dot product as before.

A slight variant of vector reduction is the @dfn{accumulate} operation,
@kbd{V U}.  This produces a vector of the intermediate results from
a corresponding reduction.  Here we compute a table of factorials:

@smallexample
@group
1:  [1, 2, 3, 4, 5, 6]    1:  [1, 2, 6, 24, 120, 720]
    .                         .

    v x 6 @key{RET}                 V U *
@end group
@end smallexample

Calc allows vectors to grow as large as you like, although it gets
rather slow if vectors have more than about a hundred elements.
Actually, most of the time is spent formatting these large vectors
for display, not calculating on them.  Try the following experiment
(if your computer is very fast you may need to substitute a larger
vector size).

@smallexample
@group
1:  [1, 2, 3, 4, ...      1:  [2, 3, 4, 5, ...
    .                         .

    v x 500 @key{RET}               1 V M +
@end group
@end smallexample

Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
experiment again.  In @kbd{v .} mode, long vectors are displayed
``abbreviated'' like this:

@smallexample
@group
1:  [1, 2, 3, ..., 500]   1:  [2, 3, 4, ..., 501]
    .                         .

    v x 500 @key{RET}               1 V M +
@end group
@end smallexample

@noindent
(where now the @samp{...} is actually part of the Calc display).
You will find both operations are now much faster.  But notice that
even in @w{@kbd{v .}} mode, the full vectors are still shown in the Trail.
Type @w{@kbd{t .}} to cause the trail to abbreviate as well, and try the
experiment one more time.  Operations on long vectors are now quite
fast!  (But of course if you use @kbd{t .} you will lose the ability
to get old vectors back using the @kbd{t y} command.)

An easy way to view a full vector when @kbd{v .} mode is active is
to press @kbd{`} (back-quote) to edit the vector; editing always works
with the full, unabbreviated value.

@cindex Least-squares for fitting a straight line
@cindex Fitting data to a line
@cindex Line, fitting data to
@cindex Data, extracting from buffers
@cindex Columns of data, extracting
As a larger example, let's try to fit a straight line to some data,
using the method of least squares.  (Calc has a built-in command for
least-squares curve fitting, but we'll do it by hand here just to
practice working with vectors.)  Suppose we have the following list
of values in a file we have loaded into Emacs:

@smallexample
  x        y
 ---      ---
 1.34    0.234
 1.41    0.298
 1.49    0.402
 1.56    0.412
 1.64    0.466
 1.73    0.473
 1.82    0.601
 1.91    0.519
 2.01    0.603
 2.11    0.637
 2.22    0.645
 2.33    0.705
 2.45    0.917
 2.58    1.009
 2.71    0.971
 2.85    1.062
 3.00    1.148
 3.15    1.157
 3.32    1.354
@end smallexample

@noindent
If you are reading this tutorial in printed form, you will find it
easiest to press @kbd{C-x * i} to enter the on-line Info version of
the manual and find this table there.  (Press @kbd{g}, then type
@kbd{List Tutorial}, to jump straight to this section.)

Position the cursor at the upper-left corner of this table, just
to the left of the @expr{1.34}.  Press @kbd{C-@@} to set the mark.
(On your system this may be @kbd{C-2}, @kbd{C-@key{SPC}}, or @kbd{NUL}.)
Now position the cursor to the lower-right, just after the @expr{1.354}.
You have now defined this region as an Emacs ``rectangle.''  Still
in the Info buffer, type @kbd{C-x * r}.  This command
(@code{calc-grab-rectangle}) will pop you back into the Calculator, with
the contents of the rectangle you specified in the form of a matrix.

@smallexample
@group
1:  [ [ 1.34, 0.234 ]
      [ 1.41, 0.298 ]
      @dots{}
@end group
@end smallexample

@noindent
(You may wish to use @kbd{v .} mode to abbreviate the display of this
large matrix.)

We want to treat this as a pair of lists.  The first step is to
transpose this matrix into a pair of rows.  Remember, a matrix is
just a vector of vectors.  So we can unpack the matrix into a pair
of row vectors on the stack.

@smallexample
@group
1:  [ [ 1.34,  1.41,  1.49,  ... ]     2:  [1.34, 1.41, 1.49, ... ]
      [ 0.234, 0.298, 0.402, ... ] ]   1:  [0.234, 0.298, 0.402, ... ]
    .                                      .

    v t                                    v u
@end group
@end smallexample

@noindent
Let's store these in quick variables 1 and 2, respectively.

@smallexample
@group
1:  [1.34, 1.41, 1.49, ... ]        .
    .

    t 2                             t 1
@end group
@end smallexample

@noindent
(Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
stored value from the stack.)

In a least squares fit, the slope @expr{m} is given by the formula

@ifinfo
@example
m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ m = {N \sum x y - \sum x \sum y  \over
        N \sum x^2 - \left( \sum x \right)^2} $$
\afterdisplay
@end tex

@noindent
where 
@texline @math{\sum x}
@infoline @expr{sum(x)} 
represents the sum of all the values of @expr{x}.  While there is an
actual @code{sum} function in Calc, it's easier to sum a vector using a
simple reduction.  First, let's compute the four different sums that
this formula uses.

@smallexample
@group
1:  41.63                 1:  98.0003
    .                         .

 r 1 V R +   t 3           r 1 2 V M ^ V R +   t 4

@end group
@end smallexample
@noindent
@smallexample
@group
1:  13.613                1:  33.36554
    .                         .

 r 2 V R +   t 5           r 1 r 2 V M * V R +   t 6
@end group
@end smallexample

@ifinfo
@noindent
These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
respectively.  (We could have used @kbd{*} to compute @samp{sum(x^2)} and
@samp{sum(x y)}.)
@end ifinfo
@tex
\turnoffactive
These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
respectively.  (We could have used \kbd{*} to compute $\sum x^2$ and
$\sum x y$.)
@end tex

Finally, we also need @expr{N}, the number of data points.  This is just
the length of either of our lists.

@smallexample
@group
1:  19
    .

 r 1 v l   t 7
@end group
@end smallexample

@noindent
(That's @kbd{v} followed by a lower-case @kbd{l}.)

Now we grind through the formula:

@smallexample
@group
1:  633.94526  2:  633.94526  1:  67.23607
    .          1:  566.70919      .
                   .

 r 7 r 6 *      r 3 r 5 *         -

@end group
@end smallexample
@noindent
@smallexample
@group
2:  67.23607   3:  67.23607   2:  67.23607   1:  0.52141679
1:  1862.0057  2:  1862.0057  1:  128.9488       .
    .          1:  1733.0569      .
                   .

 r 7 r 4 *      r 3 2 ^           -              /   t 8
@end group
@end smallexample

That gives us the slope @expr{m}.  The y-intercept @expr{b} can now
be found with the simple formula,

@ifinfo
@example
b = (sum(y) - m sum(x)) / N
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ b = {\sum y - m \sum x \over N} $$
\afterdisplay
\vskip10pt
@end tex

@smallexample
@group
1:  13.613     2:  13.613     1:  -8.09358   1:  -0.425978
    .          1:  21.70658       .              .
                   .

   r 5            r 8 r 3 *       -              r 7 /   t 9
@end group
@end smallexample

Let's ``plot'' this straight line approximation, 
@texline @math{y \approx m x + b},
@infoline @expr{m x + b}, 
and compare it with the original data.

@smallexample
@group
1:  [0.699, 0.735, ... ]    1:  [0.273, 0.309, ... ]
    .                           .

    r 1 r 8 *                   r 9 +    s 0
@end group
@end smallexample

@noindent
Notice that multiplying a vector by a constant, and adding a constant
to a vector, can be done without mapping commands since these are
common operations from vector algebra.  As far as Calc is concerned,
we've just been doing geometry in 19-dimensional space!

We can subtract this vector from our original @expr{y} vector to get
a feel for the error of our fit.  Let's find the maximum error:

@smallexample
@group
1:  [0.0387, 0.0112, ... ]   1:  [0.0387, 0.0112, ... ]   1:  0.0897
    .                            .                            .

    r 2 -                        V M A                        V R X
@end group
@end smallexample

@noindent
First we compute a vector of differences, then we take the absolute
values of these differences, then we reduce the @code{max} function
across the vector.  (The @code{max} function is on the two-key sequence
@kbd{f x}; because it is so common to use @code{max} in a vector
operation, the letters @kbd{X} and @kbd{N} are also accepted for
@code{max} and @code{min} in this context.  In general, you answer
the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
invokes the function you want.  You could have typed @kbd{V R f x} or
even @kbd{V R x max @key{RET}} if you had preferred.)

If your system has the GNUPLOT program, you can see graphs of your
data and your straight line to see how well they match.  (If you have
GNUPLOT 3.0 or higher, the following instructions will work regardless
of the kind of display you have.  Some GNUPLOT 2.0, non-X-windows systems
may require additional steps to view the graphs.)

Let's start by plotting the original data.  Recall the ``@var{x}'' and ``@var{y}''
vectors onto the stack and press @kbd{g f}.  This ``fast'' graphing
command does everything you need to do for simple, straightforward
plotting of data.

@smallexample
@group
2:  [1.34, 1.41, 1.49, ... ]
1:  [0.234, 0.298, 0.402, ... ]
    .

    r 1 r 2    g f
@end group
@end smallexample

If all goes well, you will shortly get a new window containing a graph
of the data.  (If not, contact your GNUPLOT or Calc installer to find
out what went wrong.)  In the X window system, this will be a separate
graphics window.  For other kinds of displays, the default is to
display the graph in Emacs itself using rough character graphics.
Press @kbd{q} when you are done viewing the character graphics.

Next, let's add the line we got from our least-squares fit.
@ifinfo
(If you are reading this tutorial on-line while running Calc, typing
@kbd{g a} may cause the tutorial to disappear from its window and be
replaced by a buffer named @samp{*Gnuplot Commands*}.  The tutorial
will reappear when you terminate GNUPLOT by typing @kbd{g q}.) 
@end ifinfo

@smallexample
@group
2:  [1.34, 1.41, 1.49, ... ]
1:  [0.273, 0.309, 0.351, ... ]
    .

    @key{DEL} r 0    g a  g p
@end group
@end smallexample

It's not very useful to get symbols to mark the data points on this
second curve; you can type @kbd{g S g p} to remove them.  Type @kbd{g q}
when you are done to remove the X graphics window and terminate GNUPLOT.

(@bullet{}) @strong{Exercise 2.}  An earlier exercise showed how to do
least squares fitting to a general system of equations.  Our 19 data
points are really 19 equations of the form @expr{y_i = m x_i + b} for
different pairs of @expr{(x_i,y_i)}.  Use the matrix-transpose method
to solve for @expr{m} and @expr{b}, duplicating the above result.
@xref{List Answer 2, 2}. (@bullet{})

@cindex Geometric mean
(@bullet{}) @strong{Exercise 3.}  If the input data do not form a
rectangle, you can use @w{@kbd{C-x * g}} (@code{calc-grab-region})
to grab the data the way Emacs normally works with regions---it reads
left-to-right, top-to-bottom, treating line breaks the same as spaces.
Use this command to find the geometric mean of the following numbers.
(The geometric mean is the @var{n}th root of the product of @var{n} numbers.)

@example
2.3  6  22  15.1  7
  15  14  7.5
  2.5
@end example

@noindent
The @kbd{C-x * g} command accepts numbers separated by spaces or commas,
with or without surrounding vector brackets.
@xref{List Answer 3, 3}. (@bullet{})

@ifinfo
As another example, a theorem about binomial coefficients tells
us that the alternating sum of binomial coefficients
@var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
on up to @var{n}-choose-@var{n},
always comes out to zero.  Let's verify this
for @expr{n=6}.
@end ifinfo
@tex
As another example, a theorem about binomial coefficients tells
us that the alternating sum of binomial coefficients
${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
always comes out to zero.  Let's verify this
for \cite{n=6}.
@end tex

@smallexample
@group
1:  [1, 2, 3, 4, 5, 6, 7]     1:  [0, 1, 2, 3, 4, 5, 6]
    .                             .

    v x 7 @key{RET}                     1 -

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [1, -6, 15, -20, 15, -6, 1]          1:  0
    .                                        .

    V M ' (-1)^$ choose(6,$) @key{RET}             V R +
@end group
@end smallexample

The @kbd{V M '} command prompts you to enter any algebraic expression
to define the function to map over the vector.  The symbol @samp{$}
inside this expression represents the argument to the function.
The Calculator applies this formula to each element of the vector,
substituting each element's value for the @samp{$} sign(s) in turn.

To define a two-argument function, use @samp{$$} for the first
argument and @samp{$} for the second:  @kbd{V M ' $$-$ @key{RET}} is
equivalent to @kbd{V M -}.  This is analogous to regular algebraic
entry, where @samp{$$} would refer to the next-to-top stack entry
and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
would act exactly like @kbd{-}.

Notice that the @kbd{V M '} command has recorded two things in the
trail:  The result, as usual, and also a funny-looking thing marked
@samp{oper} that represents the operator function you typed in.
The function is enclosed in @samp{< >} brackets, and the argument is
denoted by a @samp{#} sign.  If there were several arguments, they
would be shown as @samp{#1}, @samp{#2}, and so on.  (For example,
@kbd{V M ' $$-$} will put the function @samp{<#1 - #2>} on the
trail.)  This object is a ``nameless function''; you can use nameless
@w{@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
Nameless function notation has the interesting, occasionally useful
property that a nameless function is not actually evaluated until
it is used.  For example, @kbd{V M ' $+random(2.0)} evaluates
@samp{random(2.0)} once and adds that random number to all elements
of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
@samp{random(2.0)} separately for each vector element.

Another group of operators that are often useful with @kbd{V M} are
the relational operators:  @kbd{a =}, for example, compares two numbers
and gives the result 1 if they are equal, or 0 if not.  Similarly,
@w{@kbd{a <}} checks for one number being less than another.

Other useful vector operations include @kbd{v v}, to reverse a
vector end-for-end; @kbd{V S}, to sort the elements of a vector
into increasing order; and @kbd{v r} and @w{@kbd{v c}}, to extract
one row or column of a matrix, or (in both cases) to extract one
element of a plain vector.  With a negative argument, @kbd{v r}
and @kbd{v c} instead delete one row, column, or vector element.

@cindex Divisor functions
(@bullet{}) @strong{Exercise 4.}  The @expr{k}th @dfn{divisor function}
@tex
$\sigma_k(n)$
@end tex
is the sum of the @expr{k}th powers of all the divisors of an
integer @expr{n}.  Figure out a method for computing the divisor
function for reasonably small values of @expr{n}.  As a test,
the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
@xref{List Answer 4, 4}. (@bullet{})

@cindex Square-free numbers
@cindex Duplicate values in a list
(@bullet{}) @strong{Exercise 5.}  The @kbd{k f} command produces a
list of prime factors for a number.  Sometimes it is important to
know that a number is @dfn{square-free}, i.e., that no prime occurs
more than once in its list of prime factors.  Find a sequence of
keystrokes to tell if a number is square-free; your method should
leave 1 on the stack if it is, or 0 if it isn't.
@xref{List Answer 5, 5}. (@bullet{})

@cindex Triangular lists
(@bullet{}) @strong{Exercise 6.}  Build a list of lists that looks
like the following diagram.  (You may wish to use the @kbd{v /}
command to enable multi-line display of vectors.)

@smallexample
@group
1:  [ [1],
      [1, 2],
      [1, 2, 3],
      [1, 2, 3, 4],
      [1, 2, 3, 4, 5],
      [1, 2, 3, 4, 5, 6] ]
@end group
@end smallexample

@noindent
@xref{List Answer 6, 6}. (@bullet{})

(@bullet{}) @strong{Exercise 7.}  Build the following list of lists.

@smallexample
@group
1:  [ [0],
      [1, 2],
      [3, 4, 5],
      [6, 7, 8, 9],
      [10, 11, 12, 13, 14],
      [15, 16, 17, 18, 19, 20] ]
@end group
@end smallexample

@noindent
@xref{List Answer 7, 7}. (@bullet{})

@cindex Maximizing a function over a list of values
@c [fix-ref Numerical Solutions]
(@bullet{}) @strong{Exercise 8.}  Compute a list of values of Bessel's
@texline @math{J_1(x)}
@infoline @expr{J1} 
function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
Find the value of @expr{x} (from among the above set of values) for
which @samp{besJ(1,x)} is a maximum.  Use an ``automatic'' method,
i.e., just reading along the list by hand to find the largest value
is not allowed!  (There is an @kbd{a X} command which does this kind
of thing automatically; @pxref{Numerical Solutions}.)
@xref{List Answer 8, 8}. (@bullet{})

@cindex Digits, vectors of
(@bullet{}) @strong{Exercise 9.}  You are given an integer in the range
@texline @math{0 \le N < 10^m}
@infoline @expr{0 <= N < 10^m} 
for @expr{m=12} (i.e., an integer of less than
twelve digits).  Convert this integer into a vector of @expr{m}
digits, each in the range from 0 to 9.  In vector-of-digits notation,
add one to this integer to produce a vector of @expr{m+1} digits
(since there could be a carry out of the most significant digit).
Convert this vector back into a regular integer.  A good integer
to try is 25129925999.  @xref{List Answer 9, 9}. (@bullet{})

(@bullet{}) @strong{Exercise 10.}  Your friend Joe tried to use
@kbd{V R a =} to test if all numbers in a list were equal.  What
happened?  How would you do this test?  @xref{List Answer 10, 10}. (@bullet{})

(@bullet{}) @strong{Exercise 11.}  The area of a circle of radius one
is @cpi{}.  The area of the 
@texline @math{2\times2}
@infoline 2x2
square that encloses that circle is 4.  So if we throw @var{n} darts at
random points in the square, about @cpiover{4} of them will land inside
the circle.  This gives us an entertaining way to estimate the value of 
@cpi{}.  The @w{@kbd{k r}}
command picks a random number between zero and the value on the stack.
We could get a random floating-point number between @mathit{-1} and 1 by typing
@w{@kbd{2.0 k r 1 -}}.  Build a vector of 100 random @expr{(x,y)} points in
this square, then use vector mapping and reduction to count how many
points lie inside the unit circle.  Hint:  Use the @kbd{v b} command.
@xref{List Answer 11, 11}. (@bullet{})

@cindex Matchstick problem
(@bullet{}) @strong{Exercise 12.}  The @dfn{matchstick problem} provides
another way to calculate @cpi{}.  Say you have an infinite field
of vertical lines with a spacing of one inch.  Toss a one-inch matchstick
onto the field.  The probability that the matchstick will land crossing
a line turns out to be 
@texline @math{2/\pi}.
@infoline @expr{2/pi}.  
Toss 100 matchsticks to estimate @cpi{}.  (If you want still more fun,
the probability that the GCD (@w{@kbd{k g}}) of two large integers is
one turns out to be 
@texline @math{6/\pi^2}.
@infoline @expr{6/pi^2}.
That provides yet another way to estimate @cpi{}.)
@xref{List Answer 12, 12}. (@bullet{})

(@bullet{}) @strong{Exercise 13.}  An algebraic entry of a string in
double-quote marks, @samp{"hello"}, creates a vector of the numerical
(ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
Sometimes it is convenient to compute a @dfn{hash code} of a string,
which is just an integer that represents the value of that string.
Two equal strings have the same hash code; two different strings
@dfn{probably} have different hash codes.  (For example, Calc has
over 400 function names, but Emacs can quickly find the definition for
any given name because it has sorted the functions into ``buckets'' by
their hash codes.  Sometimes a few names will hash into the same bucket,
but it is easier to search among a few names than among all the names.)
One popular hash function is computed as follows:  First set @expr{h = 0}.
Then, for each character from the string in turn, set @expr{h = 3h + c_i}
where @expr{c_i} is the character's ASCII code.  If we have 511 buckets,
we then take the hash code modulo 511 to get the bucket number.  Develop a
simple command or commands for converting string vectors into hash codes.
The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
511 is 121.  @xref{List Answer 13, 13}. (@bullet{})

(@bullet{}) @strong{Exercise 14.}  The @kbd{H V R} and @kbd{H V U}
commands do nested function evaluations.  @kbd{H V U} takes a starting
value and a number of steps @var{n} from the stack; it then applies the
function you give to the starting value 0, 1, 2, up to @var{n} times
and returns a vector of the results.  Use this command to create a
``random walk'' of 50 steps.  Start with the two-dimensional point
@expr{(0,0)}; then take one step a random distance between @mathit{-1} and 1
in both @expr{x} and @expr{y}; then take another step, and so on.  Use the
@kbd{g f} command to display this random walk.  Now modify your random
walk to walk a unit distance, but in a random direction, at each step.
(Hint:  The @code{sincos} function returns a vector of the cosine and
sine of an angle.)  @xref{List Answer 14, 14}. (@bullet{})

@node Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial, Tutorial
@section Types Tutorial

@noindent
Calc understands a variety of data types as well as simple numbers.
In this section, we'll experiment with each of these types in turn.

The numbers we've been using so far have mainly been either @dfn{integers}
or @dfn{floats}.  We saw that floats are usually a good approximation to
the mathematical concept of real numbers, but they are only approximations
and are susceptible to roundoff error.  Calc also supports @dfn{fractions},
which can exactly represent any rational number.

@smallexample
@group
1:  3628800    2:  3628800    1:  518400:7   1:  518414:7   1:  7:518414
    .          1:  49             .              .              .
                   .

    10 !           49 @key{RET}         :              2 +            &
@end group
@end smallexample

@noindent
The @kbd{:} command divides two integers to get a fraction; @kbd{/}
would normally divide integers to get a floating-point result.
Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
since the @kbd{:} would otherwise be interpreted as part of a
fraction beginning with 49.

You can convert between floating-point and fractional format using
@kbd{c f} and @kbd{c F}:

@smallexample
@group
1:  1.35027217629e-5    1:  7:518414
    .                       .

    c f                     c F
@end group
@end smallexample

The @kbd{c F} command replaces a floating-point number with the
``simplest'' fraction whose floating-point representation is the
same, to within the current precision.

@smallexample
@group
1:  3.14159265359   1:  1146408:364913   1:  3.1416   1:  355:113
    .                   .                    .            .

    P                   c F      @key{DEL}       p 5 @key{RET} P      c F
@end group
@end smallexample

(@bullet{}) @strong{Exercise 1.}  A calculation has produced the
result 1.26508260337.  You suspect it is the square root of the
product of @cpi{} and some rational number.  Is it?  (Be sure
to allow for roundoff error!)  @xref{Types Answer 1, 1}. (@bullet{})

@dfn{Complex numbers} can be stored in both rectangular and polar form.

@smallexample
@group
1:  -9     1:  (0, 3)    1:  (3; 90.)   1:  (6; 90.)   1:  (2.4495; 45.)
    .          .             .              .              .

    9 n        Q             c p            2 *            Q
@end group
@end smallexample

@noindent
The square root of @mathit{-9} is by default rendered in rectangular form
(@w{@expr{0 + 3i}}), but we can convert it to polar form (3 with a
phase angle of 90 degrees).  All the usual arithmetic and scientific
operations are defined on both types of complex numbers.

Another generalized kind of number is @dfn{infinity}.  Infinity
isn't really a number, but it can sometimes be treated like one.
Calc uses the symbol @code{inf} to represent positive infinity,
i.e., a value greater than any real number.  Naturally, you can
also write @samp{-inf} for minus infinity, a value less than any
real number.  The word @code{inf} can only be input using
algebraic entry.

@smallexample
@group
2:  inf        2:  -inf       2:  -inf       2:  -inf       1:  nan
1:  -17        1:  -inf       1:  -inf       1:  inf            .
    .              .              .              .

' inf @key{RET} 17 n     *  @key{RET}         72 +           A              +
@end group
@end smallexample

@noindent
Since infinity is infinitely large, multiplying it by any finite
number (like @mathit{-17}) has no effect, except that since @mathit{-17}
is negative, it changes a plus infinity to a minus infinity.
(``A huge positive number, multiplied by @mathit{-17}, yields a huge
negative number.'')  Adding any finite number to infinity also
leaves it unchanged.  Taking an absolute value gives us plus
infinity again.  Finally, we add this plus infinity to the minus
infinity we had earlier.  If you work it out, you might expect
the answer to be @mathit{-72} for this.  But the 72 has been completely
lost next to the infinities; by the time we compute @w{@samp{inf - inf}}
the finite difference between them, if any, is undetectable.
So we say the result is @dfn{indeterminate}, which Calc writes
with the symbol @code{nan} (for Not A Number).

Dividing by zero is normally treated as an error, but you can get
Calc to write an answer in terms of infinity by pressing @kbd{m i}
to turn on Infinite mode.

@smallexample
@group
3:  nan        2:  nan        2:  nan        2:  nan        1:  nan
2:  1          1:  1 / 0      1:  uinf       1:  uinf           .
1:  0              .              .              .
    .

  1 @key{RET} 0          /       m i    U /            17 n *         +
@end group
@end smallexample

@noindent
Dividing by zero normally is left unevaluated, but after @kbd{m i}
it instead gives an infinite result.  The answer is actually
@code{uinf}, ``undirected infinity.''  If you look at a graph of
@expr{1 / x} around @w{@expr{x = 0}}, you'll see that it goes toward
plus infinity as you approach zero from above, but toward minus
infinity as you approach from below.  Since we said only @expr{1 / 0},
Calc knows that the answer is infinite but not in which direction.
That's what @code{uinf} means.  Notice that multiplying @code{uinf}
by a negative number still leaves plain @code{uinf}; there's no
point in saying @samp{-uinf} because the sign of @code{uinf} is
unknown anyway.  Finally, we add @code{uinf} to our @code{nan},
yielding @code{nan} again.  It's easy to see that, because
@code{nan} means ``totally unknown'' while @code{uinf} means
``unknown sign but known to be infinite,'' the more mysterious
@code{nan} wins out when it is combined with @code{uinf}, or, for
that matter, with anything else.

(@bullet{}) @strong{Exercise 2.}  Predict what Calc will answer
for each of these formulas:  @samp{inf / inf}, @samp{exp(inf)},
@samp{exp(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
@samp{abs(uinf)}, @samp{ln(0)}.
@xref{Types Answer 2, 2}. (@bullet{})

(@bullet{}) @strong{Exercise 3.}  We saw that @samp{inf - inf = nan},
which stands for an unknown value.  Can @code{nan} stand for
a complex number?  Can it stand for infinity?
@xref{Types Answer 3, 3}. (@bullet{})

@dfn{HMS forms} represent a value in terms of hours, minutes, and
seconds.

@smallexample
@group
1:  2@@ 30' 0"     1:  3@@ 30' 0"     2:  3@@ 30' 0"     1:  2.
    .                 .             1:  1@@ 45' 0."        .
                                        .

  2@@ 30' @key{RET}          1 +               @key{RET} 2 /           /
@end group
@end smallexample

HMS forms can also be used to hold angles in degrees, minutes, and
seconds.

@smallexample
@group
1:  0.5        1:  26.56505   1:  26@@ 33' 54.18"    1:  0.44721
    .              .              .                     .

    0.5            I T            c h                   S
@end group
@end smallexample

@noindent
First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
form, then we take the sine of that angle.  Note that the trigonometric
functions will accept HMS forms directly as input.

@cindex Beatles
(@bullet{}) @strong{Exercise 4.}  The Beatles' @emph{Abbey Road} is
47 minutes and 26 seconds long, and contains 17 songs.  What is the
average length of a song on @emph{Abbey Road}?  If the Extended Disco
Version of @emph{Abbey Road} added 20 seconds to the length of each
song, how long would the album be?  @xref{Types Answer 4, 4}. (@bullet{})

A @dfn{date form} represents a date, or a date and time.  Dates must
be entered using algebraic entry.  Date forms are surrounded by
@samp{< >} symbols; most standard formats for dates are recognized.

@smallexample
@group
2:  <Sun Jan 13, 1991>                    1:  2.25
1:  <6:00pm Thu Jan 10, 1991>                 .
    .

' <13 Jan 1991>, <1/10/91, 6pm> @key{RET}           -
@end group
@end smallexample

@noindent
In this example, we enter two dates, then subtract to find the
number of days between them.  It is also possible to add an
HMS form or a number (of days) to a date form to get another
date form.

@smallexample
@group
1:  <4:45:59pm Mon Jan 14, 1991>     1:  <2:50:59am Thu Jan 17, 1991>
    .                                    .

    t N                                  2 + 10@@ 5' +
@end group
@end smallexample

@c [fix-ref Date Arithmetic]
@noindent
The @kbd{t N} (``now'') command pushes the current date and time on the
stack; then we add two days, ten hours and five minutes to the date and
time.  Other date-and-time related commands include @kbd{t J}, which
does Julian day conversions, @kbd{t W}, which finds the beginning of
the week in which a date form lies, and @kbd{t I}, which increments a
date by one or several months.  @xref{Date Arithmetic}, for more.

(@bullet{}) @strong{Exercise 5.}  How many days until the next
Friday the 13th?  @xref{Types Answer 5, 5}. (@bullet{})

(@bullet{}) @strong{Exercise 6.}  How many leap years will there be
between now and the year 10001 A.D.?  @xref{Types Answer 6, 6}. (@bullet{})

@cindex Slope and angle of a line
@cindex Angle and slope of a line
An @dfn{error form} represents a mean value with an attached standard
deviation, or error estimate.  Suppose our measurements indicate that
a certain telephone pole is about 30 meters away, with an estimated
error of 1 meter, and 8 meters tall, with an estimated error of 0.2
meters.  What is the slope of a line from here to the top of the
pole, and what is the equivalent angle in degrees?

@smallexample
@group
1:  8 +/- 0.2    2:  8 +/- 0.2   1:  0.266 +/- 0.011   1:  14.93 +/- 0.594
    .            1:  30 +/- 1        .                     .
                     .

    8 p .2 @key{RET}       30 p 1          /                     I T
@end group
@end smallexample

@noindent
This means that the angle is about 15 degrees, and, assuming our
original error estimates were valid standard deviations, there is about
a 60% chance that the result is correct within 0.59 degrees.

@cindex Torus, volume of
(@bullet{}) @strong{Exercise 7.}  The volume of a torus (a donut shape) is
@texline @math{2 \pi^2 R r^2}
@infoline @w{@expr{2 pi^2 R r^2}} 
where @expr{R} is the radius of the circle that
defines the center of the tube and @expr{r} is the radius of the tube
itself.  Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
within 5 percent.  What is the volume and the relative uncertainty of
the volume?  @xref{Types Answer 7, 7}. (@bullet{})

An @dfn{interval form} represents a range of values.  While an
error form is best for making statistical estimates, intervals give
you exact bounds on an answer.  Suppose we additionally know that
our telephone pole is definitely between 28 and 31 meters away,
and that it is between 7.7 and 8.1 meters tall.

@smallexample
@group
1:  [7.7 .. 8.1]  2:  [7.7 .. 8.1]  1:  [0.24 .. 0.28]  1:  [13.9 .. 16.1]
    .             1:  [28 .. 31]        .                   .
                      .

  [ 7.7 .. 8.1 ]    [ 28 .. 31 ]        /                   I T
@end group
@end smallexample

@noindent
If our bounds were correct, then the angle to the top of the pole
is sure to lie in the range shown.

The square brackets around these intervals indicate that the endpoints
themselves are allowable values.  In other words, the distance to the
telephone pole is between 28 and 31, @emph{inclusive}.  You can also
make an interval that is exclusive of its endpoints by writing
parentheses instead of square brackets.  You can even make an interval
which is inclusive (``closed'') on one end and exclusive (``open'') on
the other.

@smallexample
@group
1:  [1 .. 10)    1:  (0.1 .. 1]   2:  (0.1 .. 1]   1:  (0.2 .. 3)
    .                .            1:  [2 .. 3)         .
                                      .

  [ 1 .. 10 )        &              [ 2 .. 3 )         *
@end group
@end smallexample

@noindent
The Calculator automatically keeps track of which end values should
be open and which should be closed.  You can also make infinite or
semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
or both endpoints.

(@bullet{}) @strong{Exercise 8.}  What answer would you expect from
@samp{@w{1 /} @w{(0 .. 10)}}?  What about @samp{@w{1 /} @w{(-10 .. 0)}}?  What
about @samp{@w{1 /} @w{[0 .. 10]}} (where the interval actually includes
zero)?  What about @samp{@w{1 /} @w{(-10 .. 10)}}?
@xref{Types Answer 8, 8}. (@bullet{})

(@bullet{}) @strong{Exercise 9.}  Two easy ways of squaring a number
are @kbd{@key{RET} *} and @w{@kbd{2 ^}}.  Normally these produce the same
answer.  Would you expect this still to hold true for interval forms?
If not, which of these will result in a larger interval?
@xref{Types Answer 9, 9}. (@bullet{})

A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
For example, arithmetic involving time is generally done modulo 12
or 24 hours.

@smallexample
@group
1:  17 mod 24    1:  3 mod 24     1:  21 mod 24    1:  9 mod 24
    .                .                .                .

    17 M 24 @key{RET}      10 +             n                5 /
@end group
@end smallexample

@noindent
In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
new number which, when multiplied by 5 modulo 24, produces the original
number, 21.  If @var{m} is prime and the divisor is not a multiple of
@var{m}, it is always possible to find such a number.  For non-prime
@var{m} like 24, it is only sometimes possible. 

@smallexample
@group
1:  10 mod 24    1:  16 mod 24    1:  1000000...   1:  16
    .                .                .                .

    10 M 24 @key{RET}      100 ^            10 @key{RET} 100 ^     24 %
@end group
@end smallexample

@noindent
These two calculations get the same answer, but the first one is
much more efficient because it avoids the huge intermediate value
that arises in the second one.

@cindex Fermat, primality test of
(@bullet{}) @strong{Exercise 10.}  A theorem of Pierre de Fermat
says that 
@texline @w{@math{x^{n-1} \bmod n = 1}}
@infoline @expr{x^(n-1) mod n = 1}
if @expr{n} is a prime number and @expr{x} is an integer less than
@expr{n}.  If @expr{n} is @emph{not} a prime number, this will
@emph{not} be true for most values of @expr{x}.  Thus we can test
informally if a number is prime by trying this formula for several
values of @expr{x}.  Use this test to tell whether the following numbers
are prime: 811749613, 15485863.  @xref{Types Answer 10, 10}. (@bullet{})

It is possible to use HMS forms as parts of error forms, intervals,
modulo forms, or as the phase part of a polar complex number.
For example, the @code{calc-time} command pushes the current time
of day on the stack as an HMS/modulo form.

@smallexample
@group
1:  17@@ 34' 45" mod 24@@ 0' 0"     1:  6@@ 22' 15" mod 24@@ 0' 0"
    .                                 .

    x time @key{RET}                        n
@end group
@end smallexample

@noindent
This calculation tells me it is six hours and 22 minutes until midnight.

(@bullet{}) @strong{Exercise 11.}  A rule of thumb is that one year
is about 
@texline @math{\pi \times 10^7}
@infoline @w{@expr{pi * 10^7}} 
seconds.  What time will it be that many seconds from right now?
@xref{Types Answer 11, 11}. (@bullet{})

(@bullet{}) @strong{Exercise 12.}  You are preparing to order packaging
for the CD release of the Extended Disco Version of @emph{Abbey Road}.
You are told that the songs will actually be anywhere from 20 to 60
seconds longer than the originals.  One CD can hold about 75 minutes
of music.  Should you order single or double packages?
@xref{Types Answer 12, 12}. (@bullet{})

Another kind of data the Calculator can manipulate is numbers with
@dfn{units}.  This isn't strictly a new data type; it's simply an
application of algebraic expressions, where we use variables with
suggestive names like @samp{cm} and @samp{in} to represent units
like centimeters and inches.

@smallexample
@group
1:  2 in        1:  5.08 cm      1:  0.027778 fath   1:  0.0508 m
    .               .                .                   .

    ' 2in @key{RET}       u c cm @key{RET}       u c fath @key{RET}        u b
@end group
@end smallexample

@noindent
We enter the quantity ``2 inches'' (actually an algebraic expression
which means two times the variable @samp{in}), then we convert it
first to centimeters, then to fathoms, then finally to ``base'' units,
which in this case means meters.

@smallexample
@group
1:  9 acre     1:  3 sqrt(acre)   1:  190.84 m   1:  190.84 m + 30 cm
    .              .                  .              .

 ' 9 acre @key{RET}      Q                  u s            ' $+30 cm @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  191.14 m     1:  36536.3046 m^2    1:  365363046 cm^2
    .                .                     .

    u s              2 ^                   u c cgs
@end group
@end smallexample

@noindent
Since units expressions are really just formulas, taking the square
root of @samp{acre} is undefined.  After all, @code{acre} might be an
algebraic variable that you will someday assign a value.  We use the
``units-simplify'' command to simplify the expression with variables
being interpreted as unit names.

In the final step, we have converted not to a particular unit, but to a
units system.  The ``cgs'' system uses centimeters instead of meters
as its standard unit of length.

There is a wide variety of units defined in the Calculator.

@smallexample
@group
1:  55 mph     1:  88.5139 kph   1:   88.5139 km / hr   1:  8.201407e-8 c
    .              .                  .                     .

 ' 55 mph @key{RET}      u c kph @key{RET}        u c km/hr @key{RET}         u c c @key{RET}
@end group
@end smallexample

@noindent
We express a speed first in miles per hour, then in kilometers per
hour, then again using a slightly more explicit notation, then
finally in terms of fractions of the speed of light.

Temperature conversions are a bit more tricky.  There are two ways to
interpret ``20 degrees Fahrenheit''---it could mean an actual
temperature, or it could mean a change in temperature.  For normal
units there is no difference, but temperature units have an offset
as well as a scale factor and so there must be two explicit commands
for them.

@smallexample
@group
1:  20 degF       1:  11.1111 degC     1:  -20:3 degC    1:  -6.666 degC
    .                 .                    .                 .

  ' 20 degF @key{RET}       u c degC @key{RET}         U u t degC @key{RET}    c f
@end group
@end smallexample

@noindent
First we convert a change of 20 degrees Fahrenheit into an equivalent
change in degrees Celsius (or Centigrade).  Then, we convert the
absolute temperature 20 degrees Fahrenheit into Celsius.  Since
this comes out as an exact fraction, we then convert to floating-point
for easier comparison with the other result.

For simple unit conversions, you can put a plain number on the stack.
Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
When you use this method, you're responsible for remembering which
numbers are in which units:

@smallexample
@group
1:  55         1:  88.5139              1:  8.201407e-8
    .              .                        .

    55             u c mph @key{RET} kph @key{RET}      u c km/hr @key{RET} c @key{RET}
@end group
@end smallexample

To see a complete list of built-in units, type @kbd{u v}.  Press
@w{@kbd{C-x * c}} again to re-enter the Calculator when you're done looking
at the units table.

(@bullet{}) @strong{Exercise 13.}  How many seconds are there really
in a year?  @xref{Types Answer 13, 13}. (@bullet{})

@cindex Speed of light
(@bullet{}) @strong{Exercise 14.}  Supercomputer designs are limited by
the speed of light (and of electricity, which is nearly as fast).
Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
cabinet is one meter across.  Is speed of light going to be a
significant factor in its design?  @xref{Types Answer 14, 14}. (@bullet{})

(@bullet{}) @strong{Exercise 15.}  Sam the Slug normally travels about
five yards in an hour.  He has obtained a supply of Power Pills; each
Power Pill he eats doubles his speed.  How many Power Pills can he
swallow and still travel legally on most US highways?
@xref{Types Answer 15, 15}. (@bullet{})

@node Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
@section Algebra and Calculus Tutorial

@noindent
This section shows how to use Calc's algebra facilities to solve
equations, do simple calculus problems, and manipulate algebraic
formulas.

@menu
* Basic Algebra Tutorial::
* Rewrites Tutorial::
@end menu

@node Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial, Algebra Tutorial
@subsection Basic Algebra

@noindent
If you enter a formula in Algebraic mode that refers to variables,
the formula itself is pushed onto the stack.  You can manipulate
formulas as regular data objects.

@smallexample
@group
1:  2 x^2 - 6       1:  6 - 2 x^2       1:  (6 - 2 x^2) (3 x^2 + y)
    .                   .                   .

    ' 2x^2-6 @key{RET}        n                   ' 3x^2+y @key{RET} *
@end group
@end smallexample

(@bullet{}) @strong{Exercise 1.}  Do @kbd{' x @key{RET} Q 2 ^} and
@kbd{' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
Why or why not?  @xref{Algebra Answer 1, 1}. (@bullet{})

There are also commands for doing common algebraic operations on
formulas.  Continuing with the formula from the last example,

@smallexample
@group
1:  18 x^2 + 6 y - 6 x^4 - 2 x^2 y    1:  (18 - 2 y) x^2 - 6 x^4 + 6 y
    .                                     .

    a x                                   a c x @key{RET}
@end group
@end smallexample

@noindent
First we ``expand'' using the distributive law, then we ``collect''
terms involving like powers of @expr{x}.

Let's find the value of this expression when @expr{x} is 2 and @expr{y}
is one-half.

@smallexample
@group
1:  17 x^2 - 6 x^4 + 3      1:  -25
    .                           .

    1:2 s l y @key{RET}               2 s l x @key{RET}
@end group
@end smallexample

@noindent
The @kbd{s l} command means ``let''; it takes a number from the top of
the stack and temporarily assigns it as the value of the variable
you specify.  It then evaluates (as if by the @kbd{=} key) the
next expression on the stack.  After this command, the variable goes
back to its original value, if any.

(An earlier exercise in this tutorial involved storing a value in the
variable @code{x}; if this value is still there, you will have to
unstore it with @kbd{s u x @key{RET}} before the above example will work
properly.)

@cindex Maximum of a function using Calculus
Let's find the maximum value of our original expression when @expr{y}
is one-half and @expr{x} ranges over all possible values.  We can
do this by taking the derivative with respect to @expr{x} and examining
values of @expr{x} for which the derivative is zero.  If the second
derivative of the function at that value of @expr{x} is negative,
the function has a local maximum there.

@smallexample
@group
1:  17 x^2 - 6 x^4 + 3      1:  34 x - 24 x^3
    .                           .

    U @key{DEL}  s 1                  a d x @key{RET}   s 2
@end group
@end smallexample

@noindent
Well, the derivative is clearly zero when @expr{x} is zero.  To find
the other root(s), let's divide through by @expr{x} and then solve:

@smallexample
@group
1:  (34 x - 24 x^3) / x    1:  34 x / x - 24 x^3 / x    1:  34 - 24 x^2
    .                          .                            .

    ' x @key{RET} /                  a x                          a s

@end group
@end smallexample
@noindent
@smallexample
@group
1:  34 - 24 x^2 = 0        1:  x = 1.19023
    .                          .

    0 a =  s 3                 a S x @key{RET}
@end group
@end smallexample

@noindent
Notice the use of @kbd{a s} to ``simplify'' the formula.  When the
default algebraic simplifications don't do enough, you can use
@kbd{a s} to tell Calc to spend more time on the job.

Now we compute the second derivative and plug in our values of @expr{x}:

@smallexample
@group
1:  1.19023        2:  1.19023         2:  1.19023
    .              1:  34 x - 24 x^3   1:  34 - 72 x^2
                       .                   .

    a .                r 2                 a d x @key{RET} s 4
@end group
@end smallexample

@noindent
(The @kbd{a .} command extracts just the righthand side of an equation.
Another method would have been to use @kbd{v u} to unpack the equation
@w{@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M-- M-2 @key{DEL}}
to delete the @samp{x}.)

@smallexample
@group
2:  34 - 72 x^2   1:  -68.         2:  34 - 72 x^2     1:  34
1:  1.19023           .            1:  0                   .
    .                                  .

    @key{TAB}               s l x @key{RET}        U @key{DEL} 0             s l x @key{RET}
@end group
@end smallexample

@noindent
The first of these second derivatives is negative, so we know the function
has a maximum value at @expr{x = 1.19023}.  (The function also has a
local @emph{minimum} at @expr{x = 0}.)

When we solved for @expr{x}, we got only one value even though
@expr{34 - 24 x^2 = 0} is a quadratic equation that ought to have
two solutions.  The reason is that @w{@kbd{a S}} normally returns a
single ``principal'' solution.  If it needs to come up with an
arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
If it needs an arbitrary integer, it picks zero.  We can get a full
solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.

@smallexample
@group
1:  34 - 24 x^2 = 0    1:  x = 1.19023 s1      1:  x = -1.19023
    .                      .                       .

    r 3                    H a S x @key{RET}  s 5        1 n  s l s1 @key{RET}
@end group
@end smallexample

@noindent
Calc has invented the variable @samp{s1} to represent an unknown sign;
it is supposed to be either @mathit{+1} or @mathit{-1}.  Here we have used
the ``let'' command to evaluate the expression when the sign is negative.
If we plugged this into our second derivative we would get the same,
negative, answer, so @expr{x = -1.19023} is also a maximum.

To find the actual maximum value, we must plug our two values of @expr{x}
into the original formula.

@smallexample
@group
2:  17 x^2 - 6 x^4 + 3    1:  24.08333 s1^2 - 12.04166 s1^4 + 3
1:  x = 1.19023 s1            .
    .

    r 1 r 5                   s l @key{RET}
@end group
@end smallexample

@noindent
(Here we see another way to use @kbd{s l}; if its input is an equation
with a variable on the lefthand side, then @kbd{s l} treats the equation
like an assignment to that variable if you don't give a variable name.)

It's clear that this will have the same value for either sign of
@code{s1}, but let's work it out anyway, just for the exercise:

@smallexample
@group
2:  [-1, 1]              1:  [15.04166, 15.04166]
1:  24.08333 s1^2 ...        .
    .

  [ 1 n , 1 ] @key{TAB}            V M $ @key{RET}
@end group
@end smallexample

@noindent
Here we have used a vector mapping operation to evaluate the function
at several values of @samp{s1} at once.  @kbd{V M $} is like @kbd{V M '}
except that it takes the formula from the top of the stack.  The
formula is interpreted as a function to apply across the vector at the
next-to-top stack level.  Since a formula on the stack can't contain
@samp{$} signs, Calc assumes the variables in the formula stand for
different arguments.  It prompts you for an @dfn{argument list}, giving
the list of all variables in the formula in alphabetical order as the
default list.  In this case the default is @samp{(s1)}, which is just
what we want so we simply press @key{RET} at the prompt.

If there had been several different values, we could have used
@w{@kbd{V R X}} to find the global maximum.

Calc has a built-in @kbd{a P} command that solves an equation using
@w{@kbd{H a S}} and returns a vector of all the solutions.  It simply
automates the job we just did by hand.  Applied to our original
cubic polynomial, it would produce the vector of solutions
@expr{[1.19023, -1.19023, 0]}.  (There is also an @kbd{a X} command
which finds a local maximum of a function.  It uses a numerical search
method rather than examining the derivatives, and thus requires you
to provide some kind of initial guess to show it where to look.)

(@bullet{}) @strong{Exercise 2.}  Given a vector of the roots of a
polynomial (such as the output of an @kbd{a P} command), what
sequence of commands would you use to reconstruct the original
polynomial?  (The answer will be unique to within a constant
multiple; choose the solution where the leading coefficient is one.)
@xref{Algebra Answer 2, 2}. (@bullet{})

The @kbd{m s} command enables Symbolic mode, in which formulas
like @samp{sqrt(5)} that can't be evaluated exactly are left in
symbolic form rather than giving a floating-point approximate answer.
Fraction mode (@kbd{m f}) is also useful when doing algebra.

@smallexample
@group
2:  34 x - 24 x^3        2:  34 x - 24 x^3
1:  34 x - 24 x^3        1:  [sqrt(51) / 6, sqrt(51) / -6, 0]
    .                        .

    r 2  @key{RET}     m s  m f    a P x @key{RET}
@end group
@end smallexample

One more mode that makes reading formulas easier is Big mode.

@smallexample
@group
               3
2:  34 x - 24 x

      ____   ____
     V 51   V 51
1:  [-----, -----, 0]
       6     -6

    .

    d B
@end group
@end smallexample

Here things like powers, square roots, and quotients and fractions
are displayed in a two-dimensional pictorial form.  Calc has other
language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
and La@TeX{} mode.

@smallexample
@group
2:  34*x - 24*pow(x, 3)               2:  34*x - 24*x**3
1:  @{sqrt(51) / 6, sqrt(51) / -6, 0@}  1:  /sqrt(51) / 6, sqrt(51) / -6, 0/
    .                                     .

    d C                                   d F

@end group
@end smallexample
@noindent
@smallexample
@group
3:  34 x - 24 x^3
2:  [@{\sqrt@{51@} \over 6@}, @{\sqrt@{51@} \over -6@}, 0]
1:  @{2 \over 3@} \sqrt@{5@}
    .

    d T   ' 2 \sqrt@{5@} \over 3 @key{RET}
@end group
@end smallexample

@noindent
As you can see, language modes affect both entry and display of
formulas.  They affect such things as the names used for built-in
functions, the set of arithmetic operators and their precedences,
and notations for vectors and matrices.

Notice that @samp{sqrt(51)} may cause problems with older
implementations of C and FORTRAN, which would require something more
like @samp{sqrt(51.0)}.  It is always wise to check over the formulas
produced by the various language modes to make sure they are fully
correct.

Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes.  (You
may prefer to remain in Big mode, but all the examples in the tutorial
are shown in normal mode.)

@cindex Area under a curve
What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
This is simply the integral of the function:

@smallexample
@group
1:  17 x^2 - 6 x^4 + 3     1:  5.6666 x^3 - 1.2 x^5 + 3 x
    .                          .

    r 1                        a i x
@end group
@end smallexample

@noindent
We want to evaluate this at our two values for @expr{x} and subtract.
One way to do it is again with vector mapping and reduction:

@smallexample
@group
2:  [2, 1]            1:  [12.93333, 7.46666]    1:  5.46666
1:  5.6666 x^3 ...        .                          .

   [ 2 , 1 ] @key{TAB}          V M $ @key{RET}                  V R -
@end group
@end smallexample

(@bullet{}) @strong{Exercise 3.}  Find the integral from 1 to @expr{y}
of 
@texline @math{x \sin \pi x}
@infoline @w{@expr{x sin(pi x)}} 
(where the sine is calculated in radians).  Find the values of the
integral for integers @expr{y} from 1 to 5.  @xref{Algebra Answer 3,
3}. (@bullet{})

Calc's integrator can do many simple integrals symbolically, but many
others are beyond its capabilities.  Suppose we wish to find the area
under the curve 
@texline @math{\sin x \ln x}
@infoline @expr{sin(x) ln(x)} 
over the same range of @expr{x}.  If you entered this formula and typed
@kbd{a i x @key{RET}} (don't bother to try this), Calc would work for a
long time but would be unable to find a solution.  In fact, there is no
closed-form solution to this integral.  Now what do we do?

@cindex Integration, numerical
@cindex Numerical integration
One approach would be to do the integral numerically.  It is not hard
to do this by hand using vector mapping and reduction.  It is rather
slow, though, since the sine and logarithm functions take a long time.
We can save some time by reducing the working precision.

@smallexample
@group
3:  10                  1:  [1, 1.1, 1.2,  ...  , 1.8, 1.9]
2:  1                       .
1:  0.1
    .

 10 @key{RET} 1 @key{RET} .1 @key{RET}        C-u v x
@end group
@end smallexample

@noindent
(Note that we have used the extended version of @kbd{v x}; we could
also have used plain @kbd{v x} as follows:  @kbd{v x 10 @key{RET} 9 + .1 *}.)

@smallexample
@group
2:  [1, 1.1, ... ]              1:  [0., 0.084941, 0.16993, ... ]
1:  sin(x) ln(x)                    .
    .

    ' sin(x) ln(x) @key{RET}  s 1    m r  p 5 @key{RET}   V M $ @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  3.4195     0.34195
    .          .

    V R +      0.1 *
@end group
@end smallexample

@noindent
(If you got wildly different results, did you remember to switch
to Radians mode?)

Here we have divided the curve into ten segments of equal width;
approximating these segments as rectangular boxes (i.e., assuming
the curve is nearly flat at that resolution), we compute the areas
of the boxes (height times width), then sum the areas.  (It is
faster to sum first, then multiply by the width, since the width
is the same for every box.)

The true value of this integral turns out to be about 0.374, so
we're not doing too well.  Let's try another approach.

@smallexample
@group
1:  sin(x) ln(x)    1:  0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ...
    .                   .

    r 1                 a t x=1 @key{RET} 4 @key{RET}
@end group
@end smallexample

@noindent
Here we have computed the Taylor series expansion of the function
about the point @expr{x=1}.  We can now integrate this polynomial
approximation, since polynomials are easy to integrate.

@smallexample
@group
1:  0.42074 x^2 + ...    1:  [-0.0446, -0.42073]      1:  0.3761
    .                        .                            .

    a i x @key{RET}            [ 2 , 1 ] @key{TAB}  V M $ @key{RET}         V R -
@end group
@end smallexample

@noindent
Better!  By increasing the precision and/or asking for more terms
in the Taylor series, we can get a result as accurate as we like.
(Taylor series converge better away from singularities in the
function such as the one at @code{ln(0)}, so it would also help to
expand the series about the points @expr{x=2} or @expr{x=1.5} instead
of @expr{x=1}.)

@cindex Simpson's rule
@cindex Integration by Simpson's rule
(@bullet{}) @strong{Exercise 4.}  Our first method approximated the
curve by stairsteps of width 0.1; the total area was then the sum
of the areas of the rectangles under these stairsteps.  Our second
method approximated the function by a polynomial, which turned out
to be a better approximation than stairsteps.  A third method is
@dfn{Simpson's rule}, which is like the stairstep method except
that the steps are not required to be flat.  Simpson's rule boils
down to the formula,

@ifinfo
@example
(h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
              + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \displaylines{
      \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
   \hfill \cr \hfill    {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
} $$
\afterdisplay
@end tex

@noindent
where @expr{n} (which must be even) is the number of slices and @expr{h}
is the width of each slice.  These are 10 and 0.1 in our example.
For reference, here is the corresponding formula for the stairstep
method:

@ifinfo
@example
h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
          + f(a+(n-2)*h) + f(a+(n-1)*h))
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
           + f(a+(n-2)h) + f(a+(n-1)h)) $$
\afterdisplay
@end tex

Compute the integral from 1 to 2 of 
@texline @math{\sin x \ln x}
@infoline @expr{sin(x) ln(x)} 
using Simpson's rule with 10 slices.  
@xref{Algebra Answer 4, 4}. (@bullet{})

Calc has a built-in @kbd{a I} command for doing numerical integration.
It uses @dfn{Romberg's method}, which is a more sophisticated cousin
of Simpson's rule.  In particular, it knows how to keep refining the
result until the current precision is satisfied.

@c [fix-ref Selecting Sub-Formulas]
Aside from the commands we've seen so far, Calc also provides a
large set of commands for operating on parts of formulas.  You
indicate the desired sub-formula by placing the cursor on any part
of the formula before giving a @dfn{selection} command.  Selections won't
be covered in the tutorial; @pxref{Selecting Subformulas}, for
details and examples.

@c hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1) 2^(n - 1)
@c                to 2^((n-1)*(r-1)).

@node Rewrites Tutorial,  , Basic Algebra Tutorial, Algebra Tutorial
@subsection Rewrite Rules

@noindent
No matter how many built-in commands Calc provided for doing algebra,
there would always be something you wanted to do that Calc didn't have
in its repertoire.  So Calc also provides a @dfn{rewrite rule} system
that you can use to define your own algebraic manipulations.

Suppose we want to simplify this trigonometric formula:

@smallexample
@group
1:  1 / cos(x) - sin(x) tan(x)
    .

    ' 1/cos(x) - sin(x) tan(x) @key{RET}   s 1
@end group
@end smallexample

@noindent
If we were simplifying this by hand, we'd probably replace the
@samp{tan} with a @samp{sin/cos} first, then combine over a common
denominator.  There is no Calc command to do the former; the @kbd{a n}
algebra command will do the latter but we'll do both with rewrite
rules just for practice.

Rewrite rules are written with the @samp{:=} symbol.

@smallexample
@group
1:  1 / cos(x) - sin(x)^2 / cos(x)
    .

    a r tan(a) := sin(a)/cos(a) @key{RET}
@end group
@end smallexample

@noindent
(The ``assignment operator'' @samp{:=} has several uses in Calc.  All
by itself the formula @samp{tan(a) := sin(a)/cos(a)} doesn't do anything,
but when it is given to the @kbd{a r} command, that command interprets
it as a rewrite rule.)

The lefthand side, @samp{tan(a)}, is called the @dfn{pattern} of the
rewrite rule.  Calc searches the formula on the stack for parts that
match the pattern.  Variables in a rewrite pattern are called
@dfn{meta-variables}, and when matching the pattern each meta-variable
can match any sub-formula.  Here, the meta-variable @samp{a} matched
the actual variable @samp{x}.

When the pattern part of a rewrite rule matches a part of the formula,
that part is replaced by the righthand side with all the meta-variables
substituted with the things they matched.  So the result is
@samp{sin(x) / cos(x)}.  Calc's normal algebraic simplifications then
mix this in with the rest of the original formula.

To merge over a common denominator, we can use another simple rule:

@smallexample
@group
1:  (1 - sin(x)^2) / cos(x)
    .

    a r a/x + b/x := (a+b)/x @key{RET}
@end group
@end smallexample

This rule points out several interesting features of rewrite patterns.
First, if a meta-variable appears several times in a pattern, it must
match the same thing everywhere.  This rule detects common denominators
because the same meta-variable @samp{x} is used in both of the
denominators.

Second, meta-variable names are independent from variables in the
target formula.  Notice that the meta-variable @samp{x} here matches
the subformula @samp{cos(x)}; Calc never confuses the two meanings of
@samp{x}.

And third, rewrite patterns know a little bit about the algebraic
properties of formulas.  The pattern called for a sum of two quotients;
Calc was able to match a difference of two quotients by matching
@samp{a = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.

@c [fix-ref Algebraic Properties of Rewrite Rules]
We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
the rule.  It would have worked just the same in all cases.  (If we
really wanted the rule to apply only to @samp{+} or only to @samp{-},
we could have used the @code{plain} symbol.  @xref{Algebraic Properties
of Rewrite Rules}, for some examples of this.)

One more rewrite will complete the job.  We want to use the identity
@samp{sin(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
the identity in a way that matches our formula.  The obvious rule
would be @samp{@w{1 - sin(x)^2} := cos(x)^2}, but a little thought shows
that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work.  The
latter rule has a more general pattern so it will work in many other
situations, too.

@smallexample
@group
1:  (1 + cos(x)^2 - 1) / cos(x)           1:  cos(x)
    .                                         .

    a r sin(x)^2 := 1 - cos(x)^2 @key{RET}          a s
@end group
@end smallexample

You may ask, what's the point of using the most general rule if you
have to type it in every time anyway?  The answer is that Calc allows
you to store a rewrite rule in a variable, then give the variable
name in the @kbd{a r} command.  In fact, this is the preferred way to
use rewrites.  For one, if you need a rule once you'll most likely
need it again later.  Also, if the rule doesn't work quite right you
can simply Undo, edit the variable, and run the rule again without
having to retype it.

@smallexample
@group
' tan(x) := sin(x)/cos(x) @key{RET}      s t tsc @key{RET}
' a/x + b/x := (a+b)/x @key{RET}         s t merge @key{RET}
' sin(x)^2 := 1 - cos(x)^2 @key{RET}     s t sinsqr @key{RET}

1:  1 / cos(x) - sin(x) tan(x)     1:  cos(x)
    .                                  .

    r 1                a r tsc @key{RET}  a r merge @key{RET}  a r sinsqr @key{RET}  a s
@end group
@end smallexample

To edit a variable, type @kbd{s e} and the variable name, use regular
Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
the edited value back into the variable. 
You can also use @w{@kbd{s e}} to create a new variable if you wish.

Notice that the first time you use each rule, Calc puts up a ``compiling''
message briefly.  The pattern matcher converts rules into a special
optimized pattern-matching language rather than using them directly.
This allows @kbd{a r} to apply even rather complicated rules very
efficiently.  If the rule is stored in a variable, Calc compiles it
only once and stores the compiled form along with the variable.  That's
another good reason to store your rules in variables rather than
entering them on the fly.

(@bullet{}) @strong{Exercise 1.}  Type @kbd{m s} to get Symbolic
mode, then enter the formula @samp{@w{(2 + sqrt(2))} / @w{(1 + sqrt(2))}}.
Using a rewrite rule, simplify this formula by multiplying the top and
bottom by the conjugate @w{@samp{1 - sqrt(2)}}.  The result will have
to be expanded by the distributive law; do this with another
rewrite.  @xref{Rewrites Answer 1, 1}. (@bullet{})

The @kbd{a r} command can also accept a vector of rewrite rules, or
a variable containing a vector of rules.

@smallexample
@group
1:  [tsc, merge, sinsqr]          1:  [tan(x) := sin(x) / cos(x), ... ]
    .                                 .

    ' [tsc,merge,sinsqr] @key{RET}          =

@end group
@end smallexample
@noindent
@smallexample
@group
1:  1 / cos(x) - sin(x) tan(x)    1:  cos(x)
    .                                 .

    s t trig @key{RET}  r 1                 a r trig @key{RET}  a s
@end group
@end smallexample

@c [fix-ref Nested Formulas with Rewrite Rules]
Calc tries all the rules you give against all parts of the formula,
repeating until no further change is possible.  (The exact order in
which things are tried is rather complex, but for simple rules like
the ones we've used here the order doesn't really matter.
@xref{Nested Formulas with Rewrite Rules}.)

Calc actually repeats only up to 100 times, just in case your rule set
has gotten into an infinite loop.  You can give a numeric prefix argument
to @kbd{a r} to specify any limit.  In particular, @kbd{M-1 a r} does
only one rewrite at a time.

@smallexample
@group
1:  1 / cos(x) - sin(x)^2 / cos(x)    1:  (1 - sin(x)^2) / cos(x)
    .                                     .

    r 1  M-1 a r trig @key{RET}                 M-1 a r trig @key{RET}
@end group
@end smallexample

You can type @kbd{M-0 a r} if you want no limit at all on the number
of rewrites that occur.

Rewrite rules can also be @dfn{conditional}.  Simply follow the rule
with a @samp{::} symbol and the desired condition.  For example,

@smallexample
@group
1:  exp(2 pi i) + exp(3 pi i) + exp(4 pi i)
    .

    ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  1 + exp(3 pi i) + 1
    .

    a r exp(k pi i) := 1 :: k % 2 = 0 @key{RET}
@end group
@end smallexample

@noindent
(Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
which will be zero only when @samp{k} is an even integer.)

An interesting point is that the variables @samp{pi} and @samp{i}
were matched literally rather than acting as meta-variables.
This is because they are special-constant variables.  The special
constants @samp{e}, @samp{phi}, and so on also match literally.
A common error with rewrite
rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
to match any @samp{f} with five arguments but in fact matching
only when the fifth argument is literally @samp{e}!

@cindex Fibonacci numbers
@ignore
@starindex
@end ignore
@tindex fib
Rewrite rules provide an interesting way to define your own functions.
Suppose we want to define @samp{fib(n)} to produce the @var{n}th
Fibonacci number.  The first two Fibonacci numbers are each 1;
later numbers are formed by summing the two preceding numbers in
the sequence.  This is easy to express in a set of three rules:

@smallexample
@group
' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET}  s t fib

1:  fib(7)               1:  13
    .                        .

    ' fib(7) @key{RET}             a r fib @key{RET}
@end group
@end smallexample

One thing that is guaranteed about the order that rewrites are tried
is that, for any given subformula, earlier rules in the rule set will
be tried for that subformula before later ones.  So even though the
first and third rules both match @samp{fib(1)}, we know the first will
be used preferentially.

This rule set has one dangerous bug:  Suppose we apply it to the
formula @samp{fib(x)}?  (Don't actually try this.)  The third rule
will match @samp{fib(x)} and replace it with @w{@samp{fib(x-1) + fib(x-2)}}.
Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
fib(x-4)}, and so on, expanding forever.  What we really want is to apply
the third rule only when @samp{n} is an integer greater than two.  Type
@w{@kbd{s e fib @key{RET}}}, then edit the third rule to:

@smallexample
fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
@end smallexample

@noindent
Now:

@smallexample
@group
1:  fib(6) + fib(x) + fib(0)      1:  8 + fib(x) + fib(0)
    .                                 .

    ' fib(6)+fib(x)+fib(0) @key{RET}        a r fib @key{RET}
@end group
@end smallexample

@noindent
We've created a new function, @code{fib}, and a new command,
@w{@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib} calls in
this formula.''  To make things easier still, we can tell Calc to
apply these rules automatically by storing them in the special
variable @code{EvalRules}.

@smallexample
@group
1:  [fib(1) := ...]    .                1:  [8, 13]
    .                                       .

    s r fib @key{RET}        s t EvalRules @key{RET}    ' [fib(6), fib(7)] @key{RET}
@end group
@end smallexample

It turns out that this rule set has the problem that it does far
more work than it needs to when @samp{n} is large.  Consider the
first few steps of the computation of @samp{fib(6)}:

@smallexample
@group
fib(6) =
fib(5)              +               fib(4) =
fib(4)     +      fib(3)     +      fib(3)     +      fib(2) =
fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
@end group
@end smallexample

@noindent
Note that @samp{fib(3)} appears three times here.  Unless Calc's
algebraic simplifier notices the multiple @samp{fib(3)}s and combines
them (and, as it happens, it doesn't), this rule set does lots of
needless recomputation.  To cure the problem, type @code{s e EvalRules}
to edit the rules (or just @kbd{s E}, a shorthand command for editing
@code{EvalRules}) and add another condition:

@smallexample
fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
@end smallexample

@noindent
If a @samp{:: remember} condition appears anywhere in a rule, then if
that rule succeeds Calc will add another rule that describes that match
to the front of the rule set.  (Remembering works in any rule set, but
for technical reasons it is most effective in @code{EvalRules}.)  For
example, if the rule rewrites @samp{fib(7)} to something that evaluates
to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.

Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
type @kbd{s E} again to see what has happened to the rule set.

With the @code{remember} feature, our rule set can now compute
@samp{fib(@var{n})} in just @var{n} steps.  In the process it builds
up a table of all Fibonacci numbers up to @var{n}.  After we have
computed the result for a particular @var{n}, we can get it back
(and the results for all smaller @var{n}) later in just one step.

All Calc operations will run somewhat slower whenever @code{EvalRules}
contains any rules.  You should type @kbd{s u EvalRules @key{RET}} now to
un-store the variable.

(@bullet{}) @strong{Exercise 2.}  Sometimes it is possible to reformulate
a problem to reduce the amount of recursion necessary to solve it.
Create a rule that, in about @var{n} simple steps and without recourse
to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
@samp{fib(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
@var{n}th and @var{n+1}st Fibonacci numbers, respectively.  This rule is
rather clunky to use, so add a couple more rules to make the ``user
interface'' the same as for our first version: enter @samp{fib(@var{n})},
get back a plain number.  @xref{Rewrites Answer 2, 2}. (@bullet{})

There are many more things that rewrites can do.  For example, there
are @samp{&&&} and @samp{|||} pattern operators that create ``and''
and ``or'' combinations of rules.  As one really simple example, we
could combine our first two Fibonacci rules thusly:

@example
[fib(1 ||| 2) := 1, fib(n) := ... ]
@end example

@noindent
That means ``@code{fib} of something matching either 1 or 2 rewrites
to 1.''

You can also make meta-variables optional by enclosing them in @code{opt}.
For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
@samp{2 + x} or @samp{3 x} or @samp{x}.  The pattern @samp{opt(a) + opt(b) x}
matches all of these forms, filling in a default of zero for @samp{a}
and one for @samp{b}.

(@bullet{}) @strong{Exercise 3.}  Your friend Joe had @samp{2 + 3 x}
on the stack and tried to use the rule
@samp{opt(a) + opt(b) x := f(a, b, x)}.  What happened?
@xref{Rewrites Answer 3, 3}. (@bullet{})

(@bullet{}) @strong{Exercise 4.}  Starting with a positive integer @expr{a},
divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
Now repeat this step over and over.  A famous unproved conjecture
is that for any starting @expr{a}, the sequence always eventually
reaches 1.  Given the formula @samp{seq(@var{a}, 0)}, write a set of
rules that convert this into @samp{seq(1, @var{n})} where @var{n}
is the number of steps it took the sequence to reach the value 1.
Now enhance the rules to accept @samp{seq(@var{a})} as a starting
configuration, and to stop with just the number @var{n} by itself.
Now make the result be a vector of values in the sequence, from @var{a}
to 1.  (The formula @samp{@var{x}|@var{y}} appends the vectors @var{x}
and @var{y}.)  For example, rewriting @samp{seq(6)} should yield the
vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
@xref{Rewrites Answer 4, 4}. (@bullet{})

(@bullet{}) @strong{Exercise 5.}  Define, using rewrite rules, a function
@samp{nterms(@var{x})} that returns the number of terms in the sum
@var{x}, or 1 if @var{x} is not a sum.  (A @dfn{sum} for our purposes
is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
@xref{Rewrites Answer 5, 5}. (@bullet{})

(@bullet{}) @strong{Exercise 6.}  A Taylor series for a function is an
infinite series that exactly equals the value of that function at
values of @expr{x} near zero.

@ifinfo
@example
cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
\afterdisplay
@end tex

The @kbd{a t} command produces a @dfn{truncated Taylor series} which
is obtained by dropping all the terms higher than, say, @expr{x^2}.
Calc represents the truncated Taylor series as a polynomial in @expr{x}.
Mathematicians often write a truncated series using a ``big-O'' notation
that records what was the lowest term that was truncated.

@ifinfo
@example
cos(x) = 1 - x^2 / 2! + O(x^3)
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
\afterdisplay
@end tex

@noindent
The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''

The exercise is to create rewrite rules that simplify sums and products of
power series represented as @samp{@var{polynomial} + O(@var{var}^@var{n})}.
For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
on the stack, we want to be able to type @kbd{*} and get the result
@samp{x - 2:3 x^3 + O(x^4)}.  Don't worry if the terms of the sum are
rearranged or if @kbd{a s} needs to be typed after rewriting.  (This one
is rather tricky; the solution at the end of this chapter uses 6 rewrite
rules.  Hint:  The @samp{constant(x)} condition tests whether @samp{x} is
a number.)  @xref{Rewrites Answer 6, 6}. (@bullet{})

Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
What happens?  (Be sure to remove this rule afterward, or you might get
a nasty surprise when you use Calc to balance your checkbook!)

@xref{Rewrite Rules}, for the whole story on rewrite rules.

@node Programming Tutorial, Answers to Exercises, Algebra Tutorial, Tutorial
@section Programming Tutorial

@noindent
The Calculator is written entirely in Emacs Lisp, a highly extensible
language.  If you know Lisp, you can program the Calculator to do
anything you like.  Rewrite rules also work as a powerful programming
system.  But Lisp and rewrite rules take a while to master, and often
all you want to do is define a new function or repeat a command a few
times.  Calc has features that allow you to do these things easily.

One very limited form of programming is defining your own functions.
Calc's @kbd{Z F} command allows you to define a function name and
key sequence to correspond to any formula.  Programming commands use
the shift-@kbd{Z} prefix; the user commands they create use the lower
case @kbd{z} prefix.

@smallexample
@group
1:  1 + x + x^2 / 2 + x^3 / 6         1:  1 + x + x^2 / 2 + x^3 / 6
    .                                     .

    ' 1 + x + x^2/2! + x^3/3! @key{RET}         Z F e myexp @key{RET} @key{RET} @key{RET} y
@end group
@end smallexample

This polynomial is a Taylor series approximation to @samp{exp(x)}.
The @kbd{Z F} command asks a number of questions.  The above answers
say that the key sequence for our function should be @kbd{z e}; the
@kbd{M-x} equivalent should be @code{calc-myexp}; the name of the
function in algebraic formulas should also be @code{myexp}; the
default argument list @samp{(x)} is acceptable; and finally @kbd{y}
answers the question ``leave it in symbolic form for non-constant
arguments?''

@smallexample
@group
1:  1.3495     2:  1.3495     3:  1.3495
    .          1:  1.34986    2:  1.34986
                   .          1:  myexp(a + 1)
                                  .

    .3 z e         .3 E           ' a+1 @key{RET} z e
@end group
@end smallexample

@noindent
First we call our new @code{exp} approximation with 0.3 as an
argument, and compare it with the true @code{exp} function.  Then
we note that, as requested, if we try to give @kbd{z e} an
argument that isn't a plain number, it leaves the @code{myexp}
function call in symbolic form.  If we had answered @kbd{n} to the
final question, @samp{myexp(a + 1)} would have evaluated by plugging
in @samp{a + 1} for @samp{x} in the defining formula.

@cindex Sine integral Si(x)
@ignore
@starindex
@end ignore
@tindex Si
(@bullet{}) @strong{Exercise 1.}  The ``sine integral'' function
@texline @math{{\rm Si}(x)}
@infoline @expr{Si(x)} 
is defined as the integral of @samp{sin(t)/t} for
@expr{t = 0} to @expr{x} in radians.  (It was invented because this
integral has no solution in terms of basic functions; if you give it
to Calc's @kbd{a i} command, it will ponder it for a long time and then
give up.)  We can use the numerical integration command, however,
which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
with any integrand @samp{f(t)}.  Define a @kbd{z s} command and
@code{Si} function that implement this.  You will need to edit the
default argument list a bit.  As a test, @samp{Si(1)} should return
0.946083. (If you don't get this answer, you might want to check that
Calc is in Radians mode.  Also, @code{ninteg} will run a lot faster if
you reduce the precision to, say, six digits beforehand.)
@xref{Programming Answer 1, 1}. (@bullet{})

The simplest way to do real ``programming'' of Emacs is to define a
@dfn{keyboard macro}.  A keyboard macro is simply a sequence of
keystrokes which Emacs has stored away and can play back on demand.
For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
you may wish to program a keyboard macro to type this for you.

@smallexample
@group
1:  y = sqrt(x)          1:  x = y^2
    .                        .

    ' y=sqrt(x) @key{RET}       C-x ( H a S x @key{RET} C-x )

1:  y = cos(x)           1:  x = s1 arccos(y) + 2 pi n1
    .                        .

    ' y=cos(x) @key{RET}           X
@end group
@end smallexample

@noindent
When you type @kbd{C-x (}, Emacs begins recording.  But it is also
still ready to execute your keystrokes, so you're really ``training''
Emacs by walking it through the procedure once.  When you type
@w{@kbd{C-x )}}, the macro is recorded.  You can now type @kbd{X} to
re-execute the same keystrokes.

You can give a name to your macro by typing @kbd{Z K}.

@smallexample
@group
1:  .              1:  y = x^4         1:  x = s2 sqrt(s1 sqrt(y))
                       .                   .

  Z K x @key{RET}            ' y=x^4 @key{RET}         z x
@end group
@end smallexample

@noindent
Notice that we use shift-@kbd{Z} to define the command, and lower-case
@kbd{z} to call it up.

Keyboard macros can call other macros.

@smallexample
@group
1:  abs(x)        1:  x = s1 y                1:  2 / x    1:  x = 2 / y
    .                 .                           .            .

 ' abs(x) @key{RET}   C-x ( ' y @key{RET} a = z x C-x )    ' 2/x @key{RET}       X
@end group
@end smallexample

(@bullet{}) @strong{Exercise 2.}  Define a keyboard macro to negate
the item in level 3 of the stack, without disturbing the rest of
the stack.  @xref{Programming Answer 2, 2}. (@bullet{})

(@bullet{}) @strong{Exercise 3.}  Define keyboard macros to compute
the following functions:

@enumerate
@item
Compute 
@texline @math{\displaystyle{\sin x \over x}},
@infoline @expr{sin(x) / x}, 
where @expr{x} is the number on the top of the stack.

@item
Compute the base-@expr{b} logarithm, just like the @kbd{B} key except
the arguments are taken in the opposite order.

@item
Produce a vector of integers from 1 to the integer on the top of
the stack.
@end enumerate
@noindent
@xref{Programming Answer 3, 3}. (@bullet{})

(@bullet{}) @strong{Exercise 4.}  Define a keyboard macro to compute
the average (mean) value of a list of numbers.
@xref{Programming Answer 4, 4}. (@bullet{})

In many programs, some of the steps must execute several times.
Calc has @dfn{looping} commands that allow this.  Loops are useful
inside keyboard macros, but actually work at any time.

@smallexample
@group
1:  x^6          2:  x^6        1: 360 x^2
    .            1:  4             .
                     .

  ' x^6 @key{RET}          4         Z < a d x @key{RET} Z >
@end group
@end smallexample

@noindent
Here we have computed the fourth derivative of @expr{x^6} by
enclosing a derivative command in a ``repeat loop'' structure.
This structure pops a repeat count from the stack, then
executes the body of the loop that many times.

If you make a mistake while entering the body of the loop,
type @w{@kbd{Z C-g}} to cancel the loop command.

@cindex Fibonacci numbers
Here's another example:

@smallexample
@group
3:  1               2:  10946
2:  1               1:  17711
1:  20                  .
    .

1 @key{RET} @key{RET} 20       Z < @key{TAB} C-j + Z >
@end group
@end smallexample

@noindent
The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
numbers, respectively.  (To see what's going on, try a few repetitions
of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
key if you have one, makes a copy of the number in level 2.)

@cindex Golden ratio
@cindex Phi, golden ratio
A fascinating property of the Fibonacci numbers is that the @expr{n}th
Fibonacci number can be found directly by computing 
@texline @math{\phi^n / \sqrt{5}}
@infoline @expr{phi^n / sqrt(5)}
and then rounding to the nearest integer, where 
@texline @math{\phi} (``phi''),
@infoline @expr{phi}, 
the ``golden ratio,'' is 
@texline @math{(1 + \sqrt{5}) / 2}.
@infoline @expr{(1 + sqrt(5)) / 2}. 
(For convenience, this constant is available from the @code{phi}
variable, or the @kbd{I H P} command.)

@smallexample
@group
1:  1.61803         1:  24476.0000409    1:  10945.9999817    1:  10946
    .                   .                    .                    .

    I H P               21 ^                 5 Q /                R
@end group
@end smallexample

@cindex Continued fractions
(@bullet{}) @strong{Exercise 5.}  The @dfn{continued fraction}
representation of 
@texline @math{\phi}
@infoline @expr{phi} 
is 
@texline @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
@infoline @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
We can compute an approximate value by carrying this however far
and then replacing the innermost 
@texline @math{1/( \ldots )}
@infoline @expr{1/( ...@: )} 
by 1.  Approximate
@texline @math{\phi}
@infoline @expr{phi} 
using a twenty-term continued fraction.
@xref{Programming Answer 5, 5}. (@bullet{})

(@bullet{}) @strong{Exercise 6.}  Linear recurrences like the one for
Fibonacci numbers can be expressed in terms of matrices.  Given a
vector @w{@expr{[a, b]}} determine a matrix which, when multiplied by this
vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
@expr{c} are three successive Fibonacci numbers.  Now write a program
that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
using matrix arithmetic.  @xref{Programming Answer 6, 6}. (@bullet{})

@cindex Harmonic numbers
A more sophisticated kind of loop is the @dfn{for} loop.  Suppose
we wish to compute the 20th ``harmonic'' number, which is equal to
the sum of the reciprocals of the integers from 1 to 20.

@smallexample
@group
3:  0               1:  3.597739
2:  1                   .
1:  20
    .

0 @key{RET} 1 @key{RET} 20         Z ( & + 1 Z )
@end group
@end smallexample

@noindent
The ``for'' loop pops two numbers, the lower and upper limits, then
repeats the body of the loop as an internal counter increases from
the lower limit to the upper one.  Just before executing the loop
body, it pushes the current loop counter.  When the loop body
finishes, it pops the ``step,'' i.e., the amount by which to
increment the loop counter.  As you can see, our loop always
uses a step of one.

This harmonic number function uses the stack to hold the running
total as well as for the various loop housekeeping functions.  If
you find this disorienting, you can sum in a variable instead:

@smallexample
@group
1:  0         2:  1                  .            1:  3.597739
    .         1:  20                                  .
                  .

    0 t 7       1 @key{RET} 20      Z ( & s + 7 1 Z )       r 7
@end group
@end smallexample

@noindent
The @kbd{s +} command adds the top-of-stack into the value in a
variable (and removes that value from the stack).

It's worth noting that many jobs that call for a ``for'' loop can
also be done more easily by Calc's high-level operations.  Two
other ways to compute harmonic numbers are to use vector mapping
and reduction (@kbd{v x 20}, then @w{@kbd{V M &}}, then @kbd{V R +}),
or to use the summation command @kbd{a +}.  Both of these are
probably easier than using loops.  However, there are some
situations where loops really are the way to go:

(@bullet{}) @strong{Exercise 7.}  Use a ``for'' loop to find the first
harmonic number which is greater than 4.0.
@xref{Programming Answer 7, 7}. (@bullet{})

Of course, if we're going to be using variables in our programs,
we have to worry about the programs clobbering values that the
caller was keeping in those same variables.  This is easy to
fix, though:

@smallexample
@group
    .        1:  0.6667       1:  0.6667     3:  0.6667
                 .                .          2:  3.597739
                                             1:  0.6667
                                                 .

   Z `    p 4 @key{RET} 2 @key{RET} 3 /   s 7 s s a @key{RET}    Z '  r 7 s r a @key{RET}
@end group
@end smallexample

@noindent
When we type @kbd{Z `} (that's a back-quote character), Calc saves
its mode settings and the contents of the ten ``quick variables''
for later reference.  When we type @kbd{Z '} (that's an apostrophe
now), Calc restores those saved values.  Thus the @kbd{p 4} and
@kbd{s 7} commands have no effect outside this sequence.  Wrapping
this around the body of a keyboard macro ensures that it doesn't
interfere with what the user of the macro was doing.  Notice that
the contents of the stack, and the values of named variables,
survive past the @kbd{Z '} command.

@cindex Bernoulli numbers, approximate
The @dfn{Bernoulli numbers} are a sequence with the interesting
property that all of the odd Bernoulli numbers are zero, and the
even ones, while difficult to compute, can be roughly approximated
by the formula 
@texline @math{\displaystyle{2 n! \over (2 \pi)^n}}.
@infoline @expr{2 n!@: / (2 pi)^n}.  
Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
(Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
this command is very slow for large @expr{n} since the higher Bernoulli
numbers are very large fractions.)

@smallexample
@group
1:  10               1:  0.0756823
    .                    .

    10     C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET} = Z ] C-x )
@end group
@end smallexample

@noindent
You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
@kbd{Z ]} as ``end-if.''  There is no need for an explicit ``if''
command.  For the purposes of @w{@kbd{Z [}}, the condition is ``true''
if the value it pops from the stack is a nonzero number, or ``false''
if it pops zero or something that is not a number (like a formula).
Here we take our integer argument modulo 2; this will be nonzero
if we're asking for an odd Bernoulli number.

The actual tenth Bernoulli number is @expr{5/66}.

@smallexample
@group
3:  0.0756823    1:  0          1:  0.25305    1:  0          1:  1.16659
2:  5:66             .              .              .              .
1:  0.0757575
    .

10 k b @key{RET} c f   M-0 @key{DEL} 11 X   @key{DEL} 12 X       @key{DEL} 13 X       @key{DEL} 14 X
@end group
@end smallexample

Just to exercise loops a bit more, let's compute a table of even
Bernoulli numbers.

@smallexample
@group
3:  []             1:  [0.10132, 0.03079, 0.02340, 0.033197, ...]
2:  2                  .
1:  30
    .

 [ ] 2 @key{RET} 30          Z ( X | 2 Z )
@end group
@end smallexample

@noindent
The vertical-bar @kbd{|} is the vector-concatenation command.  When
we execute it, the list we are building will be in stack level 2
(initially this is an empty list), and the next Bernoulli number
will be in level 1.  The effect is to append the Bernoulli number
onto the end of the list.  (To create a table of exact fractional
Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
sequence of keystrokes.)

With loops and conditionals, you can program essentially anything
in Calc.  One other command that makes looping easier is @kbd{Z /},
which takes a condition from the stack and breaks out of the enclosing
loop if the condition is true (non-zero).  You can use this to make
``while'' and ``until'' style loops.

If you make a mistake when entering a keyboard macro, you can edit
it using @kbd{Z E}.  First, you must attach it to a key with @kbd{Z K}.
One technique is to enter a throwaway dummy definition for the macro,
then enter the real one in the edit command.

@smallexample
@group
1:  3                   1:  3           Calc Macro Edit Mode.
    .                       .           Original keys: 1 <return> 2 +

                                        1                          ;; calc digits
                                        RET                        ;; calc-enter
                                        2                          ;; calc digits
                                        +                          ;; calc-plus

C-x ( 1 @key{RET} 2 + C-x )    Z K h @key{RET}      Z E h
@end group
@end smallexample

@noindent
A keyboard macro is stored as a pure keystroke sequence.  The
@file{edmacro} package (invoked by @kbd{Z E}) scans along the
macro and tries to decode it back into human-readable steps.
Descriptions of the keystrokes are given as comments, which begin with
@samp{;;}, and which are ignored when the edited macro is saved.
Spaces and line breaks are also ignored when the edited macro is saved.
To enter a space into the macro, type @code{SPC}.  All the special
characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
and @code{NUL} must be written in all uppercase, as must the prefixes
@code{C-} and @code{M-}.

Let's edit in a new definition, for computing harmonic numbers.
First, erase the four lines of the old definition.  Then, type
in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
to copy it from this page of the Info file; you can of course skip
typing the comments, which begin with @samp{;;}).

@smallexample
Z`                      ;; calc-kbd-push     (Save local values)
0                       ;; calc digits       (Push a zero onto the stack)
st                      ;; calc-store-into   (Store it in the following variable)
1                       ;; calc quick variable  (Quick variable q1)
1                       ;; calc digits       (Initial value for the loop) 
TAB                     ;; calc-roll-down    (Swap initial and final)
Z(                      ;; calc-kbd-for      (Begin the "for" loop)
&                       ;; calc-inv          (Take the reciprocal)
s+                      ;; calc-store-plus   (Add to the following variable)
1                       ;; calc quick variable  (Quick variable q1)
1                       ;; calc digits       (The loop step is 1)
Z)                      ;; calc-kbd-end-for  (End the "for" loop)
sr                      ;; calc-recall       (Recall the final accumulated value)
1                       ;; calc quick variable (Quick variable q1)
Z'                      ;; calc-kbd-pop      (Restore values)
@end smallexample

@noindent
Press @kbd{C-c C-c} to finish editing and return to the Calculator.

@smallexample
@group
1:  20         1:  3.597739
    .              .

    20             z h
@end group
@end smallexample

The @file{edmacro} package defines a handy @code{read-kbd-macro} command
which reads the current region of the current buffer as a sequence of
keystroke names, and defines that sequence on the @kbd{X} 
(and @kbd{C-x e}) key.  Because this is so useful, Calc puts this
command on the @kbd{C-x * m} key.  Try reading in this macro in the
following form:  Press @kbd{C-@@} (or @kbd{C-@key{SPC}}) at 
one end of the text below, then type @kbd{C-x * m} at the other.

@example
@group
Z ` 0 t 1
    1 TAB
    Z (  & s + 1  1 Z )
    r 1
Z '
@end group
@end example

(@bullet{}) @strong{Exercise 8.}  A general algorithm for solving
equations numerically is @dfn{Newton's Method}.  Given the equation
@expr{f(x) = 0} for any function @expr{f}, and an initial guess
@expr{x_0} which is reasonably close to the desired solution, apply
this formula over and over:

@ifinfo
@example
new_x = x - f(x)/f'(x)
@end example
@end ifinfo
@tex
\beforedisplay
$$ x_{\rm new} = x - {f(x) \over f'(x)} $$
\afterdisplay
@end tex

@noindent
where @expr{f'(x)} is the derivative of @expr{f}.  The @expr{x}
values will quickly converge to a solution, i.e., eventually
@texline @math{x_{\rm new}}
@infoline @expr{new_x} 
and @expr{x} will be equal to within the limits
of the current precision.  Write a program which takes a formula
involving the variable @expr{x}, and an initial guess @expr{x_0},
on the stack, and produces a value of @expr{x} for which the formula
is zero.  Use it to find a solution of 
@texline @math{\sin(\cos x) = 0.5}
@infoline @expr{sin(cos(x)) = 0.5}
near @expr{x = 4.5}.  (Use angles measured in radians.)  Note that
the built-in @w{@kbd{a R}} (@code{calc-find-root}) command uses Newton's
method when it is able.  @xref{Programming Answer 8, 8}. (@bullet{})

@cindex Digamma function
@cindex Gamma constant, Euler's
@cindex Euler's gamma constant
(@bullet{}) @strong{Exercise 9.}  The @dfn{digamma} function 
@texline @math{\psi(z) (``psi'')}
@infoline @expr{psi(z)}
is defined as the derivative of 
@texline @math{\ln \Gamma(z)}.
@infoline @expr{ln(gamma(z))}.  
For large values of @expr{z}, it can be approximated by the infinite sum

@ifinfo
@example
psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
@end example
@end ifinfo
@tex
\beforedisplay
$$ \psi(z) \approx \ln z - {1\over2z} -
   \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
$$
\afterdisplay
@end tex

@noindent
where 
@texline @math{\sum}
@infoline @expr{sum} 
represents the sum over @expr{n} from 1 to infinity
(or to some limit high enough to give the desired accuracy), and
the @code{bern} function produces (exact) Bernoulli numbers.
While this sum is not guaranteed to converge, in practice it is safe.
An interesting mathematical constant is Euler's gamma, which is equal
to about 0.5772.  One way to compute it is by the formula,
@texline @math{\gamma = -\psi(1)}.
@infoline @expr{gamma = -psi(1)}.  
Unfortunately, 1 isn't a large enough argument
for the above formula to work (5 is a much safer value for @expr{z}).
Fortunately, we can compute 
@texline @math{\psi(1)}
@infoline @expr{psi(1)} 
from 
@texline @math{\psi(5)}
@infoline @expr{psi(5)} 
using the recurrence 
@texline @math{\psi(z+1) = \psi(z) + {1 \over z}}.
@infoline @expr{psi(z+1) = psi(z) + 1/z}.  
Your task:  Develop a program to compute 
@texline @math{\psi(z)};
@infoline @expr{psi(z)}; 
it should ``pump up'' @expr{z}
if necessary to be greater than 5, then use the above summation
formula.  Use looping commands to compute the sum.  Use your function
to compute 
@texline @math{\gamma}
@infoline @expr{gamma} 
to twelve decimal places.  (Calc has a built-in command
for Euler's constant, @kbd{I P}, which you can use to check your answer.)
@xref{Programming Answer 9, 9}. (@bullet{})

@cindex Polynomial, list of coefficients
(@bullet{}) @strong{Exercise 10.}  Given a polynomial in @expr{x} and
a number @expr{m} on the stack, where the polynomial is of degree
@expr{m} or less (i.e., does not have any terms higher than @expr{x^m}),
write a program to convert the polynomial into a list-of-coefficients
notation.  For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}.  Also develop
a way to convert from this form back to the standard algebraic form.
@xref{Programming Answer 10, 10}. (@bullet{})

@cindex Recursion
(@bullet{}) @strong{Exercise 11.}  The @dfn{Stirling numbers of the
first kind} are defined by the recurrences,

@ifinfo
@example
s(n,n) = 1   for n >= 0,
s(n,0) = 0   for n > 0,
s(n+1,m) = s(n,m-1) - n s(n,m)   for n >= m >= 1.
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \eqalign{ s(n,n)   &= 1 \qquad \hbox{for } n \ge 0,  \cr
             s(n,0)   &= 0 \qquad \hbox{for } n > 0, \cr
             s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
                          \hbox{for } n \ge m \ge 1.}
$$
\afterdisplay
\vskip5pt
(These numbers are also sometimes written $\displaystyle{n \brack m}$.)
@end tex

This can be implemented using a @dfn{recursive} program in Calc; the
program must invoke itself in order to calculate the two righthand
terms in the general formula.  Since it always invokes itself with
``simpler'' arguments, it's easy to see that it must eventually finish
the computation.  Recursion is a little difficult with Emacs keyboard
macros since the macro is executed before its definition is complete.
So here's the recommended strategy:  Create a ``dummy macro'' and assign
it to a key with, e.g., @kbd{Z K s}.  Now enter the true definition,
using the @kbd{z s} command to call itself recursively, then assign it
to the same key with @kbd{Z K s}.  Now the @kbd{z s} command will run
the complete recursive program.  (Another way is to use @w{@kbd{Z E}}
or @kbd{C-x * m} (@code{read-kbd-macro}) to read the whole macro at once,
thus avoiding the ``training'' phase.)  The task:  Write a program
that computes Stirling numbers of the first kind, given @expr{n} and
@expr{m} on the stack.  Test it with @emph{small} inputs like
@expr{s(4,2)}.  (There is a built-in command for Stirling numbers,
@kbd{k s}, which you can use to check your answers.)
@xref{Programming Answer 11, 11}. (@bullet{})

The programming commands we've seen in this part of the tutorial
are low-level, general-purpose operations.  Often you will find
that a higher-level function, such as vector mapping or rewrite
rules, will do the job much more easily than a detailed, step-by-step
program can:

(@bullet{}) @strong{Exercise 12.}  Write another program for
computing Stirling numbers of the first kind, this time using
rewrite rules.  Once again, @expr{n} and @expr{m} should be taken
from the stack.  @xref{Programming Answer 12, 12}. (@bullet{})

@example

@end example
This ends the tutorial section of the Calc manual.  Now you know enough
about Calc to use it effectively for many kinds of calculations.  But
Calc has many features that were not even touched upon in this tutorial.
@c [not-split]
The rest of this manual tells the whole story.
@c [when-split]
@c Volume II of this manual, the @dfn{Calc Reference}, tells the whole story.

@page
@node Answers to Exercises,  , Programming Tutorial, Tutorial
@section Answers to Exercises

@noindent
This section includes answers to all the exercises in the Calc tutorial.

@menu
* RPN Answer 1::           1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
* RPN Answer 2::           2*4 + 7*9.5 + 5/4
* RPN Answer 3::           Operating on levels 2 and 3
* RPN Answer 4::           Joe's complex problems
* Algebraic Answer 1::     Simulating Q command
* Algebraic Answer 2::     Joe's algebraic woes
* Algebraic Answer 3::     1 / 0
* Modes Answer 1::         3#0.1 = 3#0.0222222?
* Modes Answer 2::         16#f.e8fe15
* Modes Answer 3::         Joe's rounding bug
* Modes Answer 4::         Why floating point?
* Arithmetic Answer 1::    Why the \ command?
* Arithmetic Answer 2::    Tripping up the B command
* Vector Answer 1::        Normalizing a vector
* Vector Answer 2::        Average position
* Matrix Answer 1::        Row and column sums
* Matrix Answer 2::        Symbolic system of equations
* Matrix Answer 3::        Over-determined system
* List Answer 1::          Powers of two
* List Answer 2::          Least-squares fit with matrices
* List Answer 3::          Geometric mean
* List Answer 4::          Divisor function
* List Answer 5::          Duplicate factors
* List Answer 6::          Triangular list
* List Answer 7::          Another triangular list
* List Answer 8::          Maximum of Bessel function
* List Answer 9::          Integers the hard way
* List Answer 10::         All elements equal
* List Answer 11::         Estimating pi with darts
* List Answer 12::         Estimating pi with matchsticks
* List Answer 13::         Hash codes
* List Answer 14::         Random walk
* Types Answer 1::         Square root of pi times rational
* Types Answer 2::         Infinities
* Types Answer 3::         What can "nan" be?
* Types Answer 4::         Abbey Road
* Types Answer 5::         Friday the 13th
* Types Answer 6::         Leap years
* Types Answer 7::         Erroneous donut
* Types Answer 8::         Dividing intervals
* Types Answer 9::         Squaring intervals
* Types Answer 10::        Fermat's primality test
* Types Answer 11::        pi * 10^7 seconds
* Types Answer 12::        Abbey Road on CD
* Types Answer 13::        Not quite pi * 10^7 seconds
* Types Answer 14::        Supercomputers and c
* Types Answer 15::        Sam the Slug
* Algebra Answer 1::       Squares and square roots
* Algebra Answer 2::       Building polynomial from roots
* Algebra Answer 3::       Integral of x sin(pi x)
* Algebra Answer 4::       Simpson's rule
* Rewrites Answer 1::      Multiplying by conjugate
* Rewrites Answer 2::      Alternative fib rule
* Rewrites Answer 3::      Rewriting opt(a) + opt(b) x
* Rewrites Answer 4::      Sequence of integers
* Rewrites Answer 5::      Number of terms in sum
* Rewrites Answer 6::      Truncated Taylor series
* Programming Answer 1::   Fresnel's C(x)
* Programming Answer 2::   Negate third stack element
* Programming Answer 3::   Compute sin(x) / x, etc.
* Programming Answer 4::   Average value of a list
* Programming Answer 5::   Continued fraction phi
* Programming Answer 6::   Matrix Fibonacci numbers
* Programming Answer 7::   Harmonic number greater than 4
* Programming Answer 8::   Newton's method
* Programming Answer 9::   Digamma function
* Programming Answer 10::  Unpacking a polynomial
* Programming Answer 11::  Recursive Stirling numbers
* Programming Answer 12::  Stirling numbers with rewrites
@end menu

@c The following kludgery prevents the individual answers from
@c being entered on the table of contents.
@tex
\global\let\oldwrite=\write
\gdef\skipwrite#1#2{\let\write=\oldwrite}
\global\let\oldchapternofonts=\chapternofonts
\gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
@end tex

@node RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to Exercises
@subsection RPN Tutorial Exercise 1

@noindent
@kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}

The result is 
@texline @math{1 - (2 \times (3 + 4)) = -13}.
@infoline @expr{1 - (2 * (3 + 4)) = -13}.

@node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
@subsection RPN Tutorial Exercise 2

@noindent
@texline @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
@infoline @expr{2*4 + 7*9.5 + 5/4 = 75.75}

After computing the intermediate term 
@texline @math{2\times4 = 8},
@infoline @expr{2*4 = 8}, 
you can leave that result on the stack while you compute the second
term.  With both of these results waiting on the stack you can then
compute the final term, then press @kbd{+ +} to add everything up.

@smallexample
@group
2:  2          1:  8          3:  8          2:  8
1:  4              .          2:  7          1:  66.5
    .                         1:  9.5            .
                                  .

  2 @key{RET} 4          *          7 @key{RET} 9.5          *

@end group
@end smallexample
@noindent
@smallexample
@group
4:  8          3:  8          2:  8          1:  75.75
3:  66.5       2:  66.5       1:  67.75          .
2:  5          1:  1.25           .
1:  4              .
    .

  5 @key{RET} 4          /              +              +
@end group
@end smallexample

Alternatively, you could add the first two terms before going on
with the third term.

@smallexample
@group
2:  8          1:  74.5       3:  74.5       2:  74.5       1:  75.75
1:  66.5           .          2:  5          1:  1.25           .
    .                         1:  4              .
                                  .

   ...             +            5 @key{RET} 4          /              +
@end group
@end smallexample

On an old-style RPN calculator this second method would have the
advantage of using only three stack levels.  But since Calc's stack
can grow arbitrarily large this isn't really an issue.  Which method
you choose is purely a matter of taste.

@node RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
@subsection RPN Tutorial Exercise 3

@noindent
The @key{TAB} key provides a way to operate on the number in level 2.

@smallexample
@group
3:  10         3:  10         4:  10         3:  10         3:  10
2:  20         2:  30         3:  30         2:  30         2:  21
1:  30         1:  20         2:  20         1:  21         1:  30
    .              .          1:  1              .              .
                                  .

                  @key{TAB}             1              +             @key{TAB}
@end group
@end smallexample

Similarly, @kbd{M-@key{TAB}} gives you access to the number in level 3.

@smallexample
@group
3:  10         3:  21         3:  21         3:  30         3:  11
2:  21         2:  30         2:  30         2:  11         2:  21
1:  30         1:  10         1:  11         1:  21         1:  30
    .              .              .              .              .

                  M-@key{TAB}           1 +           M-@key{TAB}          M-@key{TAB}
@end group
@end smallexample

@node RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to Exercises
@subsection RPN Tutorial Exercise 4

@noindent
Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
but using both the comma and the space at once yields:

@smallexample
@group
1:  ( ...      2:  ( ...      1:  (2, ...    2:  (2, ...    2:  (2, ...
    .          1:  2              .          1:  (2, ...    1:  (2, 3)
                   .                             .              .

    (              2              ,             @key{SPC}            3 )
@end group
@end smallexample

Joe probably tried to type @kbd{@key{TAB} @key{DEL}} to swap the
extra incomplete object to the top of the stack and delete it.
But a feature of Calc is that @key{DEL} on an incomplete object
deletes just one component out of that object, so he had to press
@key{DEL} twice to finish the job.

@smallexample
@group
2:  (2, ...    2:  (2, 3)     2:  (2, 3)     1:  (2, 3)
1:  (2, 3)     1:  (2, ...    1:  ( ...          .
    .              .              .

                  @key{TAB}            @key{DEL}            @key{DEL}
@end group
@end smallexample

(As it turns out, deleting the second-to-top stack entry happens often
enough that Calc provides a special key, @kbd{M-@key{DEL}}, to do just that.
@kbd{M-@key{DEL}} is just like @kbd{@key{TAB} @key{DEL}}, except that it doesn't exhibit
the ``feature'' that tripped poor Joe.)

@node Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to Exercises
@subsection Algebraic Entry Tutorial Exercise 1

@noindent
Type @kbd{' sqrt($) @key{RET}}.

If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
Or, RPN style, @kbd{0.5 ^}.

(Actually, @samp{$^1:2}, using the fraction one-half as the power, is
a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
@samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)

@node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises
@subsection Algebraic Entry Tutorial Exercise 2

@noindent
In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
name with @samp{1+y} as its argument.  Assigning a value to a variable
has no relation to a function by the same name.  Joe needed to use an
explicit @samp{*} symbol here:  @samp{2 x*(1+y)}.

@node Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers to Exercises
@subsection Algebraic Entry Tutorial Exercise 3

@noindent
The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
The ``function'' @samp{/} cannot be evaluated when its second argument
is zero, so it is left in symbolic form.  When you now type @kbd{0 *},
the result will be zero because Calc uses the general rule that ``zero
times anything is zero.''

@c [fix-ref Infinities]
The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
results in a special symbol that represents ``infinity.''  If you
multiply infinity by zero, Calc uses another special new symbol to
show that the answer is ``indeterminate.''  @xref{Infinities}, for
further discussion of infinite and indeterminate values.

@node Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to Exercises
@subsection Modes Tutorial Exercise 1

@noindent
Calc always stores its numbers in decimal, so even though one-third has
an exact base-3 representation (@samp{3#0.1}), it is still stored as
0.3333333 (chopped off after 12 or however many decimal digits) inside
the calculator's memory.  When this inexact number is converted back
to base 3 for display, it may still be slightly inexact.  When we
multiply this number by 3, we get 0.999999, also an inexact value.

When Calc displays a number in base 3, it has to decide how many digits
to show.  If the current precision is 12 (decimal) digits, that corresponds
to @samp{12 / log10(3) = 25.15} base-3 digits.  Because 25.15 is not an
exact integer, Calc shows only 25 digits, with the result that stored
numbers carry a little bit of extra information that may not show up on
the screen.  When Joe entered @samp{3#0.2}, the stored number 0.666666
happened to round to a pleasing value when it lost that last 0.15 of a
digit, but it was still inexact in Calc's memory.  When he divided by 2,
he still got the dreaded inexact value 0.333333.  (Actually, he divided
0.666667 by 2 to get 0.333334, which is why he got something a little
higher than @code{3#0.1} instead of a little lower.)

If Joe didn't want to be bothered with all this, he could have typed
@kbd{M-24 d n} to display with one less digit than the default.  (If
you give @kbd{d n} a negative argument, it uses default-minus-that,
so @kbd{M-- d n} would be an easier way to get the same effect.)  Those
inexact results would still be lurking there, but they would now be
rounded to nice, natural-looking values for display purposes.  (Remember,
@samp{0.022222} in base 3 is like @samp{0.099999} in base 10; rounding
off one digit will round the number up to @samp{0.1}.)  Depending on the
nature of your work, this hiding of the inexactness may be a benefit or
a danger.  With the @kbd{d n} command, Calc gives you the choice.

Incidentally, another consequence of all this is that if you type
@kbd{M-30 d n} to display more digits than are ``really there,''
you'll see garbage digits at the end of the number.  (In decimal
display mode, with decimally-stored numbers, these garbage digits are
always zero so they vanish and you don't notice them.)  Because Calc
rounds off that 0.15 digit, there is the danger that two numbers could
be slightly different internally but still look the same.  If you feel
uneasy about this, set the @kbd{d n} precision to be a little higher
than normal; you'll get ugly garbage digits, but you'll always be able
to tell two distinct numbers apart.

An interesting side note is that most computers store their
floating-point numbers in binary, and convert to decimal for display.
Thus everyday programs have the same problem:  Decimal 0.1 cannot be
represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
comes out as an inexact approximation to 1 on some machines (though
they generally arrange to hide it from you by rounding off one digit as
we did above).  Because Calc works in decimal instead of binary, you can
be sure that numbers that look exact @emph{are} exact as long as you stay
in decimal display mode.

It's not hard to show that any number that can be represented exactly
in binary, octal, or hexadecimal is also exact in decimal, so the kinds
of problems we saw in this exercise are likely to be severe only when
you use a relatively unusual radix like 3.

@node Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to Exercises
@subsection Modes Tutorial Exercise 2

If the radix is 15 or higher, we can't use the letter @samp{e} to mark
the exponent because @samp{e} is interpreted as a digit.  When Calc
needs to display scientific notation in a high radix, it writes
@samp{16#F.E8F*16.^15}.  You can enter a number like this as an
algebraic entry.  Also, pressing @kbd{e} without any digits before it
normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
puts you in algebraic entry:  @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is another
way to enter this number.

The reason Calc puts a decimal point in the @samp{16.^} is to prevent
huge integers from being generated if the exponent is large (consider
@samp{16#1.23*16^1000}, where we compute @samp{16^1000} as a giant
exact integer and then throw away most of the digits when we multiply
it by the floating-point @samp{16#1.23}).  While this wouldn't normally
matter for display purposes, it could give you a nasty surprise if you
copied that number into a file and later moved it back into Calc.

@node Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to Exercises
@subsection Modes Tutorial Exercise 3

@noindent
The answer he got was @expr{0.5000000000006399}.

The problem is not that the square operation is inexact, but that the
sine of 45 that was already on the stack was accurate to only 12 places.
Arbitrary-precision calculations still only give answers as good as
their inputs.

The real problem is that there is no 12-digit number which, when
squared, comes out to 0.5 exactly.  The @kbd{f [} and @kbd{f ]}
commands decrease or increase a number by one unit in the last
place (according to the current precision).  They are useful for
determining facts like this.

@smallexample
@group
1:  0.707106781187      1:  0.500000000001
    .                       .

    45 S                    2 ^

@end group
@end smallexample
@noindent
@smallexample
@group
1:  0.707106781187      1:  0.707106781186      1:  0.499999999999
    .                       .                       .

    U  @key{DEL}                  f [                     2 ^
@end group
@end smallexample

A high-precision calculation must be carried out in high precision
all the way.  The only number in the original problem which was known
exactly was the quantity 45 degrees, so the precision must be raised
before anything is done after the number 45 has been entered in order
for the higher precision to be meaningful.

@node Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to Exercises
@subsection Modes Tutorial Exercise 4

@noindent
Many calculations involve real-world quantities, like the width and
height of a piece of wood or the volume of a jar.  Such quantities
can't be measured exactly anyway, and if the data that is input to
a calculation is inexact, doing exact arithmetic on it is a waste
of time.

Fractions become unwieldy after too many calculations have been
done with them.  For example, the sum of the reciprocals of the
integers from 1 to 10 is 7381:2520.  The sum from 1 to 30 is
9304682830147:2329089562800.  After a point it will take a long
time to add even one more term to this sum, but a floating-point
calculation of the sum will not have this problem.

Also, rational numbers cannot express the results of all calculations.
There is no fractional form for the square root of two, so if you type
@w{@kbd{2 Q}}, Calc has no choice but to give you a floating-point answer.

@node Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4, Answers to Exercises
@subsection Arithmetic Tutorial Exercise 1

@noindent
Dividing two integers that are larger than the current precision may
give a floating-point result that is inaccurate even when rounded
down to an integer.  Consider @expr{123456789 / 2} when the current
precision is 6 digits.  The true answer is @expr{61728394.5}, but
with a precision of 6 this will be rounded to 
@texline @math{12345700.0/2.0 = 61728500.0}.
@infoline @expr{12345700.@: / 2.@: = 61728500.}.
The result, when converted to an integer, will be off by 106.

Here are two solutions:  Raise the precision enough that the
floating-point round-off error is strictly to the right of the
decimal point.  Or, convert to Fraction mode so that @expr{123456789 / 2}
produces the exact fraction @expr{123456789:2}, which can be rounded
down by the @kbd{F} command without ever switching to floating-point
format.

@node Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1, Answers to Exercises
@subsection Arithmetic Tutorial Exercise 2

@noindent
@kbd{27 @key{RET} 9 B} could give the exact result @expr{3:2}, but it
does a floating-point calculation instead and produces @expr{1.5}.

Calc will find an exact result for a logarithm if the result is an integer
or (when in Fraction mode) the reciprocal of an integer.  But there is
no efficient way to search the space of all possible rational numbers
for an exact answer, so Calc doesn't try.

@node Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers to Exercises
@subsection Vector Tutorial Exercise 1

@noindent
Duplicate the vector, compute its length, then divide the vector
by its length:  @kbd{@key{RET} A /}.

@smallexample
@group
1:  [1, 2, 3]  2:  [1, 2, 3]      1:  [0.27, 0.53, 0.80]  1:  1.
    .          1:  3.74165738677      .                       .
                   .

    r 1            @key{RET} A              /                       A
@end group
@end smallexample

The final @kbd{A} command shows that the normalized vector does
indeed have unit length.

@node Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to Exercises
@subsection Vector Tutorial Exercise 2

@noindent
The average position is equal to the sum of the products of the
positions times their corresponding probabilities.  This is the
definition of the dot product operation.  So all you need to do
is to put the two vectors on the stack and press @kbd{*}.

@node Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to Exercises
@subsection Matrix Tutorial Exercise 1

@noindent
The trick is to multiply by a vector of ones.  Use @kbd{r 4 [1 1 1] *} to
get the row sum.  Similarly, use @kbd{[1 1] r 4 *} to get the column sum.

@node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
@subsection Matrix Tutorial Exercise 2

@ifinfo
@example
@group
   x + a y = 6
   x + b y = 10
@end group
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \eqalign{ x &+ a y = 6 \cr
             x &+ b y = 10}
$$
\afterdisplay
@end tex

Just enter the righthand side vector, then divide by the lefthand side
matrix as usual.

@smallexample
@group
1:  [6, 10]    2:  [6, 10]         1:  [6 - 4 a / (b - a), 4 / (b - a) ]
    .          1:  [ [ 1, a ]          .
                     [ 1, b ] ]
                   .

' [6 10] @key{RET}     ' [1 a; 1 b] @key{RET}      /
@end group
@end smallexample

This can be made more readable using @kbd{d B} to enable Big display
mode:

@smallexample
@group
          4 a     4
1:  [6 - -----, -----]
         b - a  b - a
@end group
@end smallexample

Type @kbd{d N} to return to Normal display mode afterwards.

@node Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to Exercises
@subsection Matrix Tutorial Exercise 3

@noindent
To solve 
@texline @math{A^T A \, X = A^T B},
@infoline @expr{trn(A) * A * X = trn(A) * B}, 
first we compute
@texline @math{A' = A^T A}
@infoline @expr{A2 = trn(A) * A} 
and 
@texline @math{B' = A^T B};
@infoline @expr{B2 = trn(A) * B}; 
now, we have a system 
@texline @math{A' X = B'}
@infoline @expr{A2 * X = B2} 
which we can solve using Calc's @samp{/} command.

@ifinfo
@example
@group
    a + 2b + 3c = 6
   4a + 5b + 6c = 2
   7a + 6b      = 3
   2a + 4b + 6c = 11
@end group
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
   $\hfil#$&$\hfil{}#{}$&
   $\hfil#$&$\hfil{}#{}$&
   $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
  a&+&2b&+&3c&=6 \cr
 4a&+&5b&+&6c&=2 \cr
 7a&+&6b& &  &=3 \cr
 2a&+&4b&+&6c&=11 \cr}
$$
\afterdisplayh
@end tex

The first step is to enter the coefficient matrix.  We'll store it in
quick variable number 7 for later reference.  Next, we compute the
@texline @math{B'}
@infoline @expr{B2} 
vector.

@smallexample
@group
1:  [ [ 1, 2, 3 ]             2:  [ [ 1, 4, 7, 2 ]     1:  [57, 84, 96]
      [ 4, 5, 6 ]                   [ 2, 5, 6, 4 ]         .
      [ 7, 6, 0 ]                   [ 3, 6, 0, 6 ] ]
      [ 2, 4, 6 ] ]           1:  [6, 2, 3, 11]
    .                             .

' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET}  s 7  v t  [6 2 3 11]   *
@end group
@end smallexample

@noindent
Now we compute the matrix 
@texline @math{A'}
@infoline @expr{A2} 
and divide.

@smallexample
@group
2:  [57, 84, 96]          1:  [-11.64, 14.08, -3.64]
1:  [ [ 70, 72, 39 ]          .
      [ 72, 81, 60 ]
      [ 39, 60, 81 ] ]
    .

    r 7 v t r 7 *             /
@end group
@end smallexample

@noindent
(The actual computed answer will be slightly inexact due to
round-off error.)

Notice that the answers are similar to those for the 
@texline @math{3\times3}
@infoline 3x3
system solved in the text.  That's because the fourth equation that was 
added to the system is almost identical to the first one multiplied
by two.  (If it were identical, we would have gotten the exact same
answer since the 
@texline @math{4\times3}
@infoline 4x3
system would be equivalent to the original 
@texline @math{3\times3}
@infoline 3x3
system.)

Since the first and fourth equations aren't quite equivalent, they
can't both be satisfied at once.  Let's plug our answers back into
the original system of equations to see how well they match.

@smallexample
@group
2:  [-11.64, 14.08, -3.64]     1:  [5.6, 2., 3., 11.2]
1:  [ [ 1, 2, 3 ]                  .
      [ 4, 5, 6 ]
      [ 7, 6, 0 ]
      [ 2, 4, 6 ] ]
    .

    r 7                            @key{TAB} *
@end group
@end smallexample

@noindent
This is reasonably close to our original @expr{B} vector,
@expr{[6, 2, 3, 11]}.

@node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises
@subsection List Tutorial Exercise 1

@noindent
We can use @kbd{v x} to build a vector of integers.  This needs to be
adjusted to get the range of integers we desire.  Mapping @samp{-}
across the vector will accomplish this, although it turns out the
plain @samp{-} key will work just as well.

@smallexample
@group
2:  2                              2:  2
1:  [1, 2, 3, 4, 5, 6, 7, 8, 9]    1:  [-4, -3, -2, -1, 0, 1, 2, 3, 4]
    .                                  .

    2  v x 9 @key{RET}                       5 V M -   or   5 -
@end group
@end smallexample

@noindent
Now we use @kbd{V M ^} to map the exponentiation operator across the
vector.

@smallexample
@group
1:  [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
    .

    V M ^
@end group
@end smallexample

@node List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
@subsection List Tutorial Exercise 2

@noindent
Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
the first job is to form the matrix that describes the problem.

@ifinfo
@example
   m*x + b*1 = y
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ m \times x + b \times 1 = y $$
\afterdisplay
@end tex

Thus we want a 
@texline @math{19\times2}
@infoline 19x2
matrix with our @expr{x} vector as one column and
ones as the other column.  So, first we build the column of ones, then
we combine the two columns to form our @expr{A} matrix.

@smallexample
@group
2:  [1.34, 1.41, 1.49, ... ]    1:  [ [ 1.34, 1 ]
1:  [1, 1, 1, ...]                    [ 1.41, 1 ]
    .                                 [ 1.49, 1 ]
                                      @dots{}

    r 1 1 v b 19 @key{RET}                M-2 v p v t   s 3
@end group
@end smallexample

@noindent
Now we compute 
@texline @math{A^T y}
@infoline @expr{trn(A) * y} 
and 
@texline @math{A^T A}
@infoline @expr{trn(A) * A} 
and divide.

@smallexample
@group
1:  [33.36554, 13.613]    2:  [33.36554, 13.613]
    .                     1:  [ [ 98.0003, 41.63 ]
                                [  41.63,   19   ] ]
                              .

 v t r 2 *                    r 3 v t r 3 *
@end group
@end smallexample

@noindent
(Hey, those numbers look familiar!)

@smallexample
@group
1:  [0.52141679, -0.425978]
    .

    /
@end group
@end smallexample

Since we were solving equations of the form 
@texline @math{m \times x + b \times 1 = y},
@infoline @expr{m*x + b*1 = y}, 
these numbers should be @expr{m} and @expr{b}, respectively.  Sure
enough, they agree exactly with the result computed using @kbd{V M} and
@kbd{V R}!

The moral of this story:  @kbd{V M} and @kbd{V R} will probably solve
your problem, but there is often an easier way using the higher-level
arithmetic functions!

@c [fix-ref Curve Fitting]
In fact, there is a built-in @kbd{a F} command that does least-squares
fits.  @xref{Curve Fitting}.

@node List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
@subsection List Tutorial Exercise 3

@noindent
Move to one end of the list and press @kbd{C-@@} (or @kbd{C-@key{SPC}} or
whatever) to set the mark, then move to the other end of the list
and type @w{@kbd{C-x * g}}.

@smallexample
@group
1:  [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
    .
@end group
@end smallexample

To make things interesting, let's assume we don't know at a glance
how many numbers are in this list.  Then we could type:

@smallexample
@group
2:  [2.3, 6, 22, ... ]     2:  [2.3, 6, 22, ... ]
1:  [2.3, 6, 22, ... ]     1:  126356422.5
    .                          .

    @key{RET}                        V R *

@end group
@end smallexample
@noindent
@smallexample
@group
2:  126356422.5            2:  126356422.5     1:  7.94652913734
1:  [2.3, 6, 22, ... ]     1:  9                   .
    .                          .

    @key{TAB}                        v l                 I ^
@end group
@end smallexample

@noindent
(The @kbd{I ^} command computes the @var{n}th root of a number.
You could also type @kbd{& ^} to take the reciprocal of 9 and
then raise the number to that power.)

@node List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
@subsection List Tutorial Exercise 4

@noindent
A number @expr{j} is a divisor of @expr{n} if 
@texline @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
@infoline @samp{n % j = 0}.  
The first step is to get a vector that identifies the divisors.

@smallexample
@group
2:  30                  2:  [0, 0, 0, 2, ...]    1:  [1, 1, 1, 0, ...]
1:  [1, 2, 3, 4, ...]   1:  0                        .
    .                       .

 30 @key{RET} v x 30 @key{RET}   s 1    V M %  0                 V M a =  s 2
@end group
@end smallexample

@noindent
This vector has 1's marking divisors of 30 and 0's marking non-divisors.

The zeroth divisor function is just the total number of divisors.
The first divisor function is the sum of the divisors.

@smallexample
@group
1:  8      3:  8                    2:  8                    2:  8
           2:  [1, 2, 3, 4, ...]    1:  [1, 2, 3, 0, ...]    1:  72
           1:  [1, 1, 1, 0, ...]        .                        .
               .

   V R +       r 1 r 2                  V M *                  V R +
@end group
@end smallexample

@noindent
Once again, the last two steps just compute a dot product for which
a simple @kbd{*} would have worked equally well.

@node List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
@subsection List Tutorial Exercise 5

@noindent
The obvious first step is to obtain the list of factors with @kbd{k f}.
This list will always be in sorted order, so if there are duplicates
they will be right next to each other.  A suitable method is to compare
the list with a copy of itself shifted over by one.

@smallexample
@group
1:  [3, 7, 7, 7, 19]   2:  [3, 7, 7, 7, 19]     2:  [3, 7, 7, 7, 19, 0]
    .                  1:  [3, 7, 7, 7, 19, 0]  1:  [0, 3, 7, 7, 7, 19]
                           .                        .

    19551 k f              @key{RET} 0 |                  @key{TAB} 0 @key{TAB} |

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [0, 0, 1, 1, 0, 0]   1:  2          1:  0
    .                        .              .

    V M a =                  V R +          0 a =
@end group
@end smallexample

@noindent
Note that we have to arrange for both vectors to have the same length
so that the mapping operation works; no prime factor will ever be
zero, so adding zeros on the left and right is safe.  From then on
the job is pretty straightforward.

Incidentally, Calc provides the 
@texline @dfn{M@"obius} @math{\mu}
@infoline @dfn{Moebius mu} 
function which is zero if and only if its argument is square-free.  It
would be a much more convenient way to do the above test in practice.

@node List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
@subsection List Tutorial Exercise 6

@noindent
First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
to get a list of lists of integers!

@node List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
@subsection List Tutorial Exercise 7

@noindent
Here's one solution.  First, compute the triangular list from the previous
exercise and type @kbd{1 -} to subtract one from all the elements.

@smallexample
@group
1:  [ [0],
      [0, 1],
      [0, 1, 2],
      @dots{}

    1 -
@end group
@end smallexample

The numbers down the lefthand edge of the list we desire are called
the ``triangular numbers'' (now you know why!).  The @expr{n}th
triangular number is the sum of the integers from 1 to @expr{n}, and
can be computed directly by the formula 
@texline @math{n (n+1) \over 2}.
@infoline @expr{n * (n+1) / 2}.

@smallexample
@group
2:  [ [0], [0, 1], ... ]    2:  [ [0], [0, 1], ... ]
1:  [0, 1, 2, 3, 4, 5]      1:  [0, 1, 3, 6, 10, 15]
    .                           .

    v x 6 @key{RET} 1 -               V M ' $ ($+1)/2 @key{RET}
@end group
@end smallexample

@noindent
Adding this list to the above list of lists produces the desired
result:

@smallexample
@group
1:  [ [0],
      [1, 2],
      [3, 4, 5],
      [6, 7, 8, 9],
      [10, 11, 12, 13, 14],
      [15, 16, 17, 18, 19, 20] ]
      .

      V M +
@end group
@end smallexample

If we did not know the formula for triangular numbers, we could have
computed them using a @kbd{V U +} command.  We could also have
gotten them the hard way by mapping a reduction across the original
triangular list.

@smallexample
@group
2:  [ [0], [0, 1], ... ]    2:  [ [0], [0, 1], ... ]
1:  [ [0], [0, 1], ... ]    1:  [0, 1, 3, 6, 10, 15]
    .                           .

    @key{RET}                         V M V R +
@end group
@end smallexample

@noindent
(This means ``map a @kbd{V R +} command across the vector,'' and
since each element of the main vector is itself a small vector,
@kbd{V R +} computes the sum of its elements.)

@node List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
@subsection List Tutorial Exercise 8

@noindent
The first step is to build a list of values of @expr{x}.

@smallexample
@group
1:  [1, 2, 3, ..., 21]  1:  [0, 1, 2, ..., 20]  1:  [0, 0.25, 0.5, ..., 5]
    .                       .                       .

    v x 21 @key{RET}              1 -                     4 /  s 1
@end group
@end smallexample

Next, we compute the Bessel function values.

@smallexample
@group
1:  [0., 0.124, 0.242, ..., -0.328]
    .

    V M ' besJ(1,$) @key{RET}
@end group
@end smallexample

@noindent
(Another way to do this would be @kbd{1 @key{TAB} V M f j}.)

A way to isolate the maximum value is to compute the maximum using
@kbd{V R X}, then compare all the Bessel values with that maximum.

@smallexample
@group
2:  [0., 0.124, 0.242, ... ]   1:  [0, 0, 0, ... ]    2:  [0, 0, 0, ... ]
1:  0.5801562                      .                  1:  1
    .                                                     .

    @key{RET} V R X                      V M a =                @key{RET} V R +    @key{DEL}
@end group
@end smallexample

@noindent
It's a good idea to verify, as in the last step above, that only
one value is equal to the maximum.  (After all, a plot of 
@texline @math{\sin x}
@infoline @expr{sin(x)}
might have many points all equal to the maximum value, 1.)

The vector we have now has a single 1 in the position that indicates
the maximum value of @expr{x}.  Now it is a simple matter to convert
this back into the corresponding value itself.

@smallexample
@group
2:  [0, 0, 0, ... ]         1:  [0, 0., 0., ... ]    1:  1.75
1:  [0, 0.25, 0.5, ... ]        .                        .
    .

    r 1                         V M *                    V R +
@end group
@end smallexample

If @kbd{a =} had produced more than one @expr{1} value, this method
would have given the sum of all maximum @expr{x} values; not very
useful!  In this case we could have used @kbd{v m} (@code{calc-mask-vector})
instead.  This command deletes all elements of a ``data'' vector that
correspond to zeros in a ``mask'' vector, leaving us with, in this
example, a vector of maximum @expr{x} values.

The built-in @kbd{a X} command maximizes a function using more
efficient methods.  Just for illustration, let's use @kbd{a X}
to maximize @samp{besJ(1,x)} over this same interval.

@smallexample
@group
2:  besJ(1, x)                 1:  [1.84115, 0.581865]
1:  [0 .. 5]                       .
    .

' besJ(1,x), [0..5] @key{RET}            a X x @key{RET}
@end group
@end smallexample

@noindent
The output from @kbd{a X} is a vector containing the value of @expr{x}
that maximizes the function, and the function's value at that maximum.
As you can see, our simple search got quite close to the right answer.

@node List Answer 9, List Answer 10, List Answer 8, Answers to Exercises
@subsection List Tutorial Exercise 9

@noindent
Step one is to convert our integer into vector notation.

@smallexample
@group
1:  25129925999           3:  25129925999
    .                     2:  10
                          1:  [11, 10, 9, ..., 1, 0]
                              .

    25129925999 @key{RET}           10 @key{RET} 12 @key{RET} v x 12 @key{RET} -

@end group
@end smallexample
@noindent
@smallexample
@group
1:  25129925999              1:  [0, 2, 25, 251, 2512, ... ]
2:  [100000000000, ... ]         .
    .

    V M ^   s 1                  V M \
@end group
@end smallexample

@noindent
(Recall, the @kbd{\} command computes an integer quotient.)

@smallexample
@group
1:  [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
    .

    10 V M %   s 2
@end group
@end smallexample

Next we must increment this number.  This involves adding one to
the last digit, plus handling carries.  There is a carry to the
left out of a digit if that digit is a nine and all the digits to
the right of it are nines.

@smallexample
@group
1:  [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1]   1:  [1, 1, 1, 0, 0, 1, ... ]
    .                                          .

    9 V M a =                                  v v

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [1, 1, 1, 0, 0, 0, ... ]   1:  [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
    .                              .

    V U *                          v v 1 |
@end group
@end smallexample

@noindent
Accumulating @kbd{*} across a vector of ones and zeros will preserve
only the initial run of ones.  These are the carries into all digits
except the rightmost digit.  Concatenating a one on the right takes
care of aligning the carries properly, and also adding one to the
rightmost digit.

@smallexample
@group
2:  [0, 0, 0, 0, ... ]     1:  [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
1:  [0, 0, 2, 5, ... ]         .
    .

    0 r 2 |                    V M +  10 V M %
@end group
@end smallexample

@noindent
Here we have concatenated 0 to the @emph{left} of the original number;
this takes care of shifting the carries by one with respect to the
digits that generated them.

Finally, we must convert this list back into an integer.

@smallexample
@group
3:  [0, 0, 2, 5, ... ]        2:  [0, 0, 2, 5, ... ]
2:  1000000000000             1:  [1000000000000, 100000000000, ... ]
1:  [100000000000, ... ]          .
    .

    10 @key{RET} 12 ^  r 1              |

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [0, 0, 20000000000, 5000000000, ... ]    1:  25129926000
    .                                            .

    V M *                                        V R +
@end group
@end smallexample

@noindent
Another way to do this final step would be to reduce the formula
@w{@samp{10 $$ + $}} across the vector of digits.

@smallexample
@group
1:  [0, 0, 2, 5, ... ]        1:  25129926000
    .                             .

                                  V R ' 10 $$ + $ @key{RET}
@end group
@end smallexample

@node List Answer 10, List Answer 11, List Answer 9, Answers to Exercises
@subsection List Tutorial Exercise 10

@noindent
For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
then compared with @expr{c} to produce another 1 or 0, which is then
compared with @expr{d}.  This is not at all what Joe wanted.

Here's a more correct method:

@smallexample
@group
1:  [7, 7, 7, 8, 7]      2:  [7, 7, 7, 8, 7]
    .                    1:  7
                             .

  ' [7,7,7,8,7] @key{RET}          @key{RET} v r 1 @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [1, 1, 1, 0, 1]      1:  0
    .                        .

    V M a =                  V R *
@end group
@end smallexample

@node List Answer 11, List Answer 12, List Answer 10, Answers to Exercises
@subsection List Tutorial Exercise 11

@noindent
The circle of unit radius consists of those points @expr{(x,y)} for which
@expr{x^2 + y^2 < 1}.  We start by generating a vector of @expr{x^2}
and a vector of @expr{y^2}.

We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
commands.

@smallexample
@group
2:  [2., 2., ..., 2.]          2:  [2., 2., ..., 2.]
1:  [2., 2., ..., 2.]          1:  [1.16, 1.98, ..., 0.81]
    .                              .

 v . t .  2. v b 100 @key{RET} @key{RET}       V M k r

@end group
@end smallexample
@noindent
@smallexample
@group
2:  [2., 2., ..., 2.]          1:  [0.026, 0.96, ..., 0.036]
1:  [0.026, 0.96, ..., 0.036]  2:  [0.53, 0.81, ..., 0.094]
    .                              .

    1 -  2 V M ^                   @key{TAB}  V M k r  1 -  2 V M ^
@end group
@end smallexample

Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
get a vector of 1/0 truth values, then sum the truth values.

@smallexample
@group
1:  [0.56, 1.78, ..., 0.13]    1:  [1, 0, ..., 1]    1:  84
    .                              .                     .

    +                              1 V M a <             V R +
@end group
@end smallexample

@noindent
The ratio @expr{84/100} should approximate the ratio @cpiover{4}.

@smallexample
@group
1:  0.84       1:  3.36       2:  3.36       1:  1.0695
    .              .          1:  3.14159        .

    100 /          4 *            P              /
@end group
@end smallexample

@noindent
Our estimate, 3.36, is off by about 7%.  We could get a better estimate
by taking more points (say, 1000), but it's clear that this method is
not very efficient!

(Naturally, since this example uses random numbers your own answer
will be slightly different from the one shown here!)

If you typed @kbd{v .} and @kbd{t .} before, type them again to
return to full-sized display of vectors.

@node List Answer 12, List Answer 13, List Answer 11, Answers to Exercises
@subsection List Tutorial Exercise 12

@noindent
This problem can be made a lot easier by taking advantage of some
symmetries.  First of all, after some thought it's clear that the
@expr{y} axis can be ignored altogether.  Just pick a random @expr{x}
component for one end of the match, pick a random direction 
@texline @math{\theta},
@infoline @expr{theta},
and see if @expr{x} and 
@texline @math{x + \cos \theta}
@infoline @expr{x + cos(theta)} 
(which is the @expr{x} coordinate of the other endpoint) cross a line.
The lines are at integer coordinates, so this happens when the two
numbers surround an integer.

Since the two endpoints are equivalent, we may as well choose the leftmost
of the two endpoints as @expr{x}.  Then @expr{theta} is an angle pointing
to the right, in the range -90 to 90 degrees.  (We could use radians, but
it would feel like cheating to refer to @cpiover{2} radians while trying
to estimate @cpi{}!)

In fact, since the field of lines is infinite we can choose the
coordinates 0 and 1 for the lines on either side of the leftmost
endpoint.  The rightmost endpoint will be between 0 and 1 if the
match does not cross a line, or between 1 and 2 if it does.  So:
Pick random @expr{x} and 
@texline @math{\theta},
@infoline @expr{theta}, 
compute
@texline @math{x + \cos \theta},
@infoline @expr{x + cos(theta)},
and count how many of the results are greater than one.  Simple!

We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
commands.

@smallexample
@group
1:  [0.52, 0.71, ..., 0.72]    2:  [0.52, 0.71, ..., 0.72]
    .                          1:  [78.4, 64.5, ..., -42.9]
                                   .

v . t . 1. v b 100 @key{RET}  V M k r    180. v b 100 @key{RET}  V M k r  90 -
@end group
@end smallexample

@noindent
(The next step may be slow, depending on the speed of your computer.)

@smallexample
@group
2:  [0.52, 0.71, ..., 0.72]    1:  [0.72, 1.14, ..., 1.45]
1:  [0.20, 0.43, ..., 0.73]        .
    .

    m d  V M C                     +

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [0, 1, ..., 1]       1:  0.64            1:  3.125
    .                        .                   .

    1 V M a >                V R + 100 /         2 @key{TAB} /
@end group
@end smallexample

Let's try the third method, too.  We'll use random integers up to
one million.  The @kbd{k r} command with an integer argument picks
a random integer.

@smallexample
@group
2:  [1000000, 1000000, ..., 1000000]   2:  [78489, 527587, ..., 814975]
1:  [1000000, 1000000, ..., 1000000]   1:  [324014, 358783, ..., 955450]
    .                                      .

    1000000 v b 100 @key{RET} @key{RET}                V M k r  @key{TAB}  V M k r

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [1, 1, ..., 25]      1:  [1, 1, ..., 0]     1:  0.56
    .                        .                      .

    V M k g                  1 V M a =              V R + 100 /

@end group
@end smallexample
@noindent
@smallexample
@group
1:  10.714        1:  3.273
    .                 .

    6 @key{TAB} /           Q
@end group
@end smallexample

For a proof of this property of the GCD function, see section 4.5.2,
exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.

If you typed @kbd{v .} and @kbd{t .} before, type them again to
return to full-sized display of vectors.

@node List Answer 13, List Answer 14, List Answer 12, Answers to Exercises
@subsection List Tutorial Exercise 13

@noindent
First, we put the string on the stack as a vector of ASCII codes.

@smallexample
@group
1:  [84, 101, 115, ..., 51]
    .

    "Testing, 1, 2, 3 @key{RET}
@end group
@end smallexample

@noindent
Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
there was no need to type an apostrophe.  Also, Calc didn't mind that
we omitted the closing @kbd{"}.  (The same goes for all closing delimiters
like @kbd{)} and @kbd{]} at the end of a formula.

We'll show two different approaches here.  In the first, we note that
if the input vector is @expr{[a, b, c, d]}, then the hash code is
@expr{3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d}.  In other words,
it's a sum of descending powers of three times the ASCII codes.

@smallexample
@group
2:  [84, 101, 115, ..., 51]    2:  [84, 101, 115, ..., 51]
1:  16                         1:  [15, 14, 13, ..., 0]
    .                              .

    @key{RET} v l                        v x 16 @key{RET} -

@end group
@end smallexample
@noindent
@smallexample
@group
2:  [84, 101, 115, ..., 51]    1:  1960915098    1:  121
1:  [14348907, ..., 1]             .                 .
    .

    3 @key{TAB} V M ^                    *                 511 %
@end group
@end smallexample

@noindent
Once again, @kbd{*} elegantly summarizes most of the computation.
But there's an even more elegant approach:  Reduce the formula
@kbd{3 $$ + $} across the vector.  Recall that this represents a
function of two arguments that computes its first argument times three
plus its second argument.

@smallexample
@group
1:  [84, 101, 115, ..., 51]    1:  1960915098
    .                              .

    "Testing, 1, 2, 3 @key{RET}          V R ' 3$$+$ @key{RET}
@end group
@end smallexample

@noindent
If you did the decimal arithmetic exercise, this will be familiar.
Basically, we're turning a base-3 vector of digits into an integer,
except that our ``digits'' are much larger than real digits.

Instead of typing @kbd{511 %} again to reduce the result, we can be
cleverer still and notice that rather than computing a huge integer
and taking the modulo at the end, we can take the modulo at each step
without affecting the result.  While this means there are more
arithmetic operations, the numbers we operate on remain small so
the operations are faster.

@smallexample
@group
1:  [84, 101, 115, ..., 51]    1:  121
    .                              .

    "Testing, 1, 2, 3 @key{RET}          V R ' (3$$+$)%511 @key{RET}
@end group
@end smallexample

Why does this work?  Think about a two-step computation:
@w{@expr{3 (3a + b) + c}}.  Taking a result modulo 511 basically means
subtracting off enough 511's to put the result in the desired range.
So the result when we take the modulo after every step is,

@ifinfo
@example
3 (3 a + b - 511 m) + c - 511 n
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ 3 (3 a + b - 511 m) + c - 511 n $$
\afterdisplay
@end tex

@noindent
for some suitable integers @expr{m} and @expr{n}.  Expanding out by
the distributive law yields

@ifinfo
@example
9 a + 3 b + c - 511*3 m - 511 n
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ 9 a + 3 b + c - 511\times3 m - 511 n $$
\afterdisplay
@end tex

@noindent
The @expr{m} term in the latter formula is redundant because any
contribution it makes could just as easily be made by the @expr{n}
term.  So we can take it out to get an equivalent formula with
@expr{n' = 3m + n},

@ifinfo
@example
9 a + 3 b + c - 511 n'
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ 9 a + 3 b + c - 511 n' $$
\afterdisplay
@end tex

@noindent
which is just the formula for taking the modulo only at the end of
the calculation.  Therefore the two methods are essentially the same.

Later in the tutorial we will encounter @dfn{modulo forms}, which
basically automate the idea of reducing every intermediate result
modulo some value @var{m}.

@node List Answer 14, Types Answer 1, List Answer 13, Answers to Exercises
@subsection List Tutorial Exercise 14

We want to use @kbd{H V U} to nest a function which adds a random
step to an @expr{(x,y)} coordinate.  The function is a bit long, but
otherwise the problem is quite straightforward.

@smallexample
@group
2:  [0, 0]     1:  [ [    0,       0    ]
1:  50               [  0.4288, -0.1695 ]
    .                [ -0.4787, -0.9027 ]
                     ...

    [0,0] 50       H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
@end group
@end smallexample

Just as the text recommended, we used @samp{< >} nameless function
notation to keep the two @code{random} calls from being evaluated
before nesting even begins.

We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
rules acts like a matrix.  We can transpose this matrix and unpack
to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.

@smallexample
@group
2:  [ 0, 0.4288, -0.4787, ... ]
1:  [ 0, -0.1696, -0.9027, ... ]
    .

    v t  v u  g f
@end group
@end smallexample

Incidentally, because the @expr{x} and @expr{y} are completely
independent in this case, we could have done two separate commands
to create our @expr{x} and @expr{y} vectors of numbers directly.

To make a random walk of unit steps, we note that @code{sincos} of
a random direction exactly gives us an @expr{[x, y]} step of unit
length; in fact, the new nesting function is even briefer, though
we might want to lower the precision a bit for it.

@smallexample
@group
2:  [0, 0]     1:  [ [    0,      0    ]
1:  50               [  0.1318, 0.9912 ]
    .                [ -0.5965, 0.3061 ]
                     ...

    [0,0] 50   m d  p 6 @key{RET}   H V U ' <# + sincos(random(360.0))> @key{RET}
@end group
@end smallexample

Another @kbd{v t v u g f} sequence will graph this new random walk.

An interesting twist on these random walk functions would be to use
complex numbers instead of 2-vectors to represent points on the plane.
In the first example, we'd use something like @samp{random + random*(0,1)},
and in the second we could use polar complex numbers with random phase
angles.  (This exercise was first suggested in this form by Randal
Schwartz.)

@node Types Answer 1, Types Answer 2, List Answer 14, Answers to Exercises
@subsection Types Tutorial Exercise 1

@noindent
If the number is the square root of @cpi{} times a rational number,
then its square, divided by @cpi{}, should be a rational number.

@smallexample
@group
1:  1.26508260337    1:  0.509433962268   1:  2486645810:4881193627
    .                    .                    .

                         2 ^ P /              c F
@end group
@end smallexample

@noindent
Technically speaking this is a rational number, but not one that is
likely to have arisen in the original problem.  More likely, it just
happens to be the fraction which most closely represents some
irrational number to within 12 digits.

But perhaps our result was not quite exact.  Let's reduce the
precision slightly and try again:

@smallexample
@group
1:  0.509433962268     1:  27:53
    .                      .

    U p 10 @key{RET}             c F
@end group
@end smallexample

@noindent
Aha!  It's unlikely that an irrational number would equal a fraction
this simple to within ten digits, so our original number was probably
@texline @math{\sqrt{27 \pi / 53}}.
@infoline @expr{sqrt(27 pi / 53)}.

Notice that we didn't need to re-round the number when we reduced the
precision.  Remember, arithmetic operations always round their inputs
to the current precision before they begin.

@node Types Answer 2, Types Answer 3, Types Answer 1, Answers to Exercises
@subsection Types Tutorial Exercise 2

@noindent
@samp{inf / inf = nan}.  Perhaps @samp{1} is the ``obvious'' answer.
But if @w{@samp{17 inf = inf}}, then @samp{17 inf / inf = inf / inf = 17}, too.

@samp{exp(inf) = inf}.  It's tempting to say that the exponential
of infinity must be ``bigger'' than ``regular'' infinity, but as
far as Calc is concerned all infinities are as just as big.
In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
to infinity, but the fact the @expr{e^x} grows much faster than
@expr{x} is not relevant here.

@samp{exp(-inf) = 0}.  Here we have a finite answer even though
the input is infinite.

@samp{sqrt(-inf) = (0, 1) inf}.  Remember that @expr{(0, 1)}
represents the imaginary number @expr{i}.  Here's a derivation:
@samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
The first part is, by definition, @expr{i}; the second is @code{inf}
because, once again, all infinities are the same size.

@samp{sqrt(uinf) = uinf}.  In fact, we do know something about the
direction because @code{sqrt} is defined to return a value in the
right half of the complex plane.  But Calc has no notation for this,
so it settles for the conservative answer @code{uinf}.

@samp{abs(uinf) = inf}.  No matter which direction @expr{x} points,
@samp{abs(x)} always points along the positive real axis.

@samp{ln(0) = -inf}.  Here we have an infinite answer to a finite
input.  As in the @expr{1 / 0} case, Calc will only use infinities
here if you have turned on Infinite mode.  Otherwise, it will
treat @samp{ln(0)} as an error.

@node Types Answer 3, Types Answer 4, Types Answer 2, Answers to Exercises
@subsection Types Tutorial Exercise 3

@noindent
We can make @samp{inf - inf} be any real number we like, say,
@expr{a}, just by claiming that we added @expr{a} to the first
infinity but not to the second.  This is just as true for complex
values of @expr{a}, so @code{nan} can stand for a complex number.
(And, similarly, @code{uinf} can stand for an infinity that points
in any direction in the complex plane, such as @samp{(0, 1) inf}).

In fact, we can multiply the first @code{inf} by two.  Surely
@w{@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf = nan}.
So @code{nan} can even stand for infinity.  Obviously it's just
as easy to make it stand for minus infinity as for plus infinity.

The moral of this story is that ``infinity'' is a slippery fish
indeed, and Calc tries to handle it by having a very simple model
for infinities (only the direction counts, not the ``size''); but
Calc is careful to write @code{nan} any time this simple model is
unable to tell what the true answer is.

@node Types Answer 4, Types Answer 5, Types Answer 3, Answers to Exercises
@subsection Types Tutorial Exercise 4

@smallexample
@group
2:  0@@ 47' 26"              1:  0@@ 2' 47.411765"
1:  17                          .
    .

    0@@ 47' 26" @key{RET} 17           /
@end group
@end smallexample

@noindent
The average song length is two minutes and 47.4 seconds.

@smallexample
@group
2:  0@@ 2' 47.411765"     1:  0@@ 3' 7.411765"    1:  0@@ 53' 6.000005"
1:  0@@ 0' 20"                .                      .
    .

    20"                      +                      17 *
@end group
@end smallexample

@noindent
The album would be 53 minutes and 6 seconds long.

@node Types Answer 5, Types Answer 6, Types Answer 4, Answers to Exercises
@subsection Types Tutorial Exercise 5

@noindent
Let's suppose it's January 14, 1991.  The easiest thing to do is
to keep trying 13ths of months until Calc reports a Friday.
We can do this by manually entering dates, or by using @kbd{t I}:

@smallexample
@group
1:  <Wed Feb 13, 1991>    1:  <Wed Mar 13, 1991>   1:  <Sat Apr 13, 1991>
    .                         .                        .

    ' <2/13> @key{RET}       @key{DEL}    ' <3/13> @key{RET}             t I
@end group
@end smallexample

@noindent
(Calc assumes the current year if you don't say otherwise.)

This is getting tedious---we can keep advancing the date by typing
@kbd{t I} over and over again, but let's automate the job by using
vector mapping.  The @kbd{t I} command actually takes a second
``how-many-months'' argument, which defaults to one.  This
argument is exactly what we want to map over:

@smallexample
@group
2:  <Sat Apr 13, 1991>     1:  [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
1:  [1, 2, 3, 4, 5, 6]          <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
    .                           <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
                               .

    v x 6 @key{RET}                  V M t I
@end group
@end smallexample

@noindent
Et voil@`a, September 13, 1991 is a Friday.

@smallexample
@group
1:  242
    .

' <sep 13> - <jan 14> @key{RET}
@end group
@end smallexample

@noindent
And the answer to our original question:  242 days to go.

@node Types Answer 6, Types Answer 7, Types Answer 5, Answers to Exercises
@subsection Types Tutorial Exercise 6

@noindent
The full rule for leap years is that they occur in every year divisible
by four, except that they don't occur in years divisible by 100, except
that they @emph{do} in years divisible by 400.  We could work out the
answer by carefully counting the years divisible by four and the
exceptions, but there is a much simpler way that works even if we
don't know the leap year rule.

Let's assume the present year is 1991.  Years have 365 days, except
that leap years (whenever they occur) have 366 days.  So let's count
the number of days between now and then, and compare that to the
number of years times 365.  The number of extra days we find must be
equal to the number of leap years there were.

@smallexample
@group
1:  <Mon Jan 1, 10001>     2:  <Mon Jan 1, 10001>     1:  2925593
    .                      1:  <Tue Jan 1, 1991>          .
                               .

  ' <jan 1 10001> @key{RET}         ' <jan 1 1991> @key{RET}          -

@end group
@end smallexample
@noindent
@smallexample
@group
3:  2925593       2:  2925593     2:  2925593     1:  1943
2:  10001         1:  8010        1:  2923650         .
1:  1991              .               .
    .

  10001 @key{RET} 1991      -               365 *           -
@end group
@end smallexample

@c [fix-ref Date Forms]
@noindent
There will be 1943 leap years before the year 10001.  (Assuming,
of course, that the algorithm for computing leap years remains
unchanged for that long.  @xref{Date Forms}, for some interesting
background information in that regard.)

@node Types Answer 7, Types Answer 8, Types Answer 6, Answers to Exercises
@subsection Types Tutorial Exercise 7

@noindent
The relative errors must be converted to absolute errors so that
@samp{+/-} notation may be used.

@smallexample
@group
1:  1.              2:  1.
    .               1:  0.2
                        .

    20 @key{RET} .05 *        4 @key{RET} .05 *
@end group
@end smallexample

Now we simply chug through the formula.

@smallexample
@group
1:  19.7392088022    1:  394.78 +/- 19.739    1:  6316.5 +/- 706.21
    .                    .                        .

    2 P 2 ^ *            20 p 1 *                 4 p .2 @key{RET} 2 ^ *
@end group
@end smallexample

It turns out the @kbd{v u} command will unpack an error form as
well as a vector.  This saves us some retyping of numbers.

@smallexample
@group
3:  6316.5 +/- 706.21     2:  6316.5 +/- 706.21
2:  6316.5                1:  0.1118
1:  706.21                    .
    .

    @key{RET} v u                   @key{TAB} /
@end group
@end smallexample

@noindent
Thus the volume is 6316 cubic centimeters, within about 11 percent.

@node Types Answer 8, Types Answer 9, Types Answer 7, Answers to Exercises
@subsection Types Tutorial Exercise 8

@noindent
The first answer is pretty simple:  @samp{1 / (0 .. 10) = (0.1 .. inf)}.
Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
close to zero, its reciprocal can get arbitrarily large, so the answer
is an interval that effectively means, ``any number greater than 0.1''
but with no upper bound.

The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.

Calc normally treats division by zero as an error, so that the formula
@w{@samp{1 / 0}} is left unsimplified.  Our third problem,
@w{@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because zero
is now a member of the interval.  So Calc leaves this one unevaluated, too.

If you turn on Infinite mode by pressing @kbd{m i}, you will
instead get the answer @samp{[0.1 .. inf]}, which includes infinity
as a possible value.

The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
Zero is buried inside the interval, but it's still a possible value.
It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
will be either greater than @mathit{0.1}, or less than @mathit{-0.1}.  Thus
the interval goes from minus infinity to plus infinity, with a ``hole''
in it from @mathit{-0.1} to @mathit{0.1}.  Calc doesn't have any way to
represent this, so it just reports @samp{[-inf .. inf]} as the answer.
It may be disappointing to hear ``the answer lies somewhere between
minus infinity and plus infinity, inclusive,'' but that's the best
that interval arithmetic can do in this case.

@node Types Answer 9, Types Answer 10, Types Answer 8, Answers to Exercises
@subsection Types Tutorial Exercise 9

@smallexample
@group
1:  [-3 .. 3]       2:  [-3 .. 3]     2:  [0 .. 9]
    .               1:  [0 .. 9]      1:  [-9 .. 9]
                        .                 .

    [ 3 n .. 3 ]        @key{RET} 2 ^           @key{TAB} @key{RET} *
@end group
@end smallexample

@noindent
In the first case the result says, ``if a number is between @mathit{-3} and
3, its square is between 0 and 9.''  The second case says, ``the product
of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''

An interval form is not a number; it is a symbol that can stand for
many different numbers.  Two identical-looking interval forms can stand
for different numbers.

The same issue arises when you try to square an error form.

@node Types Answer 10, Types Answer 11, Types Answer 9, Answers to Exercises
@subsection Types Tutorial Exercise 10

@noindent
Testing the first number, we might arbitrarily choose 17 for @expr{x}.

@smallexample
@group
1:  17 mod 811749613   2:  17 mod 811749613   1:  533694123 mod 811749613
    .                      811749612              .
                           .

    17 M 811749613 @key{RET}     811749612              ^
@end group
@end smallexample

@noindent
Since 533694123 is (considerably) different from 1, the number 811749613
must not be prime.

It's awkward to type the number in twice as we did above.  There are
various ways to avoid this, and algebraic entry is one.  In fact, using
a vector mapping operation we can perform several tests at once.  Let's
use this method to test the second number.

@smallexample
@group
2:  [17, 42, 100000]               1:  [1 mod 15485863, 1 mod ... ]
1:  15485863                           .
    .

 [17 42 100000] 15485863 @key{RET}           V M ' ($$ mod $)^($-1) @key{RET}
@end group
@end smallexample

@noindent
The result is three ones (modulo @expr{n}), so it's very probable that
15485863 is prime.  (In fact, this number is the millionth prime.)

Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
would have been hopelessly inefficient, since they would have calculated
the power using full integer arithmetic.

Calc has a @kbd{k p} command that does primality testing.  For small
numbers it does an exact test; for large numbers it uses a variant
of the Fermat test we used here.  You can use @kbd{k p} repeatedly
to prove that a large integer is prime with any desired probability.

@node Types Answer 11, Types Answer 12, Types Answer 10, Answers to Exercises
@subsection Types Tutorial Exercise 11

@noindent
There are several ways to insert a calculated number into an HMS form.
One way to convert a number of seconds to an HMS form is simply to
multiply the number by an HMS form representing one second:

@smallexample
@group
1:  31415926.5359     2:  31415926.5359     1:  8726@@ 38' 46.5359"
    .                 1:  0@@ 0' 1"              .
                          .

    P 1e7 *               0@@ 0' 1"              *

@end group
@end smallexample
@noindent
@smallexample
@group
2:  8726@@ 38' 46.5359"             1:  6@@ 6' 2.5359" mod 24@@ 0' 0"
1:  15@@ 27' 16" mod 24@@ 0' 0"          .
    .

    x time @key{RET}                         +
@end group
@end smallexample

@noindent
It will be just after six in the morning.

The algebraic @code{hms} function can also be used to build an
HMS form:

@smallexample
@group
1:  hms(0, 0, 10000000. pi)       1:  8726@@ 38' 46.5359"
    .                                 .

  ' hms(0, 0, 1e7 pi) @key{RET}             =
@end group
@end smallexample

@noindent
The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
the actual number 3.14159...

@node Types Answer 12, Types Answer 13, Types Answer 11, Answers to Exercises
@subsection Types Tutorial Exercise 12

@noindent
As we recall, there are 17 songs of about 2 minutes and 47 seconds
each.

@smallexample
@group
2:  0@@ 2' 47"                    1:  [0@@ 3' 7" .. 0@@ 3' 47"]
1:  [0@@ 0' 20" .. 0@@ 1' 0"]          .
    .

    [ 0@@ 20" .. 0@@ 1' ]              +

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [0@@ 52' 59." .. 1@@ 4' 19."]
    .

    17 *
@end group
@end smallexample

@noindent
No matter how long it is, the album will fit nicely on one CD.

@node Types Answer 13, Types Answer 14, Types Answer 12, Answers to Exercises
@subsection Types Tutorial Exercise 13

@noindent
Type @kbd{' 1 yr @key{RET} u c s @key{RET}}.  The answer is 31557600 seconds.

@node Types Answer 14, Types Answer 15, Types Answer 13, Answers to Exercises
@subsection Types Tutorial Exercise 14

@noindent
How long will it take for a signal to get from one end of the computer
to the other?

@smallexample
@group
1:  m / c         1:  3.3356 ns
    .                 .

 ' 1 m / c @key{RET}        u c ns @key{RET}
@end group
@end smallexample

@noindent
(Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)

@smallexample
@group
1:  3.3356 ns     1:  0.81356 ns / ns     1:  0.81356
2:  4.1 ns            .                       .
    .

  ' 4.1 ns @key{RET}        /                       u s
@end group
@end smallexample

@noindent
Thus a signal could take up to 81 percent of a clock cycle just to
go from one place to another inside the computer, assuming the signal
could actually attain the full speed of light.  Pretty tight!

@node Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to Exercises
@subsection Types Tutorial Exercise 15

@noindent
The speed limit is 55 miles per hour on most highways.  We want to
find the ratio of Sam's speed to the US speed limit.

@smallexample
@group
1:  55 mph         2:  55 mph           3:  11 hr mph / yd
    .              1:  5 yd / hr            .
                       .

  ' 55 mph @key{RET}       ' 5 yd/hr @key{RET}          /
@end group
@end smallexample

The @kbd{u s} command cancels out these units to get a plain
number.  Now we take the logarithm base two to find the final
answer, assuming that each successive pill doubles his speed.

@smallexample
@group
1:  19360.       2:  19360.       1:  14.24
    .            1:  2                .
                     .

    u s              2                B
@end group
@end smallexample

@noindent
Thus Sam can take up to 14 pills without a worry.

@node Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to Exercises
@subsection Algebra Tutorial Exercise 1

@noindent
@c [fix-ref Declarations]
The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
Calculator, but @samp{sqrt(x^2)} is not.  (Consider what happens
if @w{@expr{x = -4}}.)  If @expr{x} is real, this formula could be
simplified to @samp{abs(x)}, but for general complex arguments even
that is not safe.  (@xref{Declarations}, for a way to tell Calc
that @expr{x} is known to be real.)

@node Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to Exercises
@subsection Algebra Tutorial Exercise 2

@noindent
Suppose our roots are @expr{[a, b, c]}.  We want a polynomial which
is zero when @expr{x} is any of these values.  The trivial polynomial
@expr{x-a} is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
will do the job.  We can use @kbd{a c x} to write this in a more
familiar form.

@smallexample
@group
1:  34 x - 24 x^3          1:  [1.19023, -1.19023, 0]
    .                          .

    r 2                        a P x @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [x - 1.19023, x + 1.19023, x]     1:  (x - 1.19023) (x + 1.19023) x
    .                                     .

    V M ' x-$ @key{RET}                         V R *

@end group
@end smallexample
@noindent
@smallexample
@group
1:  x^3 - 1.41666 x        1:  34 x - 24 x^3
    .                          .

    a c x @key{RET}                  24 n *  a x
@end group
@end smallexample

@noindent
Sure enough, our answer (multiplied by a suitable constant) is the
same as the original polynomial.

@node Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to Exercises
@subsection Algebra Tutorial Exercise 3

@smallexample
@group
1:  x sin(pi x)         1:  (sin(pi x) - pi x cos(pi x)) / pi^2
    .                       .

  ' x sin(pi x) @key{RET}   m r   a i x @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [y, 1]
2:  (sin(pi x) - pi x cos(pi x)) / pi^2
    .

  ' [y,1] @key{RET} @key{TAB}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2]
    .

    V M $ @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
1:  (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2
    .

    V R -

@end group
@end smallexample
@noindent
@smallexample
@group
1:  (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183
    .

    =

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [0., -0.95493, 0.63662, -1.5915, 1.2732]
    .

    v x 5 @key{RET}  @key{TAB}  V M $ @key{RET}
@end group
@end smallexample

@node Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers to Exercises
@subsection Algebra Tutorial Exercise 4

@noindent
The hard part is that @kbd{V R +} is no longer sufficient to add up all
the contributions from the slices, since the slices have varying
coefficients.  So first we must come up with a vector of these
coefficients.  Here's one way:

@smallexample
@group
2:  -1                 2:  3                    1:  [4, 2, ..., 4]
1:  [1, 2, ..., 9]     1:  [-1, 1, ..., -1]         .
    .                      .

    1 n v x 9 @key{RET}          V M ^  3 @key{TAB}             -

@end group
@end smallexample
@noindent
@smallexample
@group
1:  [4, 2, ..., 4, 1]      1:  [1, 4, 2, ..., 4, 1]
    .                          .

    1 |                        1 @key{TAB} |
@end group
@end smallexample

@noindent
Now we compute the function values.  Note that for this method we need
eleven values, including both endpoints of the desired interval.

@smallexample
@group
2:  [1, 4, 2, ..., 4, 1]
1:  [1, 1.1, 1.2,  ...  , 1.8, 1.9, 2.]
    .

 11 @key{RET} 1 @key{RET} .1 @key{RET}  C-u v x

@end group
@end smallexample
@noindent
@smallexample
@group
2:  [1, 4, 2, ..., 4, 1]
1:  [0., 0.084941, 0.16993, ... ]
    .

    ' sin(x) ln(x) @key{RET}   m r  p 5 @key{RET}   V M $ @key{RET}
@end group
@end smallexample

@noindent
Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
same thing.

@smallexample
@group
1:  11.22      1:  1.122      1:  0.374
    .              .              .

    *              .1 *           3 /
@end group
@end smallexample

@noindent
Wow!  That's even better than the result from the Taylor series method.

@node Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers to Exercises
@subsection Rewrites Tutorial Exercise 1

@noindent
We'll use Big mode to make the formulas more readable.

@smallexample
@group
                                               ___
                                          2 + V 2
1:  (2 + sqrt(2)) / (1 + sqrt(2))     1:  --------
    .                                          ___
                                          1 + V 2

                                          .

  ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET}         d B
@end group
@end smallexample

@noindent
Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.

@smallexample
@group
          ___    ___
1:  (2 + V 2 ) (V 2  - 1)
    .

  a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
         ___                         ___
1:  2 + V 2  - 2                1:  V 2
    .                               .

  a r a*(b+c) := a*b + a*c          a s
@end group
@end smallexample

@noindent
(We could have used @kbd{a x} instead of a rewrite rule for the
second step.)

The multiply-by-conjugate rule turns out to be useful in many
different circumstances, such as when the denominator involves
sines and cosines or the imaginary constant @code{i}.

@node Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers to Exercises
@subsection Rewrites Tutorial Exercise 2

@noindent
Here is the rule set:

@smallexample
@group
[ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
  fib(1, x, y) := x,
  fib(n, x, y) := fib(n-1, y, x+y) ]
@end group
@end smallexample

@noindent
The first rule turns a one-argument @code{fib} that people like to write
into a three-argument @code{fib} that makes computation easier.  The
second rule converts back from three-argument form once the computation
is done.  The third rule does the computation itself.  It basically
says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
numbers.

Notice that because the number @expr{n} was ``validated'' by the
conditions on the first rule, there is no need to put conditions on
the other rules because the rule set would never get that far unless
the input were valid.  That further speeds computation, since no
extra conditions need to be checked at every step.

Actually, a user with a nasty sense of humor could enter a bad
three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
which would get the rules into an infinite loop.  One thing that would
help keep this from happening by accident would be to use something like
@samp{ZzFib} instead of @code{fib} as the name of the three-argument
function.

@node Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers to Exercises
@subsection Rewrites Tutorial Exercise 3

@noindent
He got an infinite loop.  First, Calc did as expected and rewrote
@w{@samp{2 + 3 x}} to @samp{f(2, 3, x)}.  Then it looked for ways to
apply the rule again, and found that @samp{f(2, 3, x)} looks like
@samp{a + b x} with @w{@samp{a = 0}} and @samp{b = 1}, so it rewrote to
@samp{f(0, 1, f(2, 3, x))}.  It then wrapped another @samp{f(0, 1, ...)}
around that, and so on, ad infinitum.  Joe should have used @kbd{M-1 a r}
to make sure the rule applied only once.

(Actually, even the first step didn't work as he expected.  What Calc
really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
to it.  While this may seem odd, it's just as valid a solution as the
``obvious'' one.  One way to fix this would be to add the condition
@samp{:: variable(x)} to the rule, to make sure the thing that matches
@samp{x} is indeed a variable, or to change @samp{x} to @samp{quote(x)}
on the lefthand side, so that the rule matches the actual variable
@samp{x} rather than letting @samp{x} stand for something else.)

@node Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers to Exercises
@subsection Rewrites Tutorial Exercise 4

@noindent
@ignore
@starindex
@end ignore
@tindex seq
Here is a suitable set of rules to solve the first part of the problem:

@smallexample
@group
[ seq(n, c) := seq(n/2,  c+1) :: n%2 = 0,
  seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
@end group
@end smallexample

Given the initial formula @samp{seq(6, 0)}, application of these
rules produces the following sequence of formulas:

@example
seq( 3, 1)
seq(10, 2)
seq( 5, 3)
seq(16, 4)
seq( 8, 5)
seq( 4, 6)
seq( 2, 7)
seq( 1, 8)
@end example

@noindent
whereupon neither of the rules match, and rewriting stops.

We can pretty this up a bit with a couple more rules:

@smallexample
@group
[ seq(n) := seq(n, 0),
  seq(1, c) := c,
  ... ]
@end group
@end smallexample

@noindent
Now, given @samp{seq(6)} as the starting configuration, we get 8
as the result.

The change to return a vector is quite simple:

@smallexample
@group
[ seq(n) := seq(n, []) :: integer(n) :: n > 0,
  seq(1, v) := v | 1,
  seq(n, v) := seq(n/2,  v | n) :: n%2 = 0,
  seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
@end group
@end smallexample

@noindent
Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.

Notice that the @expr{n > 1} guard is no longer necessary on the last
rule since the @expr{n = 1} case is now detected by another rule.
But a guard has been added to the initial rule to make sure the
initial value is suitable before the computation begins.

While still a good idea, this guard is not as vitally important as it
was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
will not get into an infinite loop.  Calc will not be able to prove
the symbol @samp{x} is either even or odd, so none of the rules will
apply and the rewrites will stop right away.

@node Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers to Exercises
@subsection Rewrites Tutorial Exercise 5

@noindent
@ignore
@starindex
@end ignore
@tindex nterms
If @expr{x} is the sum @expr{a + b}, then `@tfn{nterms(}@var{x}@tfn{)}' must
be `@tfn{nterms(}@var{a}@tfn{)}' plus `@tfn{nterms(}@var{b}@tfn{)}'.  If @expr{x}
is not a sum, then `@tfn{nterms(}@var{x}@tfn{)}' = 1.

@smallexample
@group
[ nterms(a + b) := nterms(a) + nterms(b),
  nterms(x)     := 1 ]
@end group
@end smallexample

@noindent
Here we have taken advantage of the fact that earlier rules always
match before later rules; @samp{nterms(x)} will only be tried if we
already know that @samp{x} is not a sum.

@node Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5, Answers to Exercises
@subsection Rewrites Tutorial Exercise 6

@noindent
Here is a rule set that will do the job:

@smallexample
@group
[ a*(b + c) := a*b + a*c,
  opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
     :: constant(a) :: constant(b),
  opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
     :: constant(a) :: constant(b),
  a O(x^n) := O(x^n) :: constant(a),
  x^opt(m) O(x^n) := O(x^(n+m)),
  O(x^n) O(x^m) := O(x^(n+m)) ]
@end group
@end smallexample

If we really want the @kbd{+} and @kbd{*} keys to operate naturally
on power series, we should put these rules in @code{EvalRules}.  For
testing purposes, it is better to put them in a different variable,
say, @code{O}, first.

The first rule just expands products of sums so that the rest of the
rules can assume they have an expanded-out polynomial to work with.
Note that this rule does not mention @samp{O} at all, so it will
apply to any product-of-sum it encounters---this rule may surprise
you if you put it into @code{EvalRules}!

In the second rule, the sum of two O's is changed to the smaller O.
The optional constant coefficients are there mostly so that
@samp{O(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
as well as @samp{O(x^2) + O(x^3)}.

The third rule absorbs higher powers of @samp{x} into O's.

The fourth rule says that a constant times a negligible quantity
is still negligible.  (This rule will also match @samp{O(x^3) / 4},
with @samp{a = 1/4}.)

The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
(It is easy to see that if one of these forms is negligible, the other
is, too.)  Notice the @samp{x^opt(m)} to pick up terms like
@w{@samp{x O(x^3)}}.  Optional powers will match @samp{x} as @samp{x^1}
but not 1 as @samp{x^0}.  This turns out to be exactly what we want here.

The sixth rule is the corresponding rule for products of two O's.

Another way to solve this problem would be to create a new ``data type''
that represents truncated power series.  We might represent these as
function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
on.  Rules would exist for sums and products of such @code{series}
objects, and as an optional convenience could also know how to combine a
@code{series} object with a normal polynomial.  (With this, and with a
rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
you could still enter power series in exactly the same notation as
before.)  Operations on such objects would probably be more efficient,
although the objects would be a bit harder to read.

@c [fix-ref Compositions]
Some other symbolic math programs provide a power series data type
similar to this.  Mathematica, for example, has an object that looks
like @samp{PowerSeries[@var{x}, @var{x0}, @var{coefs}, @var{nmin},
@var{nmax}, @var{den}]}, where @var{x0} is the point about which the
power series is taken (we've been assuming this was always zero),
and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
with fractional or negative powers.  Also, the @code{PowerSeries}
objects have a special display format that makes them look like
@samp{2 x^2 + O(x^4)} when they are printed out.  (@xref{Compositions},
for a way to do this in Calc, although for something as involved as
this it would probably be better to write the formatting routine
in Lisp.)

@node Programming Answer 1, Programming Answer 2, Rewrites Answer 6, Answers to Exercises
@subsection Programming Tutorial Exercise 1

@noindent
Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
@kbd{Z F}, and answer the questions.  Since this formula contains two
variables, the default argument list will be @samp{(t x)}.  We want to
change this to @samp{(x)} since @expr{t} is really a dummy variable
to be used within @code{ninteg}.

The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL} @key{DEL} @key{RET} y}.
(The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)

@node Programming Answer 2, Programming Answer 3, Programming Answer 1, Answers to Exercises
@subsection Programming Tutorial Exercise 2

@noindent
One way is to move the number to the top of the stack, operate on
it, then move it back:  @kbd{C-x ( M-@key{TAB} n M-@key{TAB} M-@key{TAB} C-x )}.

Another way is to negate the top three stack entries, then negate
again the top two stack entries:  @kbd{C-x ( M-3 n M-2 n C-x )}.

Finally, it turns out that a negative prefix argument causes a
command like @kbd{n} to operate on the specified stack entry only,
which is just what we want:  @kbd{C-x ( M-- 3 n C-x )}.

Just for kicks, let's also do it algebraically:
@w{@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.

@node Programming Answer 3, Programming Answer 4, Programming Answer 2, Answers to Exercises
@subsection Programming Tutorial Exercise 3

@noindent
Each of these functions can be computed using the stack, or using
algebraic entry, whichever way you prefer:

@noindent
Computing 
@texline @math{\displaystyle{\sin x \over x}}:
@infoline @expr{sin(x) / x}:

Using the stack:  @kbd{C-x (  @key{RET} S @key{TAB} /  C-x )}.

Using algebraic entry:  @kbd{C-x (  ' sin($)/$ @key{RET}  C-x )}.

@noindent
Computing the logarithm:

Using the stack:  @kbd{C-x (  @key{TAB} B  C-x )}

Using algebraic entry:  @kbd{C-x (  ' log($,$$) @key{RET}  C-x )}.

@noindent
Computing the vector of integers:

Using the stack:  @kbd{C-x (  1 @key{RET} 1  C-u v x  C-x )}.  (Recall that
@kbd{C-u v x} takes the vector size, starting value, and increment
from the stack.)

Alternatively:  @kbd{C-x (  ~ v x  C-x )}.  (The @kbd{~} key pops a
number from the stack and uses it as the prefix argument for the
next command.)

Using algebraic entry:  @kbd{C-x (  ' index($) @key{RET}  C-x )}.

@node Programming Answer 4, Programming Answer 5, Programming Answer 3, Answers to Exercises
@subsection Programming Tutorial Exercise 4

@noindent
Here's one way:  @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.

@node Programming Answer 5, Programming Answer 6, Programming Answer 4, Answers to Exercises
@subsection Programming Tutorial Exercise 5

@smallexample
@group
2:  1              1:  1.61803398502         2:  1.61803398502
1:  20                 .                     1:  1.61803398875
    .                                            .

   1 @key{RET} 20         Z < & 1 + Z >                I H P
@end group
@end smallexample

@noindent
This answer is quite accurate.

@node Programming Answer 6, Programming Answer 7, Programming Answer 5, Answers to Exercises
@subsection Programming Tutorial Exercise 6

@noindent
Here is the matrix:

@example
[ [ 0, 1 ]   * [a, b] = [b, a + b]
  [ 1, 1 ] ]
@end example

@noindent
Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
and @expr{n+2}.  Here's one program that does the job:

@example
C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
@end example

@noindent
This program is quite efficient because Calc knows how to raise a
matrix (or other value) to the power @expr{n} in only 
@texline @math{\log_2 n}
@infoline @expr{log(n,2)}
steps.  For example, this program can compute the 1000th Fibonacci
number (a 209-digit integer!) in about 10 steps; even though the
@kbd{Z < ... Z >} solution had much simpler steps, it would have
required so many steps that it would not have been practical.

@node Programming Answer 7, Programming Answer 8, Programming Answer 6, Answers to Exercises
@subsection Programming Tutorial Exercise 7

@noindent
The trick here is to compute the harmonic numbers differently, so that
the loop counter itself accumulates the sum of reciprocals.  We use
a separate variable to hold the integer counter.

@smallexample
@group
1:  1          2:  1       1:  .
    .          1:  4
                   .

    1 t 1       1 @key{RET} 4      Z ( t 2 r 1 1 + s 1 & Z )
@end group
@end smallexample

@noindent
The body of the loop goes as follows:  First save the harmonic sum
so far in variable 2.  Then delete it from the stack; the for loop
itself will take care of remembering it for us.  Next, recall the
count from variable 1, add one to it, and feed its reciprocal to
the for loop to use as the step value.  The for loop will increase
the ``loop counter'' by that amount and keep going until the
loop counter exceeds 4.

@smallexample
@group
2:  31                  3:  31
1:  3.99498713092       2:  3.99498713092
    .                   1:  4.02724519544
                            .

    r 1 r 2                 @key{RET} 31 & +
@end group
@end smallexample

Thus we find that the 30th harmonic number is 3.99, and the 31st
harmonic number is 4.02.

@node Programming Answer 8, Programming Answer 9, Programming Answer 7, Answers to Exercises
@subsection Programming Tutorial Exercise 8

@noindent
The first step is to compute the derivative @expr{f'(x)} and thus
the formula 
@texline @math{\displaystyle{x - {f(x) \over f'(x)}}}.
@infoline @expr{x - f(x)/f'(x)}.

(Because this definition is long, it will be repeated in concise form
below.  You can use @w{@kbd{C-x * m}} to load it from there.  While you are
entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
keystrokes without executing them.  In the following diagrams we'll
pretend Calc actually executed the keystrokes as you typed them,
just for purposes of illustration.)

@smallexample
@group
2:  sin(cos(x)) - 0.5            3:  4.5
1:  4.5                          2:  sin(cos(x)) - 0.5
    .                            1:  -(sin(x) cos(cos(x)))
                                     .

' sin(cos(x))-0.5 @key{RET} 4.5  m r  C-x ( Z `  @key{TAB} @key{RET} a d x @key{RET}

@end group
@end smallexample
@noindent
@smallexample
@group
2:  4.5
1:  x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
    .

    /  ' x @key{RET} @key{TAB} -   t 1
@end group
@end smallexample

Now, we enter the loop.  We'll use a repeat loop with a 20-repetition
limit just in case the method fails to converge for some reason.
(Normally, the @w{@kbd{Z /}} command will stop the loop before all 20
repetitions are done.)

@smallexample
@group
1:  4.5         3:  4.5                     2:  4.5
    .           2:  x + (sin(cos(x)) ...    1:  5.24196456928
                1:  4.5                         .
                    .

  20 Z <          @key{RET} r 1 @key{TAB}                 s l x @key{RET}
@end group
@end smallexample

This is the new guess for @expr{x}.  Now we compare it with the
old one to see if we've converged.

@smallexample
@group
3:  5.24196     2:  5.24196     1:  5.24196     1:  5.26345856348
2:  5.24196     1:  0               .               .
1:  4.5             .
    .

  @key{RET} M-@key{TAB}         a =             Z /             Z > Z ' C-x )
@end group
@end smallexample

The loop converges in just a few steps to this value.  To check
the result, we can simply substitute it back into the equation.

@smallexample
@group
2:  5.26345856348
1:  0.499999999997
    .

 @key{RET} ' sin(cos($)) @key{RET}
@end group
@end smallexample

Let's test the new definition again:

@smallexample
@group
2:  x^2 - 9           1:  3.
1:  1                     .
    .

  ' x^2-9 @key{RET} 1           X
@end group
@end smallexample

Once again, here's the full Newton's Method definition:

@example
@group
C-x ( Z `  @key{TAB} @key{RET} a d x @key{RET}  /  ' x @key{RET} @key{TAB} -  t 1
           20 Z <  @key{RET} r 1 @key{TAB}  s l x @key{RET}
                   @key{RET} M-@key{TAB}  a =  Z /
              Z >
      Z '
C-x )
@end group
@end example

@c [fix-ref Nesting and Fixed Points]
It turns out that Calc has a built-in command for applying a formula
repeatedly until it converges to a number.  @xref{Nesting and Fixed Points},
to see how to use it.

@c [fix-ref Root Finding]
Also, of course, @kbd{a R} is a built-in command that uses Newton's
method (among others) to look for numerical solutions to any equation.
@xref{Root Finding}.

@node Programming Answer 9, Programming Answer 10, Programming Answer 8, Answers to Exercises
@subsection Programming Tutorial Exercise 9

@noindent
The first step is to adjust @expr{z} to be greater than 5.  A simple
``for'' loop will do the job here.  If @expr{z} is less than 5, we
reduce the problem using 
@texline @math{\psi(z) = \psi(z+1) - 1/z}.
@infoline @expr{psi(z) = psi(z+1) - 1/z}.  We go
on to compute 
@texline @math{\psi(z+1)},
@infoline @expr{psi(z+1)}, 
and remember to add back a factor of @expr{-1/z} when we're done.  This
step is repeated until @expr{z > 5}.

(Because this definition is long, it will be repeated in concise form
below.  You can use @w{@kbd{C-x * m}} to load it from there.  While you are
entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
keystrokes without executing them.  In the following diagrams we'll
pretend Calc actually executed the keystrokes as you typed them,
just for purposes of illustration.)

@smallexample
@group
1:  1.             1:  1.
    .                  .

 1.0 @key{RET}       C-x ( Z `  s 1  0 t 2
@end group
@end smallexample

Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
factor.  If @expr{z < 5}, we use a loop to increase it.

(By the way, we started with @samp{1.0} instead of the integer 1 because
otherwise the calculation below will try to do exact fractional arithmetic,
and will never converge because fractions compare equal only if they
are exactly equal, not just equal to within the current precision.)

@smallexample
@group
3:  1.      2:  1.       1:  6.
2:  1.      1:  1            .
1:  5           .
    .

  @key{RET} 5        a <    Z [  5 Z (  & s + 2  1 s + 1  1 Z ) r 1  Z ]
@end group
@end smallexample

Now we compute the initial part of the sum:  
@texline @math{\ln z - {1 \over 2z}}
@infoline @expr{ln(z) - 1/2z}
minus the adjustment factor.

@smallexample
@group
2:  1.79175946923      2:  1.7084261359      1:  -0.57490719743
1:  0.0833333333333    1:  2.28333333333         .
    .                      .

    L  r 1 2 * &           -  r 2                -
@end group
@end smallexample

Now we evaluate the series.  We'll use another ``for'' loop counting
up the value of @expr{2 n}.  (Calc does have a summation command,
@kbd{a +}, but we'll use loops just to get more practice with them.)

@smallexample
@group
3:  -0.5749       3:  -0.5749        4:  -0.5749      2:  -0.5749
2:  2             2:  1:6            3:  1:6          1:  2.3148e-3
1:  40            1:  2              2:  2                .
    .                 .              1:  36.
                                         .

   2 @key{RET} 40        Z ( @key{RET} k b @key{TAB}     @key{RET} r 1 @key{TAB} ^      * /

@end group
@end smallexample
@noindent
@smallexample
@group
3:  -0.5749       3:  -0.5772      2:  -0.5772     1:  -0.577215664892
2:  -0.5749       2:  -0.5772      1:  0               .
1:  2.3148e-3     1:  -0.5749          .
    .                 .

  @key{TAB} @key{RET} M-@key{TAB}       - @key{RET} M-@key{TAB}      a =     Z /    2  Z )  Z ' C-x )
@end group
@end smallexample

This is the value of 
@texline @math{-\gamma},
@infoline @expr{- gamma}, 
with a slight bit of roundoff error.  To get a full 12 digits, let's use
a higher precision:

@smallexample
@group
2:  -0.577215664892      2:  -0.577215664892
1:  1.                   1:  -0.577215664901532

    1. @key{RET}                   p 16 @key{RET} X
@end group
@end smallexample

Here's the complete sequence of keystrokes:

@example
@group
C-x ( Z `  s 1  0 t 2
           @key{RET} 5 a <  Z [  5 Z (  & s + 2  1 s + 1  1 Z ) r 1  Z ]
           L r 1 2 * & - r 2 -
           2 @key{RET} 40  Z (  @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
                          @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z /
                  2  Z )
      Z '
C-x )
@end group
@end example

@node Programming Answer 10, Programming Answer 11, Programming Answer 9, Answers to Exercises
@subsection Programming Tutorial Exercise 10

@noindent
Taking the derivative of a term of the form @expr{x^n} will produce
a term like 
@texline @math{n x^{n-1}}.
@infoline @expr{n x^(n-1)}.  
Taking the derivative of a constant
produces zero.  From this it is easy to see that the @expr{n}th
derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
coefficient on the @expr{x^n} term times @expr{n!}.

(Because this definition is long, it will be repeated in concise form
below.  You can use @w{@kbd{C-x * m}} to load it from there.  While you are
entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
keystrokes without executing them.  In the following diagrams we'll
pretend Calc actually executed the keystrokes as you typed them,
just for purposes of illustration.)

@smallexample
@group
2:  5 x^4 + (x + 1)^2          3:  5 x^4 + (x + 1)^2
1:  6                          2:  0
    .                          1:  6
                                   .

  ' 5 x^4 + (x+1)^2 @key{RET} 6        C-x ( Z `  [ ] t 1  0 @key{TAB}
@end group
@end smallexample

@noindent
Variable 1 will accumulate the vector of coefficients.

@smallexample
@group
2:  0              3:  0                  2:  5 x^4 + ...
1:  5 x^4 + ...    2:  5 x^4 + ...        1:  1
    .              1:  1                      .
                       .

   Z ( @key{TAB}         @key{RET} 0 s l x @key{RET}            M-@key{TAB} ! /  s | 1
@end group
@end smallexample

@noindent
Note that @kbd{s | 1} appends the top-of-stack value to the vector
in a variable; it is completely analogous to @kbd{s + 1}.  We could
have written instead, @kbd{r 1 @key{TAB} | t 1}.

@smallexample
@group
1:  20 x^3 + 2 x + 2      1:  0         1:  [1, 2, 1, 0, 5, 0, 0]
    .                         .             .

    a d x @key{RET}                 1 Z )         @key{DEL} r 1  Z ' C-x )
@end group
@end smallexample

To convert back, a simple method is just to map the coefficients
against a table of powers of @expr{x}.

@smallexample
@group
2:  [1, 2, 1, 0, 5, 0, 0]    2:  [1, 2, 1, 0, 5, 0, 0]
1:  6                        1:  [0, 1, 2, 3, 4, 5, 6]
    .                            .

    6 @key{RET}                        1 + 0 @key{RET} 1 C-u v x

@end group
@end smallexample
@noindent
@smallexample
@group
2:  [1, 2, 1, 0, 5, 0, 0]    2:  1 + 2 x + x^2 + 5 x^4
1:  [1, x, x^2, x^3, ... ]       .
    .

    ' x @key{RET} @key{TAB} V M ^            *
@end group
@end smallexample

Once again, here are the whole polynomial to/from vector programs:

@example
@group
C-x ( Z `  [ ] t 1  0 @key{TAB}
           Z (  @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! /  s | 1
                a d x @key{RET}
         1 Z ) r 1
      Z '
C-x )

C-x (  1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ *  C-x )
@end group
@end example

@node Programming Answer 11, Programming Answer 12, Programming Answer 10, Answers to Exercises
@subsection Programming Tutorial Exercise 11

@noindent
First we define a dummy program to go on the @kbd{z s} key.  The true
@w{@kbd{z s}} key is supposed to take two numbers from the stack and
return one number, so @key{DEL} as a dummy definition will make
sure the stack comes out right.

@smallexample
@group
2:  4          1:  4                         2:  4
1:  2              .                         1:  2
    .                                            .

  4 @key{RET} 2       C-x ( @key{DEL} C-x )  Z K s @key{RET}       2
@end group
@end smallexample

The last step replaces the 2 that was eaten during the creation
of the dummy @kbd{z s} command.  Now we move on to the real
definition.  The recurrence needs to be rewritten slightly,
to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.

(Because this definition is long, it will be repeated in concise form
below.  You can use @kbd{C-x * m} to load it from there.)

@smallexample
@group
2:  4        4:  4       3:  4       2:  4
1:  2        3:  2       2:  2       1:  2
    .        2:  4       1:  0           .
             1:  2           .
                 .

  C-x (       M-2 @key{RET}        a =         Z [  @key{DEL} @key{DEL} 1  Z :

@end group
@end smallexample
@noindent
@smallexample
@group
4:  4       2:  4                     2:  3      4:  3    4:  3    3:  3
3:  2       1:  2                     1:  2      3:  2    3:  2    2:  2
2:  2           .                         .      2:  3    2:  3    1:  3
1:  0                                            1:  2    1:  1        .
    .                                                .        .

  @key{RET} 0   a = Z [  @key{DEL} @key{DEL} 0  Z :  @key{TAB} 1 - @key{TAB}   M-2 @key{RET}     1 -      z s
@end group
@end smallexample

@noindent
(Note that the value 3 that our dummy @kbd{z s} produces is not correct;
it is merely a placeholder that will do just as well for now.)

@smallexample
@group
3:  3               4:  3           3:  3       2:  3      1:  -6
2:  3               3:  3           2:  3       1:  9          .
1:  2               2:  3           1:  3           .
    .               1:  2               .
                        .

 M-@key{TAB} M-@key{TAB}     @key{TAB} @key{RET} M-@key{TAB}         z s          *          -

@end group
@end smallexample
@noindent
@smallexample
@group
1:  -6                          2:  4          1:  11      2:  11
    .                           1:  2              .       1:  11
                                    .                          .

  Z ] Z ] C-x )   Z K s @key{RET}      @key{DEL} 4 @key{RET} 2       z s      M-@key{RET} k s
@end group
@end smallexample

Even though the result that we got during the definition was highly
bogus, once the definition is complete the @kbd{z s} command gets
the right answers.

Here's the full program once again:

@example
@group
C-x (  M-2 @key{RET} a =
       Z [  @key{DEL} @key{DEL} 1
       Z :  @key{RET} 0 a =
            Z [  @key{DEL} @key{DEL} 0
            Z :  @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
                 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
            Z ]
       Z ]
C-x )
@end group
@end example

You can read this definition using @kbd{C-x * m} (@code{read-kbd-macro})
followed by @kbd{Z K s}, without having to make a dummy definition
first, because @code{read-kbd-macro} doesn't need to execute the
definition as it reads it in.  For this reason, @code{C-x * m} is often
the easiest way to create recursive programs in Calc.

@node Programming Answer 12,  , Programming Answer 11, Answers to Exercises
@subsection Programming Tutorial Exercise 12

@noindent
This turns out to be a much easier way to solve the problem.  Let's
denote Stirling numbers as calls of the function @samp{s}.

First, we store the rewrite rules corresponding to the definition of
Stirling numbers in a convenient variable:

@smallexample
s e StirlingRules @key{RET}
[ s(n,n) := 1  :: n >= 0,
  s(n,0) := 0  :: n > 0,
  s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
C-c C-c
@end smallexample

Now, it's just a matter of applying the rules:

@smallexample
@group
2:  4          1:  s(4, 2)              1:  11
1:  2              .                        .
    .

  4 @key{RET} 2       C-x (  ' s($$,$) @key{RET}     a r StirlingRules @key{RET}  C-x )
@end group
@end smallexample

As in the case of the @code{fib} rules, it would be useful to put these
rules in @code{EvalRules} and to add a @samp{:: remember} condition to
the last rule.

@c This ends the table-of-contents kludge from above:
@tex
\global\let\chapternofonts=\oldchapternofonts
@end tex

@c [reference]

@node Introduction, Data Types, Tutorial, Top
@chapter Introduction

@noindent
This chapter is the beginning of the Calc reference manual.
It covers basic concepts such as the stack, algebraic and
numeric entry, undo, numeric prefix arguments, etc.

@c [when-split]
@c (Chapter 2, the Tutorial, has been printed in a separate volume.)

@menu
* Basic Commands::
* Help Commands::
* Stack Basics::
* Numeric Entry::
* Algebraic Entry::
* Quick Calculator::
* Prefix Arguments::
* Undo::
* Error Messages::
* Multiple Calculators::
* Troubleshooting Commands::
@end menu

@node Basic Commands, Help Commands, Introduction, Introduction
@section Basic Commands

@noindent
@pindex calc
@pindex calc-mode
@cindex Starting the Calculator
@cindex Running the Calculator
To start the Calculator in its standard interface, type @kbd{M-x calc}.
By default this creates a pair of small windows, @samp{*Calculator*}
and @samp{*Calc Trail*}.  The former displays the contents of the
Calculator stack and is manipulated exclusively through Calc commands.
It is possible (though not usually necessary) to create several Calc
mode buffers each of which has an independent stack, undo list, and
mode settings.  There is exactly one Calc Trail buffer; it records a
list of the results of all calculations that have been done.  The
Calc Trail buffer uses a variant of Calc mode, so Calculator commands
still work when the trail buffer's window is selected.  It is possible
to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
still exists and is updated silently.  @xref{Trail Commands}.

@kindex C-x * c
@kindex C-x * *
@ignore
@mindex @null
@end ignore
In most installations, the @kbd{C-x * c} key sequence is a more
convenient way to start the Calculator.  Also, @kbd{C-x * *} 
is a synonym for @kbd{C-x * c} unless you last used Calc
in its Keypad mode.

@kindex x
@kindex M-x
@pindex calc-execute-extended-command
Most Calc commands use one or two keystrokes.  Lower- and upper-case
letters are distinct.  Commands may also be entered in full @kbd{M-x} form;
for some commands this is the only form.  As a convenience, the @kbd{x}
key (@code{calc-execute-extended-command})
is like @kbd{M-x} except that it enters the initial string @samp{calc-}
for you.  For example, the following key sequences are equivalent:
@kbd{S}, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.

@cindex Extensions module
@cindex @file{calc-ext} module
The Calculator exists in many parts.  When you type @kbd{C-x * c}, the
Emacs ``auto-load'' mechanism will bring in only the first part, which
contains the basic arithmetic functions.  The other parts will be
auto-loaded the first time you use the more advanced commands like trig
functions or matrix operations.  This is done to improve the response time
of the Calculator in the common case when all you need to do is a
little arithmetic.  If for some reason the Calculator fails to load an
extension module automatically, you can force it to load all the
extensions by using the @kbd{C-x * L} (@code{calc-load-everything})
command.  @xref{Mode Settings}.

If you type @kbd{M-x calc} or @kbd{C-x * c} with any numeric prefix argument,
the Calculator is loaded if necessary, but it is not actually started.
If the argument is positive, the @file{calc-ext} extensions are also
loaded if necessary.  User-written Lisp code that wishes to make use
of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
to auto-load the Calculator.

@kindex C-x * b
@pindex full-calc
If you type @kbd{C-x * b}, then next time you use @kbd{C-x * c} you
will get a Calculator that uses the full height of the Emacs screen.
When full-screen mode is on, @kbd{C-x * c} runs the @code{full-calc}
command instead of @code{calc}.  From the Unix shell you can type
@samp{emacs -f full-calc} to start a new Emacs specifically for use
as a calculator.  When Calc is started from the Emacs command line
like this, Calc's normal ``quit'' commands actually quit Emacs itself.

@kindex C-x * o
@pindex calc-other-window
The @kbd{C-x * o} command is like @kbd{C-x * c} except that the Calc
window is not actually selected.  If you are already in the Calc
window, @kbd{C-x * o} switches you out of it.  (The regular Emacs
@kbd{C-x o} command would also work for this, but it has a
tendency to drop you into the Calc Trail window instead, which
@kbd{C-x * o} takes care not to do.)

@ignore
@mindex C-x * q
@end ignore
For one quick calculation, you can type @kbd{C-x * q} (@code{quick-calc})
which prompts you for a formula (like @samp{2+3/4}).  The result is
displayed at the bottom of the Emacs screen without ever creating
any special Calculator windows.  @xref{Quick Calculator}.

@ignore
@mindex C-x * k
@end ignore
Finally, if you are using the X window system you may want to try
@kbd{C-x * k} (@code{calc-keypad}) which runs Calc with a
``calculator keypad'' picture as well as a stack display.  Click on
the keys with the mouse to operate the calculator.  @xref{Keypad Mode}.

@kindex q
@pindex calc-quit
@cindex Quitting the Calculator
@cindex Exiting the Calculator
The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
Calculator's window(s).  It does not delete the Calculator buffers.
If you type @kbd{M-x calc} again, the Calculator will reappear with the
contents of the stack intact.  Typing @kbd{C-x * c} or @kbd{C-x * *}
again from inside the Calculator buffer is equivalent to executing
@code{calc-quit}; you can think of @kbd{C-x * *} as toggling the
Calculator on and off.

@kindex C-x * x
The @kbd{C-x * x} command also turns the Calculator off, no matter which
user interface (standard, Keypad, or Embedded) is currently active.
It also cancels @code{calc-edit} mode if used from there.

@kindex d @key{SPC}
@pindex calc-refresh
@cindex Refreshing a garbled display
@cindex Garbled displays, refreshing
The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
of the Calculator buffer from memory.  Use this if the contents of the
buffer have been damaged somehow.

@ignore
@mindex o
@end ignore
The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
``home'' position at the bottom of the Calculator buffer.

@kindex <
@kindex >
@pindex calc-scroll-left
@pindex calc-scroll-right
@cindex Horizontal scrolling
@cindex Scrolling
@cindex Wide text, scrolling
The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
@code{calc-scroll-right}.  These are just like the normal horizontal
scrolling commands except that they scroll one half-screen at a time by
default.  (Calc formats its output to fit within the bounds of the
window whenever it can.)

@kindex @{
@kindex @}
@pindex calc-scroll-down
@pindex calc-scroll-up
@cindex Vertical scrolling
The @kbd{@{} and @kbd{@}} keys are bound to @code{calc-scroll-down}
and @code{calc-scroll-up}.  They scroll up or down by one-half the
height of the Calc window.

@kindex C-x * 0
@pindex calc-reset
The @kbd{C-x * 0} command (@code{calc-reset}; that's @kbd{C-x *} followed
by a zero) resets the Calculator to its initial state.  This clears
the stack, resets all the modes to their initial values (the values
that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
caches (@pxref{Caches}), and so on.  (It does @emph{not} erase the
values of any variables.) With an argument of 0, Calc will be reset to
its default state; namely, the modes will be given their default values.
With a positive prefix argument, @kbd{C-x * 0} preserves the contents of
the stack but resets everything else to its initial state; with a
negative prefix argument, @kbd{C-x * 0} preserves the contents of the
stack but resets everything else to its default state.

@pindex calc-version
The @kbd{M-x calc-version} command displays the current version number
of Calc and the name of the person who installed it on your system.
(This information is also present in the @samp{*Calc Trail*} buffer,
and in the output of the @kbd{h h} command.)

@node Help Commands, Stack Basics, Basic Commands, Introduction
@section Help Commands

@noindent
@cindex Help commands
@kindex ?
@pindex calc-help
The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs'
@key{ESC} and @kbd{C-x} prefixes.  You can type
@kbd{?} after a prefix to see a list of commands beginning with that
prefix.  (If the message includes @samp{[MORE]}, press @kbd{?} again
to see additional commands for that prefix.)

@kindex h h
@pindex calc-full-help
The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
responses at once.  When printed, this makes a nice, compact (three pages)
summary of Calc keystrokes.

In general, the @kbd{h} key prefix introduces various commands that
provide help within Calc.  Many of the @kbd{h} key functions are
Calc-specific analogues to the @kbd{C-h} functions for Emacs help.

@kindex h i
@kindex C-x * i
@kindex i
@pindex calc-info
The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
to read this manual on-line.  This is basically the same as typing
@kbd{C-h i} (the regular way to run the Info system), then, if Info
is not already in the Calc manual, selecting the beginning of the
manual.  The @kbd{C-x * i} command is another way to read the Calc
manual; it is different from @kbd{h i} in that it works any time,
not just inside Calc.  The plain @kbd{i} key is also equivalent to
@kbd{h i}, though this key is obsolete and may be replaced with a
different command in a future version of Calc.

@kindex h t
@kindex C-x * t
@pindex calc-tutorial
The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
the Tutorial section of the Calc manual.  It is like @kbd{h i},
except that it selects the starting node of the tutorial rather
than the beginning of the whole manual.  (It actually selects the
node ``Interactive Tutorial'' which tells a few things about
using the Info system before going on to the actual tutorial.)
The @kbd{C-x * t} key is equivalent to @kbd{h t} (but it works at
all times).

@kindex h s
@kindex C-x * s
@pindex calc-info-summary
The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
on the Summary node of the Calc manual.  @xref{Summary}.  The @kbd{C-x * s}
key is equivalent to @kbd{h s}.

@kindex h k
@pindex calc-describe-key
The @kbd{h k} (@code{calc-describe-key}) command looks up a key
sequence in the Calc manual.  For example, @kbd{h k H a S} looks
up the documentation on the @kbd{H a S} (@code{calc-solve-for})
command.  This works by looking up the textual description of
the key(s) in the Key Index of the manual, then jumping to the
node indicated by the index.

Most Calc commands do not have traditional Emacs documentation
strings, since the @kbd{h k} command is both more convenient and
more instructive.  This means the regular Emacs @kbd{C-h k}
(@code{describe-key}) command will not be useful for Calc keystrokes.

@kindex h c
@pindex calc-describe-key-briefly
The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
key sequence and displays a brief one-line description of it at
the bottom of the screen.  It looks for the key sequence in the
Summary node of the Calc manual; if it doesn't find the sequence
there, it acts just like its regular Emacs counterpart @kbd{C-h c}
(@code{describe-key-briefly}).  For example, @kbd{h c H a S}
gives the description:

@smallexample
H a S runs calc-solve-for:  a `H a S' v  => fsolve(a,v)  (?=notes)
@end smallexample

@noindent
which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
takes a value @expr{a} from the stack, prompts for a value @expr{v},
then applies the algebraic function @code{fsolve} to these values.
The @samp{?=notes} message means you can now type @kbd{?} to see
additional notes from the summary that apply to this command.

@kindex h f
@pindex calc-describe-function
The @kbd{h f} (@code{calc-describe-function}) command looks up an
algebraic function or a command name in the Calc manual.  Enter an
algebraic function name to look up that function in the Function
Index or enter a command name beginning with @samp{calc-} to look it 
up in the Command Index.  This command will also look up operator
symbols that can appear in algebraic formulas, like @samp{%} and 
@samp{=>}.

@kindex h v
@pindex calc-describe-variable
The @kbd{h v} (@code{calc-describe-variable}) command looks up a
variable in the Calc manual.  Enter a variable name like @code{pi} or
@code{PlotRejects}.

@kindex h b
@pindex describe-bindings
The @kbd{h b} (@code{calc-describe-bindings}) command is just like
@kbd{C-h b}, except that only local (Calc-related) key bindings are
listed.

@kindex h n
The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
the ``news'' or change history of Calc.  This is kept in the file
@file{README}, which Calc looks for in the same directory as the Calc
source files.

@kindex h C-c
@kindex h C-d
@kindex h C-w
The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
distribution, and warranty information about Calc.  These work by
pulling up the appropriate parts of the ``Copying'' or ``Reporting
Bugs'' sections of the manual.

@node Stack Basics, Numeric Entry, Help Commands, Introduction
@section Stack Basics

@noindent
@cindex Stack basics
@c [fix-tut RPN Calculations and the Stack]
Calc uses RPN notation.  If you are not familiar with RPN, @pxref{RPN
Tutorial}.

To add the numbers 1 and 2 in Calc you would type the keys:
@kbd{1 @key{RET} 2 +}.
(@key{RET} corresponds to the @key{ENTER} key on most calculators.)
The first three keystrokes ``push'' the numbers 1 and 2 onto the stack.  The
@kbd{+} key always ``pops'' the top two numbers from the stack, adds them,
and pushes the result (3) back onto the stack.  This number is ready for
further calculations:  @kbd{5 -} pushes 5 onto the stack, then pops the
3 and 5, subtracts them, and pushes the result (@mathit{-2}).

Note that the ``top'' of the stack actually appears at the @emph{bottom}
of the buffer.  A line containing a single @samp{.} character signifies
the end of the buffer; Calculator commands operate on the number(s)
directly above this line.  The @kbd{d t} (@code{calc-truncate-stack})
command allows you to move the @samp{.} marker up and down in the stack;
@pxref{Truncating the Stack}.

@kindex d l
@pindex calc-line-numbering
Stack elements are numbered consecutively, with number 1 being the top of
the stack.  These line numbers are ordinarily displayed on the lefthand side
of the window.  The @kbd{d l} (@code{calc-line-numbering}) command controls
whether these numbers appear.  (Line numbers may be turned off since they
slow the Calculator down a bit and also clutter the display.)

@kindex o
@pindex calc-realign
The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
the cursor to its top-of-stack ``home'' position.  It also undoes any
horizontal scrolling in the window.  If you give it a numeric prefix
argument, it instead moves the cursor to the specified stack element.

The @key{RET} (or equivalent @key{SPC}) key is only required to separate
two consecutive numbers.
(After all, if you typed @kbd{1 2} by themselves the Calculator
would enter the number 12.)  If you press @key{RET} or @key{SPC} @emph{not}
right after typing a number, the key duplicates the number on the top of
the stack.  @kbd{@key{RET} *} is thus a handy way to square a number.

The @key{DEL} key pops and throws away the top number on the stack.
The @key{TAB} key swaps the top two objects on the stack.
@xref{Stack and Trail}, for descriptions of these and other stack-related
commands.

@node Numeric Entry, Algebraic Entry, Stack Basics, Introduction
@section Numeric Entry

@noindent
@kindex 0-9
@kindex .
@kindex e
@cindex Numeric entry
@cindex Entering numbers
Pressing a digit or other numeric key begins numeric entry using the
minibuffer.  The number is pushed on the stack when you press the @key{RET}
or @key{SPC} keys.  If you press any other non-numeric key, the number is
pushed onto the stack and the appropriate operation is performed.  If
you press a numeric key which is not valid, the key is ignored.

@cindex Minus signs
@cindex Negative numbers, entering
@kindex _
There are three different concepts corresponding to the word ``minus,''
typified by @expr{a-b} (subtraction), @expr{-x}
(change-sign), and @expr{-5} (negative number).  Calc uses three
different keys for these operations, respectively:
@kbd{-}, @kbd{n}, and @kbd{_} (the underscore).  The @kbd{-} key subtracts
the two numbers on the top of the stack.  The @kbd{n} key changes the sign
of the number on the top of the stack or the number currently being entered.
The @kbd{_} key begins entry of a negative number or changes the sign of
the number currently being entered.  The following sequences all enter the
number @mathit{-5} onto the stack:  @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
@kbd{5 @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.

Some other keys are active during numeric entry, such as @kbd{#} for
non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
These notations are described later in this manual with the corresponding
data types.  @xref{Data Types}.

During numeric entry, the only editing key available is @key{DEL}.

@node Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
@section Algebraic Entry

@noindent
@kindex '
@pindex calc-algebraic-entry
@cindex Algebraic notation
@cindex Formulas, entering
Calculations can also be entered in algebraic form.  This is accomplished
by typing the apostrophe key, @kbd{'}, followed by the expression in
standard format:  @kbd{@key{'} 2+3*4 @key{RET}} computes
@texline @math{2+(3\times4) = 14}
@infoline @expr{2+(3*4) = 14} 
and pushes that on the stack.  If you wish you can
ignore the RPN aspect of Calc altogether and simply enter algebraic
expressions in this way.  You may want to use @key{DEL} every so often to
clear previous results off the stack.

You can press the apostrophe key during normal numeric entry to switch
the half-entered number into Algebraic entry mode.  One reason to do this
would be to use the full Emacs cursor motion and editing keys, which are
available during algebraic entry but not during numeric entry.

In the same vein, during either numeric or algebraic entry you can
press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
you complete your half-finished entry in a separate buffer.
@xref{Editing Stack Entries}.

@kindex m a
@pindex calc-algebraic-mode
@cindex Algebraic Mode
If you prefer algebraic entry, you can use the command @kbd{m a}
(@code{calc-algebraic-mode}) to set Algebraic mode.  In this mode,
digits and other keys that would normally start numeric entry instead
start full algebraic entry; as long as your formula begins with a digit
you can omit the apostrophe.  Open parentheses and square brackets also
begin algebraic entry.  You can still do RPN calculations in this mode,
but you will have to press @key{RET} to terminate every number:
@kbd{2 @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
thing as @kbd{2*3+4 @key{RET}}.

@cindex Incomplete Algebraic Mode
If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
command, it enables Incomplete Algebraic mode; this is like regular
Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
only.  Numeric keys still begin a numeric entry in this mode.

@kindex m t
@pindex calc-total-algebraic-mode
@cindex Total Algebraic Mode
The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
stronger algebraic-entry mode, in which @emph{all} regular letter and
punctuation keys begin algebraic entry.  Use this if you prefer typing
@w{@kbd{sqrt( )}} instead of @kbd{Q}, @w{@kbd{factor( )}} instead of
@kbd{a f}, and so on.  To type regular Calc commands when you are in
Total Algebraic mode, hold down the @key{META} key.  Thus @kbd{M-q}
is the command to quit Calc, @kbd{M-p} sets the precision, and
@kbd{M-m t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
mode back off again.  Meta keys also terminate algebraic entry, so
that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}.  The symbol
@samp{Alg*} will appear in the mode line whenever you are in this mode.

Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
algebraic formula.  You can then use the normal Emacs editing keys to
modify this formula to your liking before pressing @key{RET}.

@kindex $
@cindex Formulas, referring to stack
Within a formula entered from the keyboard, the symbol @kbd{$}
represents the number on the top of the stack.  If an entered formula
contains any @kbd{$} characters, the Calculator replaces the top of
stack with that formula rather than simply pushing the formula onto the
stack.  Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
@key{RET}} replaces it with 6.  Note that the @kbd{$} key always
initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
first character in the new formula.

Higher stack elements can be accessed from an entered formula with the
symbols @kbd{$$}, @kbd{$$$}, and so on.  The number of stack elements
removed (to be replaced by the entered values) equals the number of dollar
signs in the longest such symbol in the formula.  For example, @samp{$$+$$$}
adds the second and third stack elements, replacing the top three elements
with the answer.  (All information about the top stack element is thus lost
since no single @samp{$} appears in this formula.)

A slightly different way to refer to stack elements is with a dollar
sign followed by a number:  @samp{$1}, @samp{$2}, and so on are much
like @samp{$}, @samp{$$}, etc., except that stack entries referred
to numerically are not replaced by the algebraic entry.  That is, while
@samp{$+1} replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
on the stack and pushes an additional 6.

If a sequence of formulas are entered separated by commas, each formula
is pushed onto the stack in turn.  For example, @samp{1,2,3} pushes
those three numbers onto the stack (leaving the 3 at the top), and
@samp{$+1,$-1} replaces a 5 on the stack with 4 followed by 6.  Also,
@samp{$,$$} exchanges the top two elements of the stack, just like the
@key{TAB} key.

You can finish an algebraic entry with @kbd{M-=} or @kbd{M-@key{RET}} instead
of @key{RET}.  This uses @kbd{=} to evaluate the variables in each
formula that goes onto the stack.  (Thus @kbd{' pi @key{RET}} pushes
the variable @samp{pi}, but @kbd{' pi M-@key{RET}} pushes 3.1415.)

If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
instead of @key{RET}, Calc disables the default simplifications
(as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
is being pushed on the stack.  Thus @kbd{' 1+2 @key{RET}} pushes 3
on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
you might then press @kbd{=} when it is time to evaluate this formula.

@node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
@section ``Quick Calculator'' Mode

@noindent
@kindex C-x * q
@pindex quick-calc
@cindex Quick Calculator
There is another way to invoke the Calculator if all you need to do
is make one or two quick calculations.  Type @kbd{C-x * q} (or
@kbd{M-x quick-calc}), then type any formula as an algebraic entry.
The Calculator will compute the result and display it in the echo
area, without ever actually putting up a Calc window.

You can use the @kbd{$} character in a Quick Calculator formula to
refer to the previous Quick Calculator result.  Older results are
not retained; the Quick Calculator has no effect on the full
Calculator's stack or trail.  If you compute a result and then
forget what it was, just run @code{C-x * q} again and enter
@samp{$} as the formula.

If this is the first time you have used the Calculator in this Emacs
session, the @kbd{C-x * q} command will create the @code{*Calculator*}
buffer and perform all the usual initializations; it simply will
refrain from putting that buffer up in a new window.  The Quick
Calculator refers to the @code{*Calculator*} buffer for all mode
settings.  Thus, for example, to set the precision that the Quick
Calculator uses, simply run the full Calculator momentarily and use
the regular @kbd{p} command.

If you use @code{C-x * q} from inside the Calculator buffer, the
effect is the same as pressing the apostrophe key (algebraic entry).

The result of a Quick calculation is placed in the Emacs ``kill ring''
as well as being displayed.  A subsequent @kbd{C-y} command will
yank the result into the editing buffer.  You can also use this
to yank the result into the next @kbd{C-x * q} input line as a more
explicit alternative to @kbd{$} notation, or to yank the result
into the Calculator stack after typing @kbd{C-x * c}.

If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
of @key{RET}, the result is inserted immediately into the current
buffer rather than going into the kill ring.

Quick Calculator results are actually evaluated as if by the @kbd{=}
key (which replaces variable names by their stored values, if any).
If the formula you enter is an assignment to a variable using the
@samp{:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
then the result of the evaluation is stored in that Calc variable.
@xref{Store and Recall}.

If the result is an integer and the current display radix is decimal,
the number will also be displayed in hex and octal formats.  If the
integer is in the range from 1 to 126, it will also be displayed as
an ASCII character.

For example, the quoted character @samp{"x"} produces the vector
result @samp{[120]} (because 120 is the ASCII code of the lower-case
`x'; @pxref{Strings}).  Since this is a vector, not an integer, it
is displayed only according to the current mode settings.  But
running Quick Calc again and entering @samp{120} will produce the
result @samp{120 (16#78, 8#170, x)} which shows the number in its
decimal, hexadecimal, octal, and ASCII forms.

Please note that the Quick Calculator is not any faster at loading
or computing the answer than the full Calculator; the name ``quick''
merely refers to the fact that it's much less hassle to use for
small calculations.

@node Prefix Arguments, Undo, Quick Calculator, Introduction
@section Numeric Prefix Arguments

@noindent
Many Calculator commands use numeric prefix arguments.  Some, such as
@kbd{d s} (@code{calc-sci-notation}), set a parameter to the value of
the prefix argument or use a default if you don't use a prefix.
Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
and prompt for a number if you don't give one as a prefix.

As a rule, stack-manipulation commands accept a numeric prefix argument
which is interpreted as an index into the stack.  A positive argument
operates on the top @var{n} stack entries; a negative argument operates
on the @var{n}th stack entry in isolation; and a zero argument operates
on the entire stack.

Most commands that perform computations (such as the arithmetic and
scientific functions) accept a numeric prefix argument that allows the
operation to be applied across many stack elements.  For unary operations
(that is, functions of one argument like absolute value or complex
conjugate), a positive prefix argument applies that function to the top
@var{n} stack entries simultaneously, and a negative argument applies it
to the @var{n}th stack entry only.  For binary operations (functions of
two arguments like addition, GCD, and vector concatenation), a positive
prefix argument ``reduces'' the function across the top @var{n}
stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
@pxref{Reducing and Mapping}), and a negative argument maps the next-to-top
@var{n} stack elements with the top stack element as a second argument
(for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
This feature is not available for operations which use the numeric prefix
argument for some other purpose.

Numeric prefixes are specified the same way as always in Emacs:  Press
a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
or press @kbd{C-u} followed by digits.  Some commands treat plain
@kbd{C-u} (without any actual digits) specially.

@kindex ~
@pindex calc-num-prefix
You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
top of the stack and enter it as the numeric prefix for the next command.
For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
(silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
to the fourth power and set the precision to that value.

Conversely, if you have typed a numeric prefix argument the @kbd{~} key
pushes it onto the stack in the form of an integer.

@node Undo, Error Messages, Prefix Arguments, Introduction
@section Undoing Mistakes

@noindent
@kindex U
@kindex C-_
@pindex calc-undo
@cindex Mistakes, undoing
@cindex Undoing mistakes
@cindex Errors, undoing
The shift-@kbd{U} key (@code{calc-undo}) undoes the most recent operation.
If that operation added or dropped objects from the stack, those objects
are removed or restored.  If it was a ``store'' operation, you are
queried whether or not to restore the variable to its original value.
The @kbd{U} key may be pressed any number of times to undo successively
farther back in time; with a numeric prefix argument it undoes a
specified number of operations.  The undo history is cleared only by the
@kbd{q} (@code{calc-quit}) command.  (Recall that @kbd{C-x * c} is
synonymous with @code{calc-quit} while inside the Calculator; this
also clears the undo history.)

Currently the mode-setting commands (like @code{calc-precision}) are not
undoable.  You can undo past a point where you changed a mode, but you
will need to reset the mode yourself.

@kindex D
@pindex calc-redo
@cindex Redoing after an Undo
The shift-@kbd{D} key (@code{calc-redo}) redoes an operation that was
mistakenly undone.  Pressing @kbd{U} with a negative prefix argument is
equivalent to executing @code{calc-redo}.  You can redo any number of
times, up to the number of recent consecutive undo commands.  Redo
information is cleared whenever you give any command that adds new undo
information, i.e., if you undo, then enter a number on the stack or make
any other change, then it will be too late to redo.

@kindex M-@key{RET}
@pindex calc-last-args
@cindex Last-arguments feature
@cindex Arguments, restoring
The @kbd{M-@key{RET}} key (@code{calc-last-args}) is like undo in that
it restores the arguments of the most recent command onto the stack;
however, it does not remove the result of that command.  Given a numeric
prefix argument, this command applies to the @expr{n}th most recent
command which removed items from the stack; it pushes those items back
onto the stack.

The @kbd{K} (@code{calc-keep-args}) command provides a related function
to @kbd{M-@key{RET}}.  @xref{Stack and Trail}.

It is also possible to recall previous results or inputs using the trail.
@xref{Trail Commands}.

The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.

@node Error Messages, Multiple Calculators, Undo, Introduction
@section Error Messages

@noindent
@kindex w
@pindex calc-why
@cindex Errors, messages
@cindex Why did an error occur?
Many situations that would produce an error message in other calculators
simply create unsimplified formulas in the Emacs Calculator.  For example,
@kbd{1 @key{RET} 0 /} pushes the formula @expr{1 / 0}; @w{@kbd{0 L}} pushes
the formula @samp{ln(0)}.  Floating-point overflow and underflow are also
reasons for this to happen.

When a function call must be left in symbolic form, Calc usually
produces a message explaining why.  Messages that are probably
surprising or indicative of user errors are displayed automatically.
Other messages are simply kept in Calc's memory and are displayed only
if you type @kbd{w} (@code{calc-why}).  You can also press @kbd{w} if
the same computation results in several messages.  (The first message
will end with @samp{[w=more]} in this case.)

@kindex d w
@pindex calc-auto-why
The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
are displayed automatically.  (Calc effectively presses @kbd{w} for you
after your computation finishes.)  By default, this occurs only for
``important'' messages.  The other possible modes are to report
@emph{all} messages automatically, or to report none automatically (so
that you must always press @kbd{w} yourself to see the messages).

@node Multiple Calculators, Troubleshooting Commands, Error Messages, Introduction
@section Multiple Calculators

@noindent
@pindex another-calc
It is possible to have any number of Calc mode buffers at once.
Usually this is done by executing @kbd{M-x another-calc}, which
is similar to @kbd{C-x * c} except that if a @samp{*Calculator*}
buffer already exists, a new, independent one with a name of the
form @samp{*Calculator*<@var{n}>} is created.  You can also use the
command @code{calc-mode} to put any buffer into Calculator mode, but
this would ordinarily never be done.

The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
it only closes its window.  Use @kbd{M-x kill-buffer} to destroy a
Calculator buffer.

Each Calculator buffer keeps its own stack, undo list, and mode settings
such as precision, angular mode, and display formats.  In Emacs terms,
variables such as @code{calc-stack} are buffer-local variables.  The
global default values of these variables are used only when a new
Calculator buffer is created.  The @code{calc-quit} command saves
the stack and mode settings of the buffer being quit as the new defaults.

There is only one trail buffer, @samp{*Calc Trail*}, used by all
Calculator buffers.

@node Troubleshooting Commands,  , Multiple Calculators, Introduction
@section Troubleshooting Commands

@noindent
This section describes commands you can use in case a computation
incorrectly fails or gives the wrong answer.

@xref{Reporting Bugs}, if you find a problem that appears to be due
to a bug or deficiency in Calc.

@menu
* Autoloading Problems::
* Recursion Depth::
* Caches::
* Debugging Calc::
@end menu

@node Autoloading Problems, Recursion Depth, Troubleshooting Commands, Troubleshooting Commands
@subsection Autoloading Problems

@noindent
The Calc program is split into many component files; components are
loaded automatically as you use various commands that require them.
Occasionally Calc may lose track of when a certain component is
necessary; typically this means you will type a command and it won't
work because some function you've never heard of was undefined.

@kindex C-x * L
@pindex calc-load-everything
If this happens, the easiest workaround is to type @kbd{C-x * L}
(@code{calc-load-everything}) to force all the parts of Calc to be
loaded right away.  This will cause Emacs to take up a lot more
memory than it would otherwise, but it's guaranteed to fix the problem.

@node Recursion Depth, Caches, Autoloading Problems, Troubleshooting Commands
@subsection Recursion Depth

@noindent
@kindex M
@kindex I M
@pindex calc-more-recursion-depth
@pindex calc-less-recursion-depth
@cindex Recursion depth
@cindex ``Computation got stuck'' message
@cindex @code{max-lisp-eval-depth}
@cindex @code{max-specpdl-size}
Calc uses recursion in many of its calculations.  Emacs Lisp keeps a
variable @code{max-lisp-eval-depth} which limits the amount of recursion
possible in an attempt to recover from program bugs.  If a calculation
ever halts incorrectly with the message ``Computation got stuck or
ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
to increase this limit.  (Of course, this will not help if the
calculation really did get stuck due to some problem inside Calc.)

The limit is always increased (multiplied) by a factor of two.  There
is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
decreases this limit by a factor of two, down to a minimum value of 200.
The default value is 1000.

These commands also double or halve @code{max-specpdl-size}, another
internal Lisp recursion limit.  The minimum value for this limit is 600.

@node Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
@subsection Caches

@noindent
@cindex Caches
@cindex Flushing caches
Calc saves certain values after they have been computed once.  For
example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
constant @cpi{} to about 20 decimal places; if the current precision
is greater than this, it will recompute @cpi{} using a series
approximation.  This value will not need to be recomputed ever again
unless you raise the precision still further.  Many operations such as
logarithms and sines make use of similarly cached values such as
@cpiover{4} and 
@texline @math{\ln 2}.
@infoline @expr{ln(2)}.  
The visible effect of caching is that
high-precision computations may seem to do extra work the first time.
Other things cached include powers of two (for the binary arithmetic
functions), matrix inverses and determinants, symbolic integrals, and
data points computed by the graphing commands.

@pindex calc-flush-caches
If you suspect a Calculator cache has become corrupt, you can use the
@code{calc-flush-caches} command to reset all caches to the empty state.
(This should only be necessary in the event of bugs in the Calculator.)
The @kbd{C-x * 0} (with the zero key) command also resets caches along
with all other aspects of the Calculator's state.

@node Debugging Calc,  , Caches, Troubleshooting Commands
@subsection Debugging Calc

@noindent
A few commands exist to help in the debugging of Calc commands.
@xref{Programming}, to see the various ways that you can write
your own Calc commands.

@kindex Z T
@pindex calc-timing
The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
in which the timing of slow commands is reported in the Trail.
Any Calc command that takes two seconds or longer writes a line
to the Trail showing how many seconds it took.  This value is
accurate only to within one second.

All steps of executing a command are included; in particular, time
taken to format the result for display in the stack and trail is
counted.  Some prompts also count time taken waiting for them to
be answered, while others do not; this depends on the exact
implementation of the command.  For best results, if you are timing
a sequence that includes prompts or multiple commands, define a
keyboard macro to run the whole sequence at once.  Calc's @kbd{X}
command (@pxref{Keyboard Macros}) will then report the time taken
to execute the whole macro.

Another advantage of the @kbd{X} command is that while it is
executing, the stack and trail are not updated from step to step.
So if you expect the output of your test sequence to leave a result
that may take a long time to format and you don't wish to count
this formatting time, end your sequence with a @key{DEL} keystroke
to clear the result from the stack.  When you run the sequence with
@kbd{X}, Calc will never bother to format the large result.

Another thing @kbd{Z T} does is to increase the Emacs variable
@code{gc-cons-threshold} to a much higher value (two million; the
usual default in Calc is 250,000) for the duration of each command.
This generally prevents garbage collection during the timing of
the command, though it may cause your Emacs process to grow
abnormally large.  (Garbage collection time is a major unpredictable
factor in the timing of Emacs operations.)

Another command that is useful when debugging your own Lisp
extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
the error handler that changes the ``@code{max-lisp-eval-depth}
exceeded'' message to the much more friendly ``Computation got
stuck or ran too long.''  This handler interferes with the Emacs
Lisp debugger's @code{debug-on-error} mode.  Errors are reported
in the handler itself rather than at the true location of the
error.  After you have executed @code{calc-pass-errors}, Lisp
errors will be reported correctly but the user-friendly message
will be lost.

@node Data Types, Stack and Trail, Introduction, Top
@chapter Data Types

@noindent
This chapter discusses the various types of objects that can be placed
on the Calculator stack, how they are displayed, and how they are
entered.  (@xref{Data Type Formats}, for information on how these data
types are represented as underlying Lisp objects.)

Integers, fractions, and floats are various ways of describing real
numbers.  HMS forms also for many purposes act as real numbers.  These
types can be combined to form complex numbers, modulo forms, error forms,
or interval forms.  (But these last four types cannot be combined
arbitrarily:@: error forms may not contain modulo forms, for example.)
Finally, all these types of numbers may be combined into vectors,
matrices, or algebraic formulas.

@menu
* Integers::                The most basic data type.
* Fractions::               This and above are called @dfn{rationals}.
* Floats::                  This and above are called @dfn{reals}.
* Complex Numbers::         This and above are called @dfn{numbers}.
* Infinities::
* Vectors and Matrices::
* Strings::
* HMS Forms::
* Date Forms::
* Modulo Forms::
* Error Forms::
* Interval Forms::
* Incomplete Objects::
* Variables::
* Formulas::
@end menu

@node Integers, Fractions, Data Types, Data Types
@section Integers

@noindent
@cindex Integers
The Calculator stores integers to arbitrary precision.  Addition,
subtraction, and multiplication of integers always yields an exact
integer result.  (If the result of a division or exponentiation of
integers is not an integer, it is expressed in fractional or
floating-point form according to the current Fraction mode.
@xref{Fraction Mode}.)

A decimal integer is represented as an optional sign followed by a
sequence of digits.  Grouping (@pxref{Grouping Digits}) can be used to
insert a comma at every third digit for display purposes, but you
must not type commas during the entry of numbers.

@kindex #
A non-decimal integer is represented as an optional sign, a radix
between 2 and 36, a @samp{#} symbol, and one or more digits.  For radix 11
and above, the letters A through Z (upper- or lower-case) count as
digits and do not terminate numeric entry mode.  @xref{Radix Modes}, for how
to set the default radix for display of integers.  Numbers of any radix
may be entered at any time.  If you press @kbd{#} at the beginning of a
number, the current display radix is used.

@node Fractions, Floats, Integers, Data Types
@section Fractions

@noindent
@cindex Fractions
A @dfn{fraction} is a ratio of two integers.  Fractions are traditionally
written ``2/3'' but Calc uses the notation @samp{2:3}.  (The @kbd{/} key
performs RPN division; the following two sequences push the number
@samp{2:3} on the stack:  @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3 /}
assuming Fraction mode has been enabled.)
When the Calculator produces a fractional result it always reduces it to
simplest form, which may in fact be an integer.

Fractions may also be entered in a three-part form, where @samp{2:3:4}
represents two-and-three-quarters.  @xref{Fraction Formats}, for fraction
display formats.

Non-decimal fractions are entered and displayed as
@samp{@var{radix}#@var{num}:@var{denom}} (or in the analogous three-part
form).  The numerator and denominator always use the same radix.

@node Floats, Complex Numbers, Fractions, Data Types
@section Floats

@noindent
@cindex Floating-point numbers
A floating-point number or @dfn{float} is a number stored in scientific
notation.  The number of significant digits in the fractional part is
governed by the current floating precision (@pxref{Precision}).  The
range of acceptable values is from 
@texline @math{10^{-3999999}}
@infoline @expr{10^-3999999} 
(inclusive) to 
@texline @math{10^{4000000}}
@infoline @expr{10^4000000}
(exclusive), plus the corresponding negative values and zero.

Calculations that would exceed the allowable range of values (such
as @samp{exp(exp(20))}) are left in symbolic form by Calc.  The
messages ``floating-point overflow'' or ``floating-point underflow''
indicate that during the calculation a number would have been produced
that was too large or too close to zero, respectively, to be represented
by Calc.  This does not necessarily mean the final result would have
overflowed, just that an overflow occurred while computing the result.
(In fact, it could report an underflow even though the final result
would have overflowed!)

If a rational number and a float are mixed in a calculation, the result
will in general be expressed as a float.  Commands that require an integer
value (such as @kbd{k g} [@code{gcd}]) will also accept integer-valued
floats, i.e., floating-point numbers with nothing after the decimal point.

Floats are identified by the presence of a decimal point and/or an
exponent.  In general a float consists of an optional sign, digits
including an optional decimal point, and an optional exponent consisting
of an @samp{e}, an optional sign, and up to seven exponent digits.
For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
or 0.235.

Floating-point numbers are normally displayed in decimal notation with
all significant figures shown.  Exceedingly large or small numbers are
displayed in scientific notation.  Various other display options are
available.  @xref{Float Formats}.

@cindex Accuracy of calculations
Floating-point numbers are stored in decimal, not binary.  The result
of each operation is rounded to the nearest value representable in the
number of significant digits specified by the current precision,
rounding away from zero in the case of a tie.  Thus (in the default
display mode) what you see is exactly what you get.  Some operations such
as square roots and transcendental functions are performed with several
digits of extra precision and then rounded down, in an effort to make the
final result accurate to the full requested precision.  However,
accuracy is not rigorously guaranteed.  If you suspect the validity of a
result, try doing the same calculation in a higher precision.  The
Calculator's arithmetic is not intended to be IEEE-conformant in any
way.

While floats are always @emph{stored} in decimal, they can be entered
and displayed in any radix just like integers and fractions.  Since a
float that is entered in a radix other that 10 will be converted to
decimal, the number that Calc stores may not be exactly the number that
was entered, it will be the closest decimal approximation given the
current precison.  The notation @samp{@var{radix}#@var{ddd}.@var{ddd}}
is a floating-point number whose digits are in the specified radix.
Note that the @samp{.}  is more aptly referred to as a ``radix point''
than as a decimal point in this case.  The number @samp{8#123.4567} is
defined as @samp{8#1234567 * 8^-4}.  If the radix is 14 or less, you can
use @samp{e} notation to write a non-decimal number in scientific
notation.  The exponent is written in decimal, and is considered to be a
power of the radix: @samp{8#1234567e-4}.  If the radix is 15 or above,
the letter @samp{e} is a digit, so scientific notation must be written
out, e.g., @samp{16#123.4567*16^2}.  The first two exercises of the
Modes Tutorial explore some of the properties of non-decimal floats.

@node Complex Numbers, Infinities, Floats, Data Types
@section Complex Numbers

@noindent
@cindex Complex numbers
There are two supported formats for complex numbers: rectangular and
polar.  The default format is rectangular, displayed in the form
@samp{(@var{real},@var{imag})} where @var{real} is the real part and
@var{imag} is the imaginary part, each of which may be any real number.
Rectangular complex numbers can also be displayed in @samp{@var{a}+@var{b}i}
notation; @pxref{Complex Formats}.

Polar complex numbers are displayed in the form 
@texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'
@infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'
where @var{r} is the nonnegative magnitude and 
@texline @math{\theta}
@infoline @var{theta} 
is the argument or phase angle.  The range of 
@texline @math{\theta}
@infoline @var{theta} 
depends on the current angular mode (@pxref{Angular Modes}); it is
generally between @mathit{-180} and @mathit{+180} degrees or the equivalent range
in radians. 

Complex numbers are entered in stages using incomplete objects.
@xref{Incomplete Objects}.

Operations on rectangular complex numbers yield rectangular complex
results, and similarly for polar complex numbers.  Where the two types
are mixed, or where new complex numbers arise (as for the square root of
a negative real), the current @dfn{Polar mode} is used to determine the
type.  @xref{Polar Mode}.

A complex result in which the imaginary part is zero (or the phase angle
is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
number.

@node Infinities, Vectors and Matrices, Complex Numbers, Data Types
@section Infinities

@noindent
@cindex Infinity
@cindex @code{inf} variable
@cindex @code{uinf} variable
@cindex @code{nan} variable
@vindex inf
@vindex uinf
@vindex nan
The word @code{inf} represents the mathematical concept of @dfn{infinity}.
Calc actually has three slightly different infinity-like values:
@code{inf}, @code{uinf}, and @code{nan}.  These are just regular
variable names (@pxref{Variables}); you should avoid using these
names for your own variables because Calc gives them special
treatment.  Infinities, like all variable names, are normally
entered using algebraic entry.

Mathematically speaking, it is not rigorously correct to treat
``infinity'' as if it were a number, but mathematicians often do
so informally.  When they say that @samp{1 / inf = 0}, what they
really mean is that @expr{1 / x}, as @expr{x} becomes larger and
larger, becomes arbitrarily close to zero.  So you can imagine
that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
would go all the way to zero.  Similarly, when they say that
@samp{exp(inf) = inf}, they mean that 
@texline @math{e^x}
@infoline @expr{exp(x)} 
grows without bound as @expr{x} grows.  The symbol @samp{-inf} likewise
stands for an infinitely negative real value; for example, we say that
@samp{exp(-inf) = 0}.  You can have an infinity pointing in any
direction on the complex plane:  @samp{sqrt(-inf) = i inf}.

The same concept of limits can be used to define @expr{1 / 0}.  We
really want the value that @expr{1 / x} approaches as @expr{x}
approaches zero.  But if all we have is @expr{1 / 0}, we can't
tell which direction @expr{x} was coming from.  If @expr{x} was
positive and decreasing toward zero, then we should say that
@samp{1 / 0 = inf}.  But if @expr{x} was negative and increasing
toward zero, the answer is @samp{1 / 0 = -inf}.  In fact, @expr{x}
could be an imaginary number, giving the answer @samp{i inf} or
@samp{-i inf}.  Calc uses the special symbol @samp{uinf} to mean
@dfn{undirected infinity}, i.e., a value which is infinitely
large but with an unknown sign (or direction on the complex plane).

Calc actually has three modes that say how infinities are handled.
Normally, infinities never arise from calculations that didn't
already have them.  Thus, @expr{1 / 0} is treated simply as an
error and left unevaluated.  The @kbd{m i} (@code{calc-infinite-mode})
command (@pxref{Infinite Mode}) enables a mode in which
@expr{1 / 0} evaluates to @code{uinf} instead.  There is also
an alternative type of infinite mode which says to treat zeros
as if they were positive, so that @samp{1 / 0 = inf}.  While this
is less mathematically correct, it may be the answer you want in
some cases.

Since all infinities are ``as large'' as all others, Calc simplifies,
e.g., @samp{5 inf} to @samp{inf}.  Another example is
@samp{5 - inf = -inf}, where the @samp{-inf} is so large that
adding a finite number like five to it does not affect it.
Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
that variables like @code{a} always stand for finite quantities.
Just to show that infinities really are all the same size,
note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
notation.

It's not so easy to define certain formulas like @samp{0 * inf} and
@samp{inf / inf}.  Depending on where these zeros and infinities
came from, the answer could be literally anything.  The latter
formula could be the limit of @expr{x / x} (giving a result of one),
or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
or @expr{x / x^2} (giving zero).  Calc uses the symbol @code{nan}
to represent such an @dfn{indeterminate} value.  (The name ``nan''
comes from analogy with the ``NAN'' concept of IEEE standard
arithmetic; it stands for ``Not A Number.''  This is somewhat of a
misnomer, since @code{nan} @emph{does} stand for some number or
infinity, it's just that @emph{which} number it stands for
cannot be determined.)  In Calc's notation, @samp{0 * inf = nan}
and @samp{inf / inf = nan}.  A few other common indeterminate
expressions are @samp{inf - inf} and @samp{inf ^ 0}.  Also,
@samp{0 / 0 = nan} if you have turned on Infinite mode
(as described above).

Infinities are especially useful as parts of @dfn{intervals}.
@xref{Interval Forms}.

@node Vectors and Matrices, Strings, Infinities, Data Types
@section Vectors and Matrices

@noindent
@cindex Vectors
@cindex Plain vectors
@cindex Matrices
The @dfn{vector} data type is flexible and general.  A vector is simply a
list of zero or more data objects.  When these objects are numbers, the
whole is a vector in the mathematical sense.  When these objects are
themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
A vector which is not a matrix is referred to here as a @dfn{plain vector}.

A vector is displayed as a list of values separated by commas and enclosed
in square brackets:  @samp{[1, 2, 3]}.  Thus the following is a 2 row by
3 column matrix:  @samp{[[1, 2, 3], [4, 5, 6]]}.  Vectors, like complex
numbers, are entered as incomplete objects.  @xref{Incomplete Objects}.
During algebraic entry, vectors are entered all at once in the usual
brackets-and-commas form.  Matrices may be entered algebraically as nested
vectors, or using the shortcut notation @w{@samp{[1, 2, 3; 4, 5, 6]}},
with rows separated by semicolons.  The commas may usually be omitted
when entering vectors:  @samp{[1 2 3]}.  Curly braces may be used in
place of brackets: @samp{@{1, 2, 3@}}, but the commas are required in
this case.

Traditional vector and matrix arithmetic is also supported;
@pxref{Basic Arithmetic} and @pxref{Matrix Functions}.
Many other operations are applied to vectors element-wise.  For example,
the complex conjugate of a vector is a vector of the complex conjugates
of its elements.

@ignore
@starindex
@end ignore
@tindex vec
Algebraic functions for building vectors include @samp{vec(a, b, c)}
to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an 
@texline @math{n\times m}
@infoline @var{n}x@var{m}
matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
from 1 to @samp{n}.

@node Strings, HMS Forms, Vectors and Matrices, Data Types
@section Strings

@noindent
@kindex "
@cindex Strings
@cindex Character strings
Character strings are not a special data type in the Calculator.
Rather, a string is represented simply as a vector all of whose
elements are integers in the range 0 to 255 (ASCII codes).  You can
enter a string at any time by pressing the @kbd{"} key.  Quotation
marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
inside strings.  Other notations introduced by backslashes are:

@example
@group
\a     7          \^@@    0
\b     8          \^a-z  1-26
\e     27         \^[    27
\f     12         \^\\   28
\n     10         \^]    29
\r     13         \^^    30
\t     9          \^_    31
                  \^?    127
@end group
@end example

@noindent
Finally, a backslash followed by three octal digits produces any
character from its ASCII code.

@kindex d "
@pindex calc-display-strings
Strings are normally displayed in vector-of-integers form.  The
@w{@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
which any vectors of small integers are displayed as quoted strings
instead.

The backslash notations shown above are also used for displaying
strings.  Characters 128 and above are not translated by Calc; unless
you have an Emacs modified for 8-bit fonts, these will show up in
backslash-octal-digits notation.  For characters below 32, and
for character 127, Calc uses the backslash-letter combination if
there is one, or otherwise uses a @samp{\^} sequence.

The only Calc feature that uses strings is @dfn{compositions};
@pxref{Compositions}.  Strings also provide a convenient
way to do conversions between ASCII characters and integers.

@ignore
@starindex
@end ignore
@tindex string
There is a @code{string} function which provides a different display
format for strings.  Basically, @samp{string(@var{s})}, where @var{s}
is a vector of integers in the proper range, is displayed as the
corresponding string of characters with no surrounding quotation
marks or other modifications.  Thus @samp{string("ABC")} (or
@samp{string([65 66 67])}) will look like @samp{ABC} on the stack.
This happens regardless of whether @w{@kbd{d "}} has been used.  The
only way to turn it off is to use @kbd{d U} (unformatted language
mode) which will display @samp{string("ABC")} instead.

Control characters are displayed somewhat differently by @code{string}.
Characters below 32, and character 127, are shown using @samp{^} notation
(same as shown above, but without the backslash).  The quote and
backslash characters are left alone, as are characters 128 and above.

@ignore
@starindex
@end ignore
@tindex bstring
The @code{bstring} function is just like @code{string} except that
the resulting string is breakable across multiple lines if it doesn't
fit all on one line.  Potential break points occur at every space
character in the string.

@node HMS Forms, Date Forms, Strings, Data Types
@section HMS Forms

@noindent
@cindex Hours-minutes-seconds forms
@cindex Degrees-minutes-seconds forms
@dfn{HMS} stands for Hours-Minutes-Seconds; when used as an angular
argument, the interpretation is Degrees-Minutes-Seconds.  All functions
that operate on angles accept HMS forms.  These are interpreted as
degrees regardless of the current angular mode.  It is also possible to
use HMS as the angular mode so that calculated angles are expressed in
degrees, minutes, and seconds.

@kindex @@
@ignore
@mindex @null
@end ignore
@kindex ' (HMS forms)
@ignore
@mindex @null
@end ignore
@kindex " (HMS forms)
@ignore
@mindex @null
@end ignore
@kindex h (HMS forms)
@ignore
@mindex @null
@end ignore
@kindex o (HMS forms)
@ignore
@mindex @null
@end ignore
@kindex m (HMS forms)
@ignore
@mindex @null
@end ignore
@kindex s (HMS forms)
The default format for HMS values is
@samp{@var{hours}@@ @var{mins}' @var{secs}"}.  During entry, the letters
@samp{h} (for ``hours'') or
@samp{o} (approximating the ``degrees'' symbol) are accepted as well as
@samp{@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
accepted in place of @samp{"}.
The @var{hours} value is an integer (or integer-valued float).
The @var{mins} value is an integer or integer-valued float between 0 and 59.
The @var{secs} value is a real number between 0 (inclusive) and 60
(exclusive).  A positive HMS form is interpreted as @var{hours} +
@var{mins}/60 + @var{secs}/3600.  A negative HMS form is interpreted
as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
Display format for HMS forms is quite flexible.  @xref{HMS Formats}.

HMS forms can be added and subtracted.  When they are added to numbers,
the numbers are interpreted according to the current angular mode.  HMS
forms can also be multiplied and divided by real numbers.  Dividing
two HMS forms produces a real-valued ratio of the two angles.

@pindex calc-time
@cindex Time of day
Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
the stack as an HMS form.

@node Date Forms, Modulo Forms, HMS Forms, Data Types
@section Date Forms

@noindent
@cindex Date forms
A @dfn{date form} represents a date and possibly an associated time.
Simple date arithmetic is supported:  Adding a number to a date
produces a new date shifted by that many days; adding an HMS form to
a date shifts it by that many hours.  Subtracting two date forms
computes the number of days between them (represented as a simple
number).  Many other operations, such as multiplying two date forms,
are nonsensical and are not allowed by Calc.

Date forms are entered and displayed enclosed in @samp{< >} brackets.
The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
Input is flexible; date forms can be entered in any of the usual
notations for dates and times.  @xref{Date Formats}.

Date forms are stored internally as numbers, specifically the number
of days since midnight on the morning of January 1 of the year 1 AD.
If the internal number is an integer, the form represents a date only;
if the internal number is a fraction or float, the form represents
a date and time.  For example, @samp{<6:00am Wed Jan 9, 1991>}
is represented by the number 726842.25.  The standard precision of
12 decimal digits is enough to ensure that a (reasonable) date and
time can be stored without roundoff error.

If the current precision is greater than 12, date forms will keep
additional digits in the seconds position.  For example, if the
precision is 15, the seconds will keep three digits after the
decimal point.  Decreasing the precision below 12 may cause the
time part of a date form to become inaccurate.  This can also happen
if astronomically high years are used, though this will not be an
issue in everyday (or even everymillennium) use.  Note that date
forms without times are stored as exact integers, so roundoff is
never an issue for them.

You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
(@code{calc-unpack}) commands to get at the numerical representation
of a date form.  @xref{Packing and Unpacking}.

Date forms can go arbitrarily far into the future or past.  Negative
year numbers represent years BC.  Calc uses a combination of the
Gregorian and Julian calendars, following the history of Great
Britain and the British colonies.  This is the same calendar that
is used by the @code{cal} program in most Unix implementations.

@cindex Julian calendar
@cindex Gregorian calendar
Some historical background:  The Julian calendar was created by
Julius Caesar in the year 46 BC as an attempt to fix the gradual
drift caused by the lack of leap years in the calendar used
until that time.  The Julian calendar introduced an extra day in
all years divisible by four.  After some initial confusion, the
calendar was adopted around the year we call 8 AD.  Some centuries
later it became apparent that the Julian year of 365.25 days was
itself not quite right.  In 1582 Pope Gregory XIII introduced the
Gregorian calendar, which added the new rule that years divisible
by 100, but not by 400, were not to be considered leap years
despite being divisible by four.  Many countries delayed adoption
of the Gregorian calendar because of religious differences;
in Britain it was put off until the year 1752, by which time
the Julian calendar had fallen eleven days behind the true
seasons.  So the switch to the Gregorian calendar in early
September 1752 introduced a discontinuity:  The day after
Sep 2, 1752 is Sep 14, 1752.  Calc follows this convention.
To take another example, Russia waited until 1918 before
adopting the new calendar, and thus needed to remove thirteen
days (between Feb 1, 1918 and Feb 14, 1918).  This means that
Calc's reckoning will be inconsistent with Russian history between
1752 and 1918, and similarly for various other countries.

Today's timekeepers introduce an occasional ``leap second'' as
well, but Calc does not take these minor effects into account.
(If it did, it would have to report a non-integer number of days
between, say, @samp{<12:00am Mon Jan 1, 1900>} and
@samp{<12:00am Sat Jan 1, 2000>}.)

Calc uses the Julian calendar for all dates before the year 1752,
including dates BC when the Julian calendar technically had not
yet been invented.  Thus the claim that day number @mathit{-10000} is
called ``August 16, 28 BC'' should be taken with a grain of salt.

Please note that there is no ``year 0''; the day before
@samp{<Sat Jan 1, +1>} is @samp{<Fri Dec 31, -1>}.  These are
days 0 and @mathit{-1} respectively in Calc's internal numbering scheme.

@cindex Julian day counting
Another day counting system in common use is, confusingly, also
called ``Julian.''  It was invented in 1583 by Joseph Justus
Scaliger, who named it in honor of his father Julius Caesar
Scaliger.  For obscure reasons he chose to start his day
numbering on Jan 1, 4713 BC at noon, which in Calc's scheme
is @mathit{-1721423.5} (recall that Calc starts at midnight instead
of noon).  Thus to convert a Calc date code obtained by
unpacking a date form into a Julian day number, simply add
1721423.5.  The Julian code for @samp{6:00am Jan 9, 1991}
is 2448265.75.  The built-in @kbd{t J} command performs
this conversion for you.

@cindex Unix time format
The Unix operating system measures time as an integer number of
seconds since midnight, Jan 1, 1970.  To convert a Calc date
value into a Unix time stamp, first subtract 719164 (the code
for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
seconds in a day) and press @kbd{R} to round to the nearest
integer.  If you have a date form, you can simply subtract the
day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
719164.  Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
to convert from Unix time to a Calc date form.  (Note that
Unix normally maintains the time in the GMT time zone; you may
need to subtract five hours to get New York time, or eight hours
for California time.  The same is usually true of Julian day
counts.)  The built-in @kbd{t U} command performs these
conversions.

@node Modulo Forms, Error Forms, Date Forms, Data Types
@section Modulo Forms

@noindent
@cindex Modulo forms
A @dfn{modulo form} is a real number which is taken modulo (i.e., within
an integer multiple of) some value @var{M}.  Arithmetic modulo @var{M}
often arises in number theory.  Modulo forms are written
`@var{a} @tfn{mod} @var{M}',
where @var{a} and @var{M} are real numbers or HMS forms, and
@texline @math{0 \le a < M}.
@infoline @expr{0 <= a < @var{M}}.
In many applications @expr{a} and @expr{M} will be
integers but this is not required.

@ignore
@mindex M
@end ignore
@kindex M (modulo forms)
@ignore
@mindex mod
@end ignore
@tindex mod (operator)
To create a modulo form during numeric entry, press the shift-@kbd{M}
key to enter the word @samp{mod}.  As a special convenience, pressing
shift-@kbd{M} a second time automatically enters the value of @expr{M}
that was most recently used before.  During algebraic entry, either
type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}).
Once again, pressing this a second time enters the current modulo.

Modulo forms are not to be confused with the modulo operator @samp{%}.
The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
the result 7.  Further computations treat this 7 as just a regular integer.
The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
further computations with this value are again reduced modulo 10 so that
the result always lies in the desired range.

When two modulo forms with identical @expr{M}'s are added or multiplied,
the Calculator simply adds or multiplies the values, then reduces modulo
@expr{M}.  If one argument is a modulo form and the other a plain number,
the plain number is treated like a compatible modulo form.  It is also
possible to raise modulo forms to powers; the result is the value raised
to the power, then reduced modulo @expr{M}.  (When all values involved
are integers, this calculation is done much more efficiently than
actually computing the power and then reducing.)

@cindex Modulo division
Two modulo forms `@var{a} @tfn{mod} @var{M}' and `@var{b} @tfn{mod} @var{M}'
can be divided if @expr{a}, @expr{b}, and @expr{M} are all
integers.  The result is the modulo form which, when multiplied by
`@var{b} @tfn{mod} @var{M}', produces `@var{a} @tfn{mod} @var{M}'.  If
there is no solution to this equation (which can happen only when
@expr{M} is non-prime), or if any of the arguments are non-integers, the
division is left in symbolic form.  Other operations, such as square
roots, are not yet supported for modulo forms.  (Note that, although
@w{`@tfn{(}@var{a} @tfn{mod} @var{M}@tfn{)^.5}'} will compute a ``modulo square root''
in the sense of reducing 
@texline @math{\sqrt a}
@infoline @expr{sqrt(a)} 
modulo @expr{M}, this is not a useful definition from the
number-theoretical point of view.)

It is possible to mix HMS forms and modulo forms.  For example, an
HMS form modulo 24 could be used to manipulate clock times; an HMS
form modulo 360 would be suitable for angles.  Making the modulo @expr{M}
also be an HMS form eliminates troubles that would arise if the angular
mode were inadvertently set to Radians, in which case
@w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo
24 radians!

Modulo forms cannot have variables or formulas for components.  If you
enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
to each of the coefficients:  @samp{(1 mod 5) x + (2 mod 5)}.

You can use @kbd{v p} and @kbd{%} to modify modulo forms.
@xref{Packing and Unpacking}.  @xref{Basic Arithmetic}.

@ignore
@starindex
@end ignore
@tindex makemod
The algebraic function @samp{makemod(a, m)} builds the modulo form
@w{@samp{a mod m}}.

@node Error Forms, Interval Forms, Modulo Forms, Data Types
@section Error Forms

@noindent
@cindex Error forms
@cindex Standard deviations
An @dfn{error form} is a number with an associated standard
deviation, as in @samp{2.3 +/- 0.12}.  The notation
@texline `@var{x} @tfn{+/-} @math{\sigma}' 
@infoline `@var{x} @tfn{+/-} sigma' 
stands for an uncertain value which follows
a normal or Gaussian distribution of mean @expr{x} and standard
deviation or ``error'' 
@texline @math{\sigma}.
@infoline @expr{sigma}.
Both the mean and the error can be either numbers or
formulas.  Generally these are real numbers but the mean may also be
complex.  If the error is negative or complex, it is changed to its
absolute value.  An error form with zero error is converted to a
regular number by the Calculator.

All arithmetic and transcendental functions accept error forms as input.
Operations on the mean-value part work just like operations on regular
numbers.  The error part for any function @expr{f(x)} (such as 
@texline @math{\sin x}
@infoline @expr{sin(x)})
is defined by the error of @expr{x} times the derivative of @expr{f}
evaluated at the mean value of @expr{x}.  For a two-argument function
@expr{f(x,y)} (such as addition) the error is the square root of the sum
of the squares of the errors due to @expr{x} and @expr{y}.
@tex
$$ \eqalign{
  f(x \hbox{\code{ +/- }} \sigma)
    &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
  f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
    &= f(x,y) \hbox{\code{ +/- }}
        \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
                             \right| \right)^2
             +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
                             \right| \right)^2 } \cr
} $$
@end tex
Note that this
definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
is not the same as @samp{(2 +/- 1)^2}; the former represents the product
of two independent values which happen to have the same probability
distributions, and the latter is the product of one random value with itself.
The former will produce an answer with less error, since on the average
the two independent errors can be expected to cancel out.

Consult a good text on error analysis for a discussion of the proper use
of standard deviations.  Actual errors often are neither Gaussian-distributed
nor uncorrelated, and the above formulas are valid only when errors
are small.  As an example, the error arising from
@texline `@tfn{sin(}@var{x} @tfn{+/-} @math{\sigma}@tfn{)}' 
@infoline `@tfn{sin(}@var{x} @tfn{+/-} @var{sigma}@tfn{)}' 
is 
@texline `@math{\sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.  
@infoline `@var{sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.  
When @expr{x} is close to zero,
@texline @math{\cos x}
@infoline @expr{cos(x)} 
is close to one so the error in the sine is close to 
@texline @math{\sigma};
@infoline @expr{sigma};
this makes sense, since 
@texline @math{\sin x}
@infoline @expr{sin(x)} 
is approximately @expr{x} near zero, so a given error in @expr{x} will
produce about the same error in the sine.  Likewise, near 90 degrees
@texline @math{\cos x}
@infoline @expr{cos(x)} 
is nearly zero and so the computed error is
small:  The sine curve is nearly flat in that region, so an error in @expr{x}
has relatively little effect on the value of 
@texline @math{\sin x}.
@infoline @expr{sin(x)}.  
However, consider @samp{sin(90 +/- 1000)}.  The cosine of 90 is zero, so
Calc will report zero error!  We get an obviously wrong result because
we have violated the small-error approximation underlying the error
analysis.  If the error in @expr{x} had been small, the error in
@texline @math{\sin x}
@infoline @expr{sin(x)} 
would indeed have been negligible.

@ignore
@mindex p
@end ignore
@kindex p (error forms)
@tindex +/-
To enter an error form during regular numeric entry, use the @kbd{p}
(``plus-or-minus'') key to type the @samp{+/-} symbol.  (If you try actually
typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
@kbd{+} command!)  Within an algebraic formula, you can press @kbd{M-+} to
type the @samp{+/-} symbol, or type it out by hand.

Error forms and complex numbers can be mixed; the formulas shown above
are used for complex numbers, too; note that if the error part evaluates
to a complex number its absolute value (or the square root of the sum of
the squares of the absolute values of the two error contributions) is
used.  Mathematically, this corresponds to a radially symmetric Gaussian
distribution of numbers on the complex plane.  However, note that Calc
considers an error form with real components to represent a real number,
not a complex distribution around a real mean.

Error forms may also be composed of HMS forms.  For best results, both
the mean and the error should be HMS forms if either one is.

@ignore
@starindex
@end ignore
@tindex sdev
The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.

@node Interval Forms, Incomplete Objects, Error Forms, Data Types
@section Interval Forms

@noindent
@cindex Interval forms
An @dfn{interval} is a subset of consecutive real numbers.  For example,
the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
inclusive.  If you multiply it by the interval @samp{[0.5 ..@: 2]} you
obtain @samp{[1 ..@: 8]}.  This calculation represents the fact that if
you multiply some number in the range @samp{[2 ..@: 4]} by some other
number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
from 1 to 8.  Interval arithmetic is used to get a worst-case estimate
of the possible range of values a computation will produce, given the
set of possible values of the input.

@ifinfo
Calc supports several varieties of intervals, including @dfn{closed}
intervals of the type shown above, @dfn{open} intervals such as
@samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
@emph{exclusive}, and @dfn{semi-open} intervals in which one end
uses a round parenthesis and the other a square bracket.  In mathematical
terms,
@samp{[2 ..@: 4]} means @expr{2 <= x <= 4}, whereas
@samp{[2 ..@: 4)} represents @expr{2 <= x < 4},
@samp{(2 ..@: 4]} represents @expr{2 < x <= 4}, and
@samp{(2 ..@: 4)} represents @expr{2 < x < 4}.
@end ifinfo
@tex
Calc supports several varieties of intervals, including \dfn{closed}
intervals of the type shown above, \dfn{open} intervals such as
\samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
\emph{exclusive}, and \dfn{semi-open} intervals in which one end
uses a round parenthesis and the other a square bracket.  In mathematical
terms,
$$ \eqalign{
   [2 \hbox{\cite{..}} 4]  &\quad\hbox{means}\quad  2 \le x \le 4  \cr
   [2 \hbox{\cite{..}} 4)  &\quad\hbox{means}\quad  2 \le x  <  4  \cr
   (2 \hbox{\cite{..}} 4]  &\quad\hbox{means}\quad  2  <  x \le 4  \cr
   (2 \hbox{\cite{..}} 4)  &\quad\hbox{means}\quad  2  <  x  <  4  \cr
} $$
@end tex

The lower and upper limits of an interval must be either real numbers
(or HMS or date forms), or symbolic expressions which are assumed to be
real-valued, or @samp{-inf} and @samp{inf}.  In general the lower limit
must be less than the upper limit.  A closed interval containing only
one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
automatically.  An interval containing no values at all (such as
@samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
guaranteed to behave well when used in arithmetic.  Note that the
interval @samp{[3 .. inf)} represents all real numbers greater than
or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
In fact, @samp{[-inf .. inf]} represents all real numbers including
the real infinities.

Intervals are entered in the notation shown here, either as algebraic
formulas, or using incomplete forms.  (@xref{Incomplete Objects}.)
In algebraic formulas, multiple periods in a row are collected from
left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
rather than @samp{1 ..@: 0.1e2}.  Add spaces or zeros if you want to
get the other interpretation.  If you omit the lower or upper limit,
a default of @samp{-inf} or @samp{inf} (respectively) is furnished.

Infinite mode also affects operations on intervals
(@pxref{Infinities}).  Calc will always introduce an open infinity,
as in @samp{1 / (0 .. 2] = [0.5 .. inf)}.  But closed infinities,
@w{@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
otherwise they are left unevaluated.  Note that the ``direction'' of
a zero is not an issue in this case since the zero is always assumed
to be continuous with the rest of the interval.  For intervals that
contain zero inside them Calc is forced to give the result,
@samp{1 / (-2 .. 2) = [-inf .. inf]}.

While it may seem that intervals and error forms are similar, they are
based on entirely different concepts of inexact quantities.  An error
form 
@texline `@var{x} @tfn{+/-} @math{\sigma}' 
@infoline `@var{x} @tfn{+/-} @var{sigma}' 
means a variable is random, and its value could
be anything but is ``probably'' within one 
@texline @math{\sigma} 
@infoline @var{sigma} 
of the mean value @expr{x}. An interval 
`@tfn{[}@var{a} @tfn{..@:} @var{b}@tfn{]}' means a
variable's value is unknown, but guaranteed to lie in the specified
range.  Error forms are statistical or ``average case'' approximations;
interval arithmetic tends to produce ``worst case'' bounds on an
answer.

Intervals may not contain complex numbers, but they may contain
HMS forms or date forms.

@xref{Set Operations}, for commands that interpret interval forms
as subsets of the set of real numbers.

@ignore
@starindex
@end ignore
@tindex intv
The algebraic function @samp{intv(n, a, b)} builds an interval form
from @samp{a} to @samp{b}; @samp{n} is an integer code which must
be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
3 for @samp{[..]}.

Please note that in fully rigorous interval arithmetic, care would be
taken to make sure that the computation of the lower bound rounds toward
minus infinity, while upper bound computations round toward plus
infinity.  Calc's arithmetic always uses a round-to-nearest mode,
which means that roundoff errors could creep into an interval
calculation to produce intervals slightly smaller than they ought to
be.  For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
should yield the interval @samp{[1..2]} again, but in fact it yields the
(slightly too small) interval @samp{[1..1.9999999]} due to roundoff
error.

@node Incomplete Objects, Variables, Interval Forms, Data Types
@section Incomplete Objects

@noindent
@ignore
@mindex [ ]
@end ignore
@kindex [
@ignore
@mindex ( )
@end ignore
@kindex (
@kindex ,
@ignore
@mindex @null
@end ignore
@kindex ]
@ignore
@mindex @null
@end ignore
@kindex )
@cindex Incomplete vectors
@cindex Incomplete complex numbers
@cindex Incomplete interval forms
When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
vector, respectively, the effect is to push an @dfn{incomplete} complex
number or vector onto the stack.  The @kbd{,} key adds the value(s) at
the top of the stack onto the current incomplete object.  The @kbd{)}
and @kbd{]} keys ``close'' the incomplete object after adding any values
on the top of the stack in front of the incomplete object.

As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
pushes the vector @samp{[2, 6, 9]} onto the stack.  Likewise, @kbd{( 1 , 2 Q )}
pushes the complex number @samp{(1, 1.414)} (approximately).

If several values lie on the stack in front of the incomplete object,
all are collected and appended to the object.  Thus the @kbd{,} key
is redundant:  @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}.  Some people
prefer the equivalent @key{SPC} key to @key{RET}.

As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
@kbd{,} adds a zero or duplicates the preceding value in the list being
formed.  Typing @key{DEL} during incomplete entry removes the last item
from the list.

@kindex ;
The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
numbers:  @kbd{( 1 ; 2 )}.  When entering a vector, @kbd{;} is useful for
creating a matrix.  In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.

@kindex ..
@pindex calc-dots
Incomplete entry is also used to enter intervals.  For example,
@kbd{[ 2 ..@: 4 )} enters a semi-open interval.  Note that when you type
the first period, it will be interpreted as a decimal point, but when
you type a second period immediately afterward, it is re-interpreted as
part of the interval symbol.  Typing @kbd{..} corresponds to executing
the @code{calc-dots} command.

If you find incomplete entry distracting, you may wish to enter vectors
and complex numbers as algebraic formulas by pressing the apostrophe key.

@node Variables, Formulas, Incomplete Objects, Data Types
@section Variables

@noindent
@cindex Variables, in formulas
A @dfn{variable} is somewhere between a storage register on a conventional
calculator, and a variable in a programming language.  (In fact, a Calc
variable is really just an Emacs Lisp variable that contains a Calc number
or formula.)  A variable's name is normally composed of letters and digits.
Calc also allows apostrophes and @code{#} signs in variable names.
(The Calc variable @code{foo} corresponds to the Emacs Lisp variable
@code{var-foo}, but unless you access the variable from within Emacs
Lisp, you don't need to worry about it.  Variable names in algebraic
formulas implicitly have @samp{var-} prefixed to their names.  The
@samp{#} character in variable names used in algebraic formulas
corresponds to a dash @samp{-} in the Lisp variable name.  If the name
contains any dashes, the prefix @samp{var-} is @emph{not} automatically
added.  Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
refer to the same variable.)

In a command that takes a variable name, you can either type the full
name of a variable, or type a single digit to use one of the special
convenience variables @code{q0} through @code{q9}.  For example,
@kbd{3 s s 2} stores the number 3 in variable @code{q2}, and
@w{@kbd{3 s s foo @key{RET}}} stores that number in variable
@code{foo}.

To push a variable itself (as opposed to the variable's value) on the
stack, enter its name as an algebraic expression using the apostrophe
(@key{'}) key.

@kindex =
@pindex calc-evaluate
@cindex Evaluation of variables in a formula
@cindex Variables, evaluation
@cindex Formulas, evaluation
The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
replacing all variables in the formula which have been given values by a
@code{calc-store} or @code{calc-let} command by their stored values.
Other variables are left alone.  Thus a variable that has not been
stored acts like an abstract variable in algebra; a variable that has
been stored acts more like a register in a traditional calculator.
With a positive numeric prefix argument, @kbd{=} evaluates the top
@var{n} stack entries; with a negative argument, @kbd{=} evaluates
the @var{n}th stack entry.

@cindex @code{e} variable
@cindex @code{pi} variable
@cindex @code{i} variable
@cindex @code{phi} variable
@cindex @code{gamma} variable
@vindex e
@vindex pi
@vindex i
@vindex phi
@vindex gamma
A few variables are called @dfn{special constants}.  Their names are
@samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
(@xref{Scientific Functions}.)  When they are evaluated with @kbd{=},
their values are calculated if necessary according to the current precision
or complex polar mode.  If you wish to use these symbols for other purposes,
simply undefine or redefine them using @code{calc-store}.

The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
infinite or indeterminate values.  It's best not to use them as
regular variables, since Calc uses special algebraic rules when
it manipulates them.  Calc displays a warning message if you store
a value into any of these special variables.

@xref{Store and Recall}, for a discussion of commands dealing with variables.

@node Formulas,  , Variables, Data Types
@section Formulas

@noindent
@cindex Formulas
@cindex Expressions
@cindex Operators in formulas
@cindex Precedence of operators
When you press the apostrophe key you may enter any expression or formula
in algebraic form.  (Calc uses the terms ``expression'' and ``formula''
interchangeably.)  An expression is built up of numbers, variable names,
and function calls, combined with various arithmetic operators.
Parentheses may
be used to indicate grouping.  Spaces are ignored within formulas, except
that spaces are not permitted within variable names or numbers.
Arithmetic operators, in order from highest to lowest precedence, and
with their equivalent function names, are:

@samp{_} [@code{subscr}] (subscripts);

postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25});

prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x})
and prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x});

@samp{+/-} [@code{sdev}] (the standard deviation symbol) and
@samp{mod} [@code{makemod}] (the symbol for modulo forms);

postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!})
and postfix @samp{!!} [@code{dfact}] (double factorial);

@samp{^} [@code{pow}] (raised-to-the-power-of);

@samp{*} [@code{mul}];

@samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and
@samp{\} [@code{idiv}] (integer division);

infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y});

@samp{|} [@code{vconcat}] (vector concatenation);

relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}],
@samp{>} [@code{gt}], @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}];

@samp{&&} [@code{land}] (logical ``and'');

@samp{||} [@code{lor}] (logical ``or'');

the C-style ``if'' operator @samp{a?b:c} [@code{if}];

@samp{!!!} [@code{pnot}] (rewrite pattern ``not'');

@samp{&&&} [@code{pand}] (rewrite pattern ``and'');

@samp{|||} [@code{por}] (rewrite pattern ``or'');

@samp{:=} [@code{assign}] (for assignments and rewrite rules);

@samp{::} [@code{condition}] (rewrite pattern condition);

@samp{=>} [@code{evalto}].

Note that, unlike in usual computer notation, multiplication binds more
strongly than division:  @samp{a*b/c*d} is equivalent to 
@texline @math{a b \over c d}.
@infoline @expr{(a*b)/(c*d)}.

@cindex Multiplication, implicit
@cindex Implicit multiplication
The multiplication sign @samp{*} may be omitted in many cases.  In particular,
if the righthand side is a number, variable name, or parenthesized
expression, the @samp{*} may be omitted.  Implicit multiplication has the
same precedence as the explicit @samp{*} operator.  The one exception to
the rule is that a variable name followed by a parenthesized expression,
as in @samp{f(x)},
is interpreted as a function call, not an implicit @samp{*}.  In many
cases you must use a space if you omit the @samp{*}:  @samp{2a} is the
same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
is a variable called @code{ab}, @emph{not} the product of @samp{a} and
@samp{b}!  Also note that @samp{f (x)} is still a function call.

@cindex Implicit comma in vectors
The rules are slightly different for vectors written with square brackets.
In vectors, the space character is interpreted (like the comma) as a
separator of elements of the vector.  Thus @w{@samp{[ 2a b+c d ]}} is
equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
to @samp{2*a*b + c*d}.
Note that spaces around the brackets, and around explicit commas, are
ignored.  To force spaces to be interpreted as multiplication you can
enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
interpreted as @samp{[a*b, 2*c*d]}.  An implicit comma is also inserted
between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.

Vectors that contain commas (not embedded within nested parentheses or
brackets) do not treat spaces specially:  @samp{[a b, 2 c d]} is a vector
of two elements.  Also, if it would be an error to treat spaces as
separators, but not otherwise, then Calc will ignore spaces:
@w{@samp{[a - b]}} is a vector of one element, but @w{@samp{[a -b]}} is
a vector of two elements.  Finally, vectors entered with curly braces
instead of square brackets do not give spaces any special treatment.
When Calc displays a vector that does not contain any commas, it will
insert parentheses if necessary to make the meaning clear:
@w{@samp{[(a b)]}}.

The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
or five modulo minus-two?  Calc always interprets the leftmost symbol as
an infix operator preferentially (modulo, in this case), so you would
need to write @samp{(5%)-2} to get the former interpretation.

@cindex Function call notation
A function call is, e.g., @samp{sin(1+x)}.  (The Calc algebraic function
@code{foo} corresponds to the Emacs Lisp function @code{calcFunc-foo},
but unless you access the function from within Emacs Lisp, you don't
need to worry about it.)  Most mathematical Calculator commands like
@code{calc-sin} have function equivalents like @code{sin}.
If no Lisp function is defined for a function called by a formula, the
call is left as it is during algebraic manipulation: @samp{f(x+y)} is
left alone.  Beware that many innocent-looking short names like @code{in}
and @code{re} have predefined meanings which could surprise you; however,
single letters or single letters followed by digits are always safe to
use for your own function names.  @xref{Function Index}.

In the documentation for particular commands, the notation @kbd{H S}
(@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the
command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
represent the same operation.

Commands that interpret (``parse'') text as algebraic formulas include
algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
the contents of the editing buffer when you finish, the @kbd{C-x * g}
and @w{@kbd{C-x * r}} commands, the @kbd{C-y} command, the X window system
``paste'' mouse operation, and Embedded mode.  All of these operations
use the same rules for parsing formulas; in particular, language modes
(@pxref{Language Modes}) affect them all in the same way.

When you read a large amount of text into the Calculator (say a vector
which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
you may wish to include comments in the text.  Calc's formula parser
ignores the symbol @samp{%%} and anything following it on a line:

@example
[ a + b,   %% the sum of "a" and "b"
  c + d,
  %% last line is coming up:
  e + f ]
@end example

@noindent
This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.

@xref{Syntax Tables}, for a way to create your own operators and other
input notations.  @xref{Compositions}, for a way to create new display
formats.

@xref{Algebra}, for commands for manipulating formulas symbolically.

@node Stack and Trail, Mode Settings, Data Types, Top
@chapter Stack and Trail Commands

@noindent
This chapter describes the Calc commands for manipulating objects on the
stack and in the trail buffer.  (These commands operate on objects of any
type, such as numbers, vectors, formulas, and incomplete objects.)

@menu
* Stack Manipulation::
* Editing Stack Entries::
* Trail Commands::
* Keep Arguments::
@end menu

@node Stack Manipulation, Editing Stack Entries, Stack and Trail, Stack and Trail
@section Stack Manipulation Commands

@noindent
@kindex @key{RET}
@kindex @key{SPC}
@pindex calc-enter
@cindex Duplicating stack entries
To duplicate the top object on the stack, press @key{RET} or @key{SPC}
(two equivalent keys for the @code{calc-enter} command).
Given a positive numeric prefix argument, these commands duplicate
several elements at the top of the stack.
Given a negative argument,
these commands duplicate the specified element of the stack.
Given an argument of zero, they duplicate the entire stack.
For example, with @samp{10 20 30} on the stack,
@key{RET} creates @samp{10 20 30 30},
@kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30},
@kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and
@kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.

@kindex @key{LFD}
@pindex calc-over
The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
have it, else on @kbd{C-j}) is like @code{calc-enter}
except that the sign of the numeric prefix argument is interpreted
oppositely.  Also, with no prefix argument the default argument is 2.
Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
@samp{10 20 30 20}.

@kindex @key{DEL}
@kindex C-d
@pindex calc-pop
@cindex Removing stack entries
@cindex Deleting stack entries
To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
The @kbd{C-d} key is a synonym for @key{DEL}.
(If the top element is an incomplete object with at least one element, the
last element is removed from it.)  Given a positive numeric prefix argument,
several elements are removed.  Given a negative argument, the specified
element of the stack is deleted.  Given an argument of zero, the entire
stack is emptied.
For example, with @samp{10 20 30} on the stack,
@key{DEL} leaves @samp{10 20},
@kbd{C-u 2 @key{DEL}} leaves @samp{10},
@kbd{C-u - 2 @key{DEL}} leaves @samp{10 30}, and
@kbd{C-u 0 @key{DEL}} leaves an empty stack.

@kindex M-@key{DEL}
@pindex calc-pop-above
The @kbd{M-@key{DEL}} (@code{calc-pop-above}) command is to @key{DEL} what
@key{LFD} is to @key{RET}:  It interprets the sign of the numeric
prefix argument in the opposite way, and the default argument is 2.
Thus @kbd{M-@key{DEL}} by itself removes the second-from-top stack element,
leaving the first, third, fourth, and so on; @kbd{M-3 M-@key{DEL}} deletes
the third stack element.

@kindex @key{TAB}
@pindex calc-roll-down
To exchange the top two elements of the stack, press @key{TAB}
(@code{calc-roll-down}).  Given a positive numeric prefix argument, the
specified number of elements at the top of the stack are rotated downward.
Given a negative argument, the entire stack is rotated downward the specified
number of times.  Given an argument of zero, the entire stack is reversed
top-for-bottom.
For example, with @samp{10 20 30 40 50} on the stack,
@key{TAB} creates @samp{10 20 30 50 40},
@kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40},
@kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
@kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.

@kindex M-@key{TAB}
@pindex calc-roll-up
The command @kbd{M-@key{TAB}} (@code{calc-roll-up}) is analogous to @key{TAB}
except that it rotates upward instead of downward.  Also, the default
with no prefix argument is to rotate the top 3 elements.
For example, with @samp{10 20 30 40 50} on the stack,
@kbd{M-@key{TAB}} creates @samp{10 20 40 50 30},
@kbd{C-u 4 M-@key{TAB}} creates @samp{10 30 40 50 20},
@kbd{C-u - 2 M-@key{TAB}} creates @samp{30 40 50 10 20}, and
@kbd{C-u 0 M-@key{TAB}} creates @samp{50 40 30 20 10}.

A good way to view the operation of @key{TAB} and @kbd{M-@key{TAB}} is in
terms of moving a particular element to a new position in the stack.
With a positive argument @var{n}, @key{TAB} moves the top stack
element down to level @var{n}, making room for it by pulling all the
intervening stack elements toward the top.  @kbd{M-@key{TAB}} moves the
element at level @var{n} up to the top.  (Compare with @key{LFD},
which copies instead of moving the element in level @var{n}.)

With a negative argument @mathit{-@var{n}}, @key{TAB} rotates the stack
to move the object in level @var{n} to the deepest place in the
stack, and the object in level @mathit{@var{n}+1} to the top.  @kbd{M-@key{TAB}}
rotates the deepest stack element to be in level @mathit{n}, also
putting the top stack element in level @mathit{@var{n}+1}.

@xref{Selecting Subformulas}, for a way to apply these commands to
any portion of a vector or formula on the stack.

@node Editing Stack Entries, Trail Commands, Stack Manipulation, Stack and Trail
@section Editing Stack Entries

@noindent
@kindex `
@pindex calc-edit
@pindex calc-edit-finish
@cindex Editing the stack with Emacs
The backquote, @kbd{`} (@code{calc-edit}) command creates a temporary
buffer (@samp{*Calc Edit*}) for editing the top-of-stack value using
regular Emacs commands.  With a numeric prefix argument, it edits the
specified number of stack entries at once.  (An argument of zero edits
the entire stack; a negative argument edits one specific stack entry.)

When you are done editing, press @kbd{C-c C-c} to finish and return
to Calc.  The @key{RET} and @key{LFD} keys also work to finish most
sorts of editing, though in some cases Calc leaves @key{RET} with its
usual meaning (``insert a newline'') if it's a situation where you
might want to insert new lines into the editing buffer.

When you finish editing, the Calculator parses the lines of text in
the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
original stack elements in the original buffer with these new values,
then kills the @samp{*Calc Edit*} buffer.  The original Calculator buffer
continues to exist during editing, but for best results you should be
careful not to change it until you have finished the edit.  You can
also cancel the edit by killing the buffer with @kbd{C-x k}.

The formula is normally reevaluated as it is put onto the stack.
For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
@kbd{C-c C-c} will push 5 on the stack.  If you use @key{LFD} to
finish, Calc will put the result on the stack without evaluating it.

If you give a prefix argument to @kbd{C-c C-c},
Calc will not kill the @samp{*Calc Edit*} buffer.  You can switch
back to that buffer and continue editing if you wish.  However, you
should understand that if you initiated the edit with @kbd{`}, the
@kbd{C-c C-c} operation will be programmed to replace the top of the
stack with the new edited value, and it will do this even if you have
rearranged the stack in the meanwhile.  This is not so much of a problem
with other editing commands, though, such as @kbd{s e}
(@code{calc-edit-variable}; @pxref{Operations on Variables}).

If the @code{calc-edit} command involves more than one stack entry,
each line of the @samp{*Calc Edit*} buffer is interpreted as a
separate formula.  Otherwise, the entire buffer is interpreted as
one formula, with line breaks ignored.  (You can use @kbd{C-o} or
@kbd{C-q C-j} to insert a newline in the buffer without pressing @key{RET}.)

The @kbd{`} key also works during numeric or algebraic entry.  The
text entered so far is moved to the @code{*Calc Edit*} buffer for
more extensive editing than is convenient in the minibuffer.

@node Trail Commands, Keep Arguments, Editing Stack Entries, Stack and Trail
@section Trail Commands

@noindent
@cindex Trail buffer
The commands for manipulating the Calc Trail buffer are two-key sequences
beginning with the @kbd{t} prefix.

@kindex t d
@pindex calc-trail-display
The @kbd{t d} (@code{calc-trail-display}) command turns display of the
trail on and off.  Normally the trail display is toggled on if it was off,
off if it was on.  With a numeric prefix of zero, this command always
turns the trail off; with a prefix of one, it always turns the trail on.
The other trail-manipulation commands described here automatically turn
the trail on.  Note that when the trail is off values are still recorded
there; they are simply not displayed.  To set Emacs to turn the trail
off by default, type @kbd{t d} and then save the mode settings with
@kbd{m m} (@code{calc-save-modes}).

@kindex t i
@pindex calc-trail-in
@kindex t o
@pindex calc-trail-out
The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
(@code{calc-trail-out}) commands switch the cursor into and out of the
Calc Trail window.  In practice they are rarely used, since the commands
shown below are a more convenient way to move around in the
trail, and they work ``by remote control'' when the cursor is still
in the Calculator window.

@cindex Trail pointer
There is a @dfn{trail pointer} which selects some entry of the trail at
any given time.  The trail pointer looks like a @samp{>} symbol right
before the selected number.  The following commands operate on the
trail pointer in various ways.

@kindex t y
@pindex calc-trail-yank
@cindex Retrieving previous results
The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
the trail and pushes it onto the Calculator stack.  It allows you to
re-use any previously computed value without retyping.  With a numeric
prefix argument @var{n}, it yanks the value @var{n} lines above the current
trail pointer.

@kindex t <
@pindex calc-trail-scroll-left
@kindex t >
@pindex calc-trail-scroll-right
The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
(@code{calc-trail-scroll-right}) commands horizontally scroll the trail
window left or right by one half of its width.

@kindex t n
@pindex calc-trail-next
@kindex t p
@pindex calc-trail-previous
@kindex t f
@pindex calc-trail-forward
@kindex t b
@pindex calc-trail-backward
The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
(@code{calc-trail-previous)} commands move the trail pointer down or up
one line.  The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
(@code{calc-trail-backward}) commands move the trail pointer down or up
one screenful at a time.  All of these commands accept numeric prefix
arguments to move several lines or screenfuls at a time.

@kindex t [
@pindex calc-trail-first
@kindex t ]
@pindex calc-trail-last
@kindex t h
@pindex calc-trail-here
The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
(@code{calc-trail-last}) commands move the trail pointer to the first or
last line of the trail.  The @kbd{t h} (@code{calc-trail-here}) command
moves the trail pointer to the cursor position; unlike the other trail
commands, @kbd{t h} works only when Calc Trail is the selected window.

@kindex t s
@pindex calc-trail-isearch-forward
@kindex t r
@pindex calc-trail-isearch-backward
@ifinfo
The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
(@code{calc-trail-isearch-backward}) commands perform an incremental
search forward or backward through the trail.  You can press @key{RET}
to terminate the search; the trail pointer moves to the current line.
If you cancel the search with @kbd{C-g}, the trail pointer stays where
it was when the search began.
@end ifinfo
@tex
The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
(@code{calc-trail-isearch-backward}) com\-mands perform an incremental
search forward or backward through the trail.  You can press @key{RET}
to terminate the search; the trail pointer moves to the current line.
If you cancel the search with @kbd{C-g}, the trail pointer stays where
it was when the search began.
@end tex

@kindex t m
@pindex calc-trail-marker
The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
line of text of your own choosing into the trail.  The text is inserted
after the line containing the trail pointer; this usually means it is
added to the end of the trail.  Trail markers are useful mainly as the
targets for later incremental searches in the trail.

@kindex t k
@pindex calc-trail-kill
The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
from the trail.  The line is saved in the Emacs kill ring suitable for
yanking into another buffer, but it is not easy to yank the text back
into the trail buffer.  With a numeric prefix argument, this command
kills the @var{n} lines below or above the selected one.

The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
elsewhere; @pxref{Vector and Matrix Formats}.

@node Keep Arguments,  , Trail Commands, Stack and Trail
@section Keep Arguments

@noindent
@kindex K
@pindex calc-keep-args
The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
the following command.  It prevents that command from removing its
arguments from the stack.  For example, after @kbd{2 @key{RET} 3 +},
the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
the stack contains the arguments and the result: @samp{2 3 5}.

With the exception of keyboard macros, this works for all commands that
take arguments off the stack. (To avoid potentially unpleasant behavior,
a @kbd{K} prefix before a keyboard macro will be ignored.  A @kbd{K}
prefix called @emph{within} the keyboard macro will still take effect.)  
As another example, @kbd{K a s} simplifies a formula, pushing the
simplified version of the formula onto the stack after the original
formula (rather than replacing the original formula).  Note that you
could get the same effect by typing @kbd{@key{RET} a s}, copying the
formula and then simplifying the copy. One difference is that for a very
large formula the time taken to format the intermediate copy in
@kbd{@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
extra work. 

Even stack manipulation commands are affected.  @key{TAB} works by
popping two values and pushing them back in the opposite order,
so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.

A few Calc commands provide other ways of doing the same thing.
For example, @kbd{' sin($)} replaces the number on the stack with
its sine using algebraic entry; to push the sine and keep the
original argument you could use either @kbd{' sin($1)} or
@kbd{K ' sin($)}.  @xref{Algebraic Entry}.  Also, the @kbd{s s}
command is effectively the same as @kbd{K s t}.  @xref{Storing Variables}.

If you execute a command and then decide you really wanted to keep
the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}).
This command pushes the last arguments that were popped by any command
onto the stack.  Note that the order of things on the stack will be
different than with @kbd{K}:  @kbd{2 @key{RET} 3 + M-@key{RET}} leaves
@samp{5 2 3} on the stack instead of @samp{2 3 5}.  @xref{Undo}.

@node Mode Settings, Arithmetic, Stack and Trail, Top
@chapter Mode Settings

@noindent
This chapter describes commands that set modes in the Calculator.
They do not affect the contents of the stack, although they may change
the @emph{appearance} or @emph{interpretation} of the stack's contents.

@menu
* General Mode Commands::
* Precision::
* Inverse and Hyperbolic::
* Calculation Modes::
* Simplification Modes::
* Declarations::
* Display Modes::
* Language Modes::
* Modes Variable::
* Calc Mode Line::
@end menu

@node General Mode Commands, Precision, Mode Settings, Mode Settings
@section General Mode Commands

@noindent
@kindex m m
@pindex calc-save-modes
@cindex Continuous memory
@cindex Saving mode settings
@cindex Permanent mode settings
@cindex Calc init file, mode settings
You can save all of the current mode settings in your Calc init file 
(the file given by the variable @code{calc-settings-file}, typically
@file{~/.calc.el}) with the @kbd{m m} (@code{calc-save-modes}) command.
This will cause Emacs to reestablish these modes each time it starts up.
The modes saved in the file include everything controlled by the @kbd{m}
and @kbd{d} prefix keys, the current precision and binary word size,
whether or not the trail is displayed, the current height of the Calc
window, and more.  The current interface (used when you type @kbd{C-x * *}) 
is also saved.  If there were already saved mode settings in the
file, they are replaced.  Otherwise, the new mode information is
appended to the end of the file.

@kindex m R
@pindex calc-mode-record-mode
The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
record all the mode settings (as if by pressing @kbd{m m}) every
time a mode setting changes.  If the modes are saved this way, then this
``automatic mode recording'' mode is also saved.
Type @kbd{m R} again to disable this method of recording the mode
settings.  To turn it off permanently, the @kbd{m m} command will also be
necessary.   (If Embedded mode is enabled, other options for recording
the modes are available; @pxref{Mode Settings in Embedded Mode}.)

@kindex m F
@pindex calc-settings-file-name
The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
choose a different file than the current value of @code{calc-settings-file}
for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
You are prompted for a file name.  All Calc modes are then reset to
their default values, then settings from the file you named are loaded
if this file exists, and this file becomes the one that Calc will
use in the future for commands like @kbd{m m}.  The default settings
file name is @file{~/.calc.el}.  You can see the current file name by
giving a blank response to the @kbd{m F} prompt.  See also the
discussion of the @code{calc-settings-file} variable; @pxref{Customizing Calc}.

If the file name you give is your user init file (typically
@file{~/.emacs}), @kbd{m F} will not automatically load the new file.  This
is because your user init file may contain other things you don't want
to reread.  You can give 
a numeric prefix argument of 1 to @kbd{m F} to force it to read the
file no matter what.  Conversely, an argument of @mathit{-1} tells
@kbd{m F} @emph{not} to read the new file.  An argument of 2 or @mathit{-2}
tells @kbd{m F} not to reset the modes to their defaults beforehand,
which is useful if you intend your new file to have a variant of the
modes present in the file you were using before.

@kindex m x
@pindex calc-always-load-extensions
The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
in which the first use of Calc loads the entire program, including all
extensions modules.  Otherwise, the extensions modules will not be loaded
until the various advanced Calc features are used.  Since this mode only
has effect when Calc is first loaded, @kbd{m x} is usually followed by
@kbd{m m} to make the mode-setting permanent.  To load all of Calc just
once, rather than always in the future, you can press @kbd{C-x * L}.

@kindex m S
@pindex calc-shift-prefix
The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
all of Calc's letter prefix keys may be typed shifted as well as unshifted.
If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
you might find it easier to turn this mode on so that you can type
@kbd{A S} instead.  When this mode is enabled, the commands that used to
be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
now be invoked by pressing the shifted letter twice: @kbd{A A}.  Note
that the @kbd{v} prefix key always works both shifted and unshifted, and
the @kbd{z} and @kbd{Z} prefix keys are always distinct.  Also, the @kbd{h}
prefix is not affected by this mode.  Press @kbd{m S} again to disable
shifted-prefix mode.

@node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings
@section Precision

@noindent
@kindex p
@pindex calc-precision
@cindex Precision of calculations
The @kbd{p} (@code{calc-precision}) command controls the precision to
which floating-point calculations are carried.  The precision must be
at least 3 digits and may be arbitrarily high, within the limits of
memory and time.  This affects only floats:  Integer and rational
calculations are always carried out with as many digits as necessary.

The @kbd{p} key prompts for the current precision.  If you wish you
can instead give the precision as a numeric prefix argument.

Many internal calculations are carried to one or two digits higher
precision than normal.  Results are rounded down afterward to the
current precision.  Unless a special display mode has been selected,
floats are always displayed with their full stored precision, i.e.,
what you see is what you get.  Reducing the current precision does not
round values already on the stack, but those values will be rounded
down before being used in any calculation.  The @kbd{c 0} through
@kbd{c 9} commands (@pxref{Conversions}) can be used to round an
existing value to a new precision.

@cindex Accuracy of calculations
It is important to distinguish the concepts of @dfn{precision} and
@dfn{accuracy}.  In the normal usage of these words, the number
123.4567 has a precision of 7 digits but an accuracy of 4 digits.
The precision is the total number of digits not counting leading
or trailing zeros (regardless of the position of the decimal point).
The accuracy is simply the number of digits after the decimal point
(again not counting trailing zeros).  In Calc you control the precision,
not the accuracy of computations.  If you were to set the accuracy
instead, then calculations like @samp{exp(100)} would generate many
more digits than you would typically need, while @samp{exp(-100)} would
probably round to zero!  In Calc, both these computations give you
exactly 12 (or the requested number of) significant digits.

The only Calc features that deal with accuracy instead of precision
are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
and the rounding functions like @code{floor} and @code{round}
(@pxref{Integer Truncation}).  Also, @kbd{c 0} through @kbd{c 9}
deal with both precision and accuracy depending on the magnitudes
of the numbers involved.

If you need to work with a particular fixed accuracy (say, dollars and
cents with two digits after the decimal point), one solution is to work
with integers and an ``implied'' decimal point.  For example, $8.99
divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
(actually $1.49833 with our implied decimal point); pressing @kbd{R}
would round this to 150 cents, i.e., $1.50.

@xref{Floats}, for still more on floating-point precision and related
issues.

@node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings
@section Inverse and Hyperbolic Flags

@noindent
@kindex I
@pindex calc-inverse
There is no single-key equivalent to the @code{calc-arcsin} function.
Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
The @kbd{I} key actually toggles the Inverse Flag.  When this flag
is set, the word @samp{Inv} appears in the mode line.

@kindex H
@pindex calc-hyperbolic
Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
If both of these flags are set at once, the effect will be
@code{calc-arcsinh}.  (The Hyperbolic flag is also used by some
non-trigonometric commands; for example @kbd{H L} computes a base-10,
instead of base-@mathit{e}, logarithm.)

Command names like @code{calc-arcsin} are provided for completeness, and
may be executed with @kbd{x} or @kbd{M-x}.  Their effect is simply to
toggle the Inverse and/or Hyperbolic flags and then execute the
corresponding base command (@code{calc-sin} in this case).

The Inverse and Hyperbolic flags apply only to the next Calculator
command, after which they are automatically cleared.  (They are also
cleared if the next keystroke is not a Calc command.)  Digits you
type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
arguments for the next command, not as numeric entries.  The same
is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
subtract and keep arguments).

The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
elsewhere.  @xref{Keep Arguments}.

@node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
@section Calculation Modes

@noindent
The commands in this section are two-key sequences beginning with
the @kbd{m} prefix.  (That's the letter @kbd{m}, not the @key{META} key.)
The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
(@pxref{Algebraic Entry}).

@menu
* Angular Modes::
* Polar Mode::
* Fraction Mode::
* Infinite Mode::
* Symbolic Mode::
* Matrix Mode::
* Automatic Recomputation::
* Working Message::
@end menu

@node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
@subsection Angular Modes

@noindent
@cindex Angular mode
The Calculator supports three notations for angles: radians, degrees,
and degrees-minutes-seconds.  When a number is presented to a function
like @code{sin} that requires an angle, the current angular mode is
used to interpret the number as either radians or degrees.  If an HMS
form is presented to @code{sin}, it is always interpreted as
degrees-minutes-seconds.

Functions that compute angles produce a number in radians, a number in
degrees, or an HMS form depending on the current angular mode.  If the
result is a complex number and the current mode is HMS, the number is
instead expressed in degrees.  (Complex-number calculations would
normally be done in Radians mode, though.  Complex numbers are converted
to degrees by calculating the complex result in radians and then
multiplying by 180 over @cpi{}.)

@kindex m r
@pindex calc-radians-mode
@kindex m d
@pindex calc-degrees-mode
@kindex m h
@pindex calc-hms-mode
The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
The current angular mode is displayed on the Emacs mode line.
The default angular mode is Degrees.

@node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
@subsection Polar Mode

@noindent
@cindex Polar mode
The Calculator normally ``prefers'' rectangular complex numbers in the
sense that rectangular form is used when the proper form can not be
decided from the input.  This might happen by multiplying a rectangular
number by a polar one, by taking the square root of a negative real
number, or by entering @kbd{( 2 @key{SPC} 3 )}.

@kindex m p
@pindex calc-polar-mode
The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
preference between rectangular and polar forms.  In Polar mode, all
of the above example situations would produce polar complex numbers.

@node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
@subsection Fraction Mode

@noindent
@cindex Fraction mode
@cindex Division of integers
Division of two integers normally yields a floating-point number if the
result cannot be expressed as an integer.  In some cases you would
rather get an exact fractional answer.  One way to accomplish this is
to use the @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command, which
divides the two integers on the top of the stack to produce a fraction:
@kbd{6 @key{RET} 4 :} produces @expr{3:2} even though 
@kbd{6 @key{RET} 4 /} produces @expr{1.5}.

@kindex m f
@pindex calc-frac-mode
To set the Calculator to produce fractional results for normal integer
divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
For example, @expr{8/4} produces @expr{2} in either mode,
but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
Float mode.

At any time you can use @kbd{c f} (@code{calc-float}) to convert a
fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
float to a fraction.  @xref{Conversions}.

@node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
@subsection Infinite Mode

@noindent
@cindex Infinite mode
The Calculator normally treats results like @expr{1 / 0} as errors;
formulas like this are left in unsimplified form.  But Calc can be
put into a mode where such calculations instead produce ``infinite''
results.

@kindex m i
@pindex calc-infinite-mode
The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
on and off.  When the mode is off, infinities do not arise except
in calculations that already had infinities as inputs.  (One exception
is that infinite open intervals like @samp{[0 .. inf)} can be
generated; however, intervals closed at infinity (@samp{[0 .. inf]})
will not be generated when Infinite mode is off.)

With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
an undirected infinity.  @xref{Infinities}, for a discussion of the
difference between @code{inf} and @code{uinf}.  Also, @expr{0 / 0}
evaluates to @code{nan}, the ``indeterminate'' symbol.  Various other
functions can also return infinities in this mode; for example,
@samp{ln(0) = -inf}, and @samp{gamma(-7) = uinf}.  Once again,
note that @samp{exp(inf) = inf} regardless of Infinite mode because
this calculation has infinity as an input.

@cindex Positive Infinite mode
The @kbd{m i} command with a numeric prefix argument of zero,
i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
which zero is treated as positive instead of being directionless.
Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
Note that zero never actually has a sign in Calc; there are no
separate representations for @mathit{+0} and @mathit{-0}.  Positive
Infinite mode merely changes the interpretation given to the
single symbol, @samp{0}.  One consequence of this is that, while
you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.

@node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
@subsection Symbolic Mode

@noindent
@cindex Symbolic mode
@cindex Inexact results
Calculations are normally performed numerically wherever possible.
For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
algebraic expression, produces a numeric answer if the argument is a
number or a symbolic expression if the argument is an expression:
@kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}.

@kindex m s
@pindex calc-symbolic-mode
In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
command, functions which would produce inexact, irrational results are
left in symbolic form.  Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
@samp{sqrt(2)}.

@kindex N
@pindex calc-eval-num
The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically
the expression at the top of the stack, by temporarily disabling
@code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}).
Given a numeric prefix argument, it also
sets the floating-point precision to the specified value for the duration
of the command.

To evaluate a formula numerically without expanding the variables it
contains, you can use the key sequence @kbd{m s a v m s} (this uses
@code{calc-alg-evaluate}, which resimplifies but doesn't evaluate
variables.)

@node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes
@subsection Matrix and Scalar Modes

@noindent
@cindex Matrix mode
@cindex Scalar mode
Calc sometimes makes assumptions during algebraic manipulation that
are awkward or incorrect when vectors and matrices are involved.
Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
modify its behavior around vectors in useful ways.

@kindex m v
@pindex calc-matrix-mode
Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
In this mode, all objects are assumed to be matrices unless provably
otherwise.  One major effect is that Calc will no longer consider
multiplication to be commutative.  (Recall that in matrix arithmetic,
@samp{A*B} is not the same as @samp{B*A}.)  This assumption affects
rewrite rules and algebraic simplification.  Another effect of this
mode is that calculations that would normally produce constants like
0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
produce function calls that represent ``generic'' zero or identity
matrices: @samp{idn(0)}, @samp{idn(1)}.  The @code{idn} function
@samp{idn(@var{a},@var{n})} returns @var{a} times an @var{n}x@var{n}
identity matrix; if @var{n} is omitted, it doesn't know what
dimension to use and so the @code{idn} call remains in symbolic
form.  However, if this generic identity matrix is later combined
with a matrix whose size is known, it will be converted into
a true identity matrix of the appropriate size.  On the other hand,
if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
will assume it really was a scalar after all and produce, e.g., 3.

Press @kbd{m v} a second time to get Scalar mode.  Here, objects are
assumed @emph{not} to be vectors or matrices unless provably so.
For example, normally adding a variable to a vector, as in
@samp{[x, y, z] + a}, will leave the sum in symbolic form because
as far as Calc knows, @samp{a} could represent either a number or
another 3-vector.  In Scalar mode, @samp{a} is assumed to be a
non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.

Press @kbd{m v} a third time to return to the normal mode of operation.

If you press @kbd{m v} with a numeric prefix argument @var{n}, you
get a special ``dimensioned'' Matrix mode in which matrices of
unknown size are assumed to be @var{n}x@var{n} square matrices.
Then, the function call @samp{idn(1)} will expand into an actual
matrix rather than representing a ``generic'' matrix.  Simply typing
@kbd{C-u m v} will get you a square Matrix mode, in which matrices of
unknown size are assumed to be square matrices of unspecified size.

@cindex Declaring scalar variables
Of course these modes are approximations to the true state of
affairs, which is probably that some quantities will be matrices
and others will be scalars.  One solution is to ``declare''
certain variables or functions to be scalar-valued.
@xref{Declarations}, to see how to make declarations in Calc.

There is nothing stopping you from declaring a variable to be
scalar and then storing a matrix in it; however, if you do, the
results you get from Calc may not be valid.  Suppose you let Calc
get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
@samp{[1, 2, 3]} in @samp{a}.  The result would not be the same as
for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
your earlier promise to Calc that @samp{a} would be scalar.

Another way to mix scalars and matrices is to use selections
(@pxref{Selecting Subformulas}).  Use Matrix mode when operating on
your formula normally; then, to apply Scalar mode to a certain part
of the formula without affecting the rest just select that part,
change into Scalar mode and press @kbd{=} to resimplify the part
under this mode, then change back to Matrix mode before deselecting.

@node Automatic Recomputation, Working Message, Matrix Mode, Calculation Modes
@subsection Automatic Recomputation

@noindent
The @dfn{evaluates-to} operator, @samp{=>}, has the special
property that any @samp{=>} formulas on the stack are recomputed
whenever variable values or mode settings that might affect them
are changed.  @xref{Evaluates-To Operator}.

@kindex m C
@pindex calc-auto-recompute
The @kbd{m C} (@code{calc-auto-recompute}) command turns this
automatic recomputation on and off.  If you turn it off, Calc will
not update @samp{=>} operators on the stack (nor those in the
attached Embedded mode buffer, if there is one).  They will not
be updated unless you explicitly do so by pressing @kbd{=} or until
you press @kbd{m C} to turn recomputation back on.  (While automatic
recomputation is off, you can think of @kbd{m C m C} as a command
to update all @samp{=>} operators while leaving recomputation off.)

To update @samp{=>} operators in an Embedded buffer while
automatic recomputation is off, use @w{@kbd{C-x * u}}.
@xref{Embedded Mode}.

@node Working Message,  , Automatic Recomputation, Calculation Modes
@subsection Working Messages

@noindent
@cindex Performance
@cindex Working messages
Since the Calculator is written entirely in Emacs Lisp, which is not
designed for heavy numerical work, many operations are quite slow.
The Calculator normally displays the message @samp{Working...} in the
echo area during any command that may be slow.  In addition, iterative
operations such as square roots and trigonometric functions display the
intermediate result at each step.  Both of these types of messages can
be disabled if you find them distracting.

@kindex m w
@pindex calc-working
Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
disable all ``working'' messages.  Use a numeric prefix of 1 to enable
only the plain @samp{Working...} message.  Use a numeric prefix of 2 to
see intermediate results as well.  With no numeric prefix this displays
the current mode.

While it may seem that the ``working'' messages will slow Calc down
considerably, experiments have shown that their impact is actually
quite small.  But if your terminal is slow you may find that it helps
to turn the messages off.

@node Simplification Modes, Declarations, Calculation Modes, Mode Settings
@section Simplification Modes

@noindent
The current @dfn{simplification mode} controls how numbers and formulas
are ``normalized'' when being taken from or pushed onto the stack.
Some normalizations are unavoidable, such as rounding floating-point
results to the current precision, and reducing fractions to simplest
form.  Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
are done by default but can be turned off when necessary.

When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
stack, Calc pops these numbers, normalizes them, creates the formula
@expr{2+3}, normalizes it, and pushes the result.  Of course the standard
rules for normalizing @expr{2+3} will produce the result @expr{5}.

Simplification mode commands consist of the lower-case @kbd{m} prefix key
followed by a shifted letter.

@kindex m O
@pindex calc-no-simplify-mode
The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
simplifications.  These would leave a formula like @expr{2+3} alone.  In
fact, nothing except simple numbers are ever affected by normalization
in this mode.

@kindex m N
@pindex calc-num-simplify-mode
The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
of any formulas except those for which all arguments are constants.  For
example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
simplified to @expr{a+0} but no further, since one argument of the sum
is not a constant.  Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
because the top-level @samp{-} operator's arguments are not both
constant numbers (one of them is the formula @expr{a+2}).
A constant is a number or other numeric object (such as a constant
error form or modulo form), or a vector all of whose
elements are constant.

@kindex m D
@pindex calc-default-simplify-mode
The @kbd{m D} (@code{calc-default-simplify-mode}) command restores the
default simplifications for all formulas.  This includes many easy and
fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
@expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like
@expr{@tfn{deriv}(x^2, x)} to @expr{2 x}.

@kindex m B
@pindex calc-bin-simplify-mode
The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the default
simplifications to a result and then, if the result is an integer,
uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
to the current binary word size.  @xref{Binary Functions}.  Real numbers
are rounded to the nearest integer and then clipped; other kinds of
results (after the default simplifications) are left alone.

@kindex m A
@pindex calc-alg-simplify-mode
The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does algebraic
simplification; it applies all the default simplifications, and also
the more powerful (and slower) simplifications made by @kbd{a s}
(@code{calc-simplify}).  @xref{Algebraic Simplifications}.

@kindex m E
@pindex calc-ext-simplify-mode
The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended''
algebraic simplification, as by the @kbd{a e} (@code{calc-simplify-extended})
command.  @xref{Unsafe Simplifications}.

@kindex m U
@pindex calc-units-simplify-mode
The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
simplification; it applies the command @kbd{u s}
(@code{calc-simplify-units}), which in turn
is a superset of @kbd{a s}.  In this mode, variable names which
are identifiable as unit names (like @samp{mm} for ``millimeters'')
are simplified with their unit definitions in mind.

A common technique is to set the simplification mode down to the lowest
amount of simplification you will allow to be applied automatically, then
use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
perform higher types of simplifications on demand.  @xref{Algebraic
Definitions}, for another sample use of No-Simplification mode.

@node Declarations, Display Modes, Simplification Modes, Mode Settings
@section Declarations

@noindent
A @dfn{declaration} is a statement you make that promises you will
use a certain variable or function in a restricted way.  This may
give Calc the freedom to do things that it couldn't do if it had to
take the fully general situation into account.

@menu
* Declaration Basics::
* Kinds of Declarations::
* Functions for Declarations::
@end menu

@node Declaration Basics, Kinds of Declarations, Declarations, Declarations
@subsection Declaration Basics

@noindent
@kindex s d
@pindex calc-declare-variable
The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
way to make a declaration for a variable.  This command prompts for
the variable name, then prompts for the declaration.  The default
at the declaration prompt is the previous declaration, if any.
You can edit this declaration, or press @kbd{C-k} to erase it and
type a new declaration.  (Or, erase it and press @key{RET} to clear
the declaration, effectively ``undeclaring'' the variable.)

A declaration is in general a vector of @dfn{type symbols} and
@dfn{range} values.  If there is only one type symbol or range value,
you can write it directly rather than enclosing it in a vector.
For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
declares @code{bar} to be a constant integer between 1 and 6.
(Actually, you can omit the outermost brackets and Calc will
provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)

@cindex @code{Decls} variable
@vindex Decls
Declarations in Calc are kept in a special variable called @code{Decls}.
This variable encodes the set of all outstanding declarations in
the form of a matrix.  Each row has two elements:  A variable or
vector of variables declared by that row, and the declaration
specifier as described above.  You can use the @kbd{s D} command to
edit this variable if you wish to see all the declarations at once.
@xref{Operations on Variables}, for a description of this command
and the @kbd{s p} command that allows you to save your declarations
permanently if you wish.

Items being declared can also be function calls.  The arguments in
the call are ignored; the effect is to say that this function returns
values of the declared type for any valid arguments.  The @kbd{s d}
command declares only variables, so if you wish to make a function
declaration you will have to edit the @code{Decls} matrix yourself.

For example, the declaration matrix

@smallexample
@group
[ [ foo,       real       ]
  [ [j, k, n], int        ]
  [ f(1,2,3),  [0 .. inf) ] ]
@end group
@end smallexample

@noindent
declares that @code{foo} represents a real number, @code{j}, @code{k}
and @code{n} represent integers, and the function @code{f} always
returns a real number in the interval shown.

@vindex All
If there is a declaration for the variable @code{All}, then that
declaration applies to all variables that are not otherwise declared.
It does not apply to function names.  For example, using the row
@samp{[All, real]} says that all your variables are real unless they
are explicitly declared without @code{real} in some other row.
The @kbd{s d} command declares @code{All} if you give a blank
response to the variable-name prompt.

@node Kinds of Declarations, Functions for Declarations, Declaration Basics, Declarations
@subsection Kinds of Declarations

@noindent
The type-specifier part of a declaration (that is, the second prompt
in the @kbd{s d} command) can be a type symbol, an interval, or a
vector consisting of zero or more type symbols followed by zero or
more intervals or numbers that represent the set of possible values
for the variable.

@smallexample
@group
[ [ a, [1, 2, 3, 4, 5] ]
  [ b, [1 .. 5]        ]
  [ c, [int, 1 .. 5]   ] ]
@end group
@end smallexample

Here @code{a} is declared to contain one of the five integers shown;
@code{b} is any number in the interval from 1 to 5 (any real number
since we haven't specified), and @code{c} is any integer in that
interval.  Thus the declarations for @code{a} and @code{c} are
nearly equivalent (see below).

The type-specifier can be the empty vector @samp{[]} to say that
nothing is known about a given variable's value.  This is the same
as not declaring the variable at all except that it overrides any
@code{All} declaration which would otherwise apply.

The initial value of @code{Decls} is the empty vector @samp{[]}.
If @code{Decls} has no stored value or if the value stored in it
is not valid, it is ignored and there are no declarations as far
as Calc is concerned.  (The @kbd{s d} command will replace such a
malformed value with a fresh empty matrix, @samp{[]}, before recording
the new declaration.)  Unrecognized type symbols are ignored.

The following type symbols describe what sorts of numbers will be
stored in a variable:

@table @code
@item int
Integers.
@item numint
Numerical integers.  (Integers or integer-valued floats.)
@item frac
Fractions.  (Rational numbers which are not integers.)
@item rat
Rational numbers.  (Either integers or fractions.)
@item float
Floating-point numbers.
@item real
Real numbers.  (Integers, fractions, or floats.  Actually,
intervals and error forms with real components also count as
reals here.)
@item pos
Positive real numbers.  (Strictly greater than zero.)
@item nonneg
Nonnegative real numbers.  (Greater than or equal to zero.)
@item number
Numbers.  (Real or complex.)
@end table

Calc uses this information to determine when certain simplifications
of formulas are safe.  For example, @samp{(x^y)^z} cannot be
simplified to @samp{x^(y z)} in general; for example,
@samp{((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is @mathit{-3}.
However, this simplification @emph{is} safe if @code{z} is known
to be an integer, or if @code{x} is known to be a nonnegative
real number.  If you have given declarations that allow Calc to
deduce either of these facts, Calc will perform this simplification
of the formula.

Calc can apply a certain amount of logic when using declarations.
For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
has been declared @code{int}; Calc knows that an integer times an
integer, plus an integer, must always be an integer.  (In fact,
Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
it is able to determine that @samp{2n+1} must be an odd integer.)

Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
because Calc knows that the @code{abs} function always returns a
nonnegative real.  If you had a @code{myabs} function that also had
this property, you could get Calc to recognize it by adding the row
@samp{[myabs(), nonneg]} to the @code{Decls} matrix.

One instance of this simplification is @samp{sqrt(x^2)} (since the
@code{sqrt} function is effectively a one-half power).  Normally
Calc leaves this formula alone.  After the command
@kbd{s d x @key{RET} real @key{RET}}, however, it can simplify the formula to
@samp{abs(x)}.  And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
simplify this formula all the way to @samp{x}.

If there are any intervals or real numbers in the type specifier,
they comprise the set of possible values that the variable or
function being declared can have.  In particular, the type symbol
@code{real} is effectively the same as the range @samp{[-inf .. inf]}
(note that infinity is included in the range of possible values);
@code{pos} is the same as @samp{(0 .. inf]}, and @code{nonneg} is
the same as @samp{[0 .. inf]}.  Saying @samp{[real, [-5 .. 5]]} is
redundant because the fact that the variable is real can be
deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
@samp{[rat, [-5 .. 5]]} are useful combinations.

Note that the vector of intervals or numbers is in the same format
used by Calc's set-manipulation commands.  @xref{Set Operations}.

The type specifier @samp{[1, 2, 3]} is equivalent to
@samp{[numint, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
In other words, the range of possible values means only that
the variable's value must be numerically equal to a number in
that range, but not that it must be equal in type as well.
Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''

If you use a conflicting combination of type specifiers, the
results are unpredictable.  An example is @samp{[pos, [0 .. 5]]},
where the interval does not lie in the range described by the
type symbol.

``Real'' declarations mostly affect simplifications involving powers
like the one described above.  Another case where they are used
is in the @kbd{a P} command which returns a list of all roots of a
polynomial; if the variable has been declared real, only the real
roots (if any) will be included in the list.

``Integer'' declarations are used for simplifications which are valid
only when certain values are integers (such as @samp{(x^y)^z}
shown above).

Another command that makes use of declarations is @kbd{a s}, when
simplifying equations and inequalities.  It will cancel @code{x}
from both sides of @samp{a x = b x} only if it is sure @code{x}
is non-zero, say, because it has a @code{pos} declaration.
To declare specifically that @code{x} is real and non-zero,
use @samp{[[-inf .. 0), (0 .. inf]]}.  (There is no way in the
current notation to say that @code{x} is nonzero but not necessarily
real.)  The @kbd{a e} command does ``unsafe'' simplifications,
including cancelling @samp{x} from the equation when @samp{x} is
not known to be nonzero.

Another set of type symbols distinguish between scalars and vectors.

@table @code
@item scalar
The value is not a vector.
@item vector
The value is a vector.
@item matrix
The value is a matrix (a rectangular vector of vectors).
@item sqmatrix
The value is a square matrix.
@end table

These type symbols can be combined with the other type symbols
described above; @samp{[int, matrix]} describes an object which
is a matrix of integers.

Scalar/vector declarations are used to determine whether certain
algebraic operations are safe.  For example, @samp{[a, b, c] + x}
is normally not simplified to @samp{[a + x, b + x, c + x]}, but
it will be if @code{x} has been declared @code{scalar}.  On the
other hand, multiplication is usually assumed to be commutative,
but the terms in @samp{x y} will never be exchanged if both @code{x}
and @code{y} are known to be vectors or matrices.  (Calc currently
never distinguishes between @code{vector} and @code{matrix}
declarations.)

@xref{Matrix Mode}, for a discussion of Matrix mode and
Scalar mode, which are similar to declaring @samp{[All, matrix]}
or @samp{[All, scalar]} but much more convenient.

One more type symbol that is recognized is used with the @kbd{H a d}
command for taking total derivatives of a formula.  @xref{Calculus}.

@table @code
@item const
The value is a constant with respect to other variables.
@end table

Calc does not check the declarations for a variable when you store
a value in it.  However, storing @mathit{-3.5} in a variable that has
been declared @code{pos}, @code{int}, or @code{matrix} may have
unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
if it substitutes the value first, or to @expr{-3.5} if @code{x}
was declared @code{pos} and the formula @samp{sqrt(x^2)} is
simplified to @samp{x} before the value is substituted.  Before
using a variable for a new purpose, it is best to use @kbd{s d}
or @kbd{s D} to check to make sure you don't still have an old
declaration for the variable that will conflict with its new meaning.

@node Functions for Declarations,  , Kinds of Declarations, Declarations
@subsection Functions for Declarations

@noindent
Calc has a set of functions for accessing the current declarations
in a convenient manner.  These functions return 1 if the argument
can be shown to have the specified property, or 0 if the argument
can be shown @emph{not} to have that property; otherwise they are
left unevaluated.  These functions are suitable for use with rewrite
rules (@pxref{Conditional Rewrite Rules}) or programming constructs
(@pxref{Conditionals in Macros}).  They can be entered only using
algebraic notation.  @xref{Logical Operations}, for functions
that perform other tests not related to declarations.

For example, @samp{dint(17)} returns 1 because 17 is an integer, as
do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
@code{int}, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
Calc consults knowledge of its own built-in functions as well as your
own declarations: @samp{dint(floor(x))} returns 1.

@ignore
@starindex
@end ignore
@tindex dint
@ignore
@starindex
@end ignore
@tindex dnumint
@ignore
@starindex
@end ignore
@tindex dnatnum
The @code{dint} function checks if its argument is an integer.
The @code{dnatnum} function checks if its argument is a natural
number, i.e., a nonnegative integer.  The @code{dnumint} function
checks if its argument is numerically an integer, i.e., either an
integer or an integer-valued float.  Note that these and the other
data type functions also accept vectors or matrices composed of
suitable elements, and that real infinities @samp{inf} and @samp{-inf}
are considered to be integers for the purposes of these functions.

@ignore
@starindex
@end ignore
@tindex drat
The @code{drat} function checks if its argument is rational, i.e.,
an integer or fraction.  Infinities count as rational, but intervals
and error forms do not.

@ignore
@starindex
@end ignore
@tindex dreal
The @code{dreal} function checks if its argument is real.  This
includes integers, fractions, floats, real error forms, and intervals.

@ignore
@starindex
@end ignore
@tindex dimag
The @code{dimag} function checks if its argument is imaginary,
i.e., is mathematically equal to a real number times @expr{i}.

@ignore
@starindex
@end ignore
@tindex dpos
@ignore
@starindex
@end ignore
@tindex dneg
@ignore
@starindex
@end ignore
@tindex dnonneg
The @code{dpos} function checks for positive (but nonzero) reals.
The @code{dneg} function checks for negative reals.  The @code{dnonneg}
function checks for nonnegative reals, i.e., reals greater than or
equal to zero.  Note that the @kbd{a s} command can simplify an
expression like @expr{x > 0} to 1 or 0 using @code{dpos}, and that
@kbd{a s} is effectively applied to all conditions in rewrite rules,
so the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
are rarely necessary.

@ignore
@starindex
@end ignore
@tindex dnonzero
The @code{dnonzero} function checks that its argument is nonzero.
This includes all nonzero real or complex numbers, all intervals that
do not include zero, all nonzero modulo forms, vectors all of whose
elements are nonzero, and variables or formulas whose values can be
deduced to be nonzero.  It does not include error forms, since they
represent values which could be anything including zero.  (This is
also the set of objects considered ``true'' in conditional contexts.)

@ignore
@starindex
@end ignore
@tindex deven
@ignore
@starindex
@end ignore
@tindex dodd
The @code{deven} function returns 1 if its argument is known to be
an even integer (or integer-valued float); it returns 0 if its argument
is known not to be even (because it is known to be odd or a non-integer).
The @kbd{a s} command uses this to simplify a test of the form
@samp{x % 2 = 0}.  There is also an analogous @code{dodd} function.

@ignore
@starindex
@end ignore
@tindex drange
The @code{drange} function returns a set (an interval or a vector
of intervals and/or numbers; @pxref{Set Operations}) that describes
the set of possible values of its argument.  If the argument is
a variable or a function with a declaration, the range is copied
from the declaration.  Otherwise, the possible signs of the
expression are determined using a method similar to @code{dpos},
etc., and a suitable set like @samp{[0 .. inf]} is returned.  If
the expression is not provably real, the @code{drange} function
remains unevaluated.

@ignore
@starindex
@end ignore
@tindex dscalar
The @code{dscalar} function returns 1 if its argument is provably
scalar, or 0 if its argument is provably non-scalar.  It is left
unevaluated if this cannot be determined.  (If Matrix mode or Scalar
mode is in effect, this function returns 1 or 0, respectively,
if it has no other information.)  When Calc interprets a condition
(say, in a rewrite rule) it considers an unevaluated formula to be
``false.''  Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
is provably non-scalar; both are ``false'' if there is insufficient
information to tell.

@node Display Modes, Language Modes, Declarations, Mode Settings
@section Display Modes

@noindent
The commands in this section are two-key sequences beginning with the
@kbd{d} prefix.  The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
(@code{calc-line-breaking}) commands are described elsewhere;
@pxref{Stack Basics} and @pxref{Normal Language Modes}, respectively.
Display formats for vectors and matrices are also covered elsewhere;
@pxref{Vector and Matrix Formats}.

One thing all display modes have in common is their treatment of the
@kbd{H} prefix.  This prefix causes any mode command that would normally
refresh the stack to leave the stack display alone.  The word ``Dirty''
will appear in the mode line when Calc thinks the stack display may not
reflect the latest mode settings.

@kindex d @key{RET}
@pindex calc-refresh-top
The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
top stack entry according to all the current modes.  Positive prefix
arguments reformat the top @var{n} entries; negative prefix arguments
reformat the specified entry, and a prefix of zero is equivalent to
@kbd{d @key{SPC}} (@code{calc-refresh}), which reformats the entire stack.
For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
but reformats only the top two stack entries in the new mode.

The @kbd{I} prefix has another effect on the display modes.  The mode
is set only temporarily; the top stack entry is reformatted according
to that mode, then the original mode setting is restored.  In other
words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old mode})}.

@menu
* Radix Modes::
* Grouping Digits::
* Float Formats::
* Complex Formats::
* Fraction Formats::
* HMS Formats::
* Date Formats::
* Truncating the Stack::
* Justification::
* Labels::
@end menu

@node Radix Modes, Grouping Digits, Display Modes, Display Modes
@subsection Radix Modes

@noindent
@cindex Radix display
@cindex Non-decimal numbers
@cindex Decimal and non-decimal numbers
Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
notation.  Calc can actually display in any radix from two (binary) to 36.
When the radix is above 10, the letters @code{A} to @code{Z} are used as
digits.  When entering such a number, letter keys are interpreted as
potential digits rather than terminating numeric entry mode.

@kindex d 2
@kindex d 8
@kindex d 6
@kindex d 0
@cindex Hexadecimal integers
@cindex Octal integers
The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
binary, octal, hexadecimal, and decimal as the current display radix,
respectively.  Numbers can always be entered in any radix, though the
current radix is used as a default if you press @kbd{#} without any initial
digits.  A number entered without a @kbd{#} is @emph{always} interpreted
as decimal.

@kindex d r
@pindex calc-radix
To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
an integer from 2 to 36.  You can specify the radix as a numeric prefix
argument; otherwise you will be prompted for it.

@kindex d z
@pindex calc-leading-zeros
@cindex Leading zeros
Integers normally are displayed with however many digits are necessary to
represent the integer and no more.  The @kbd{d z} (@code{calc-leading-zeros})
command causes integers to be padded out with leading zeros according to the
current binary word size.  (@xref{Binary Functions}, for a discussion of
word size.)  If the absolute value of the word size is @expr{w}, all integers
are displayed with at least enough digits to represent 
@texline @math{2^w-1}
@infoline @expr{(2^w)-1} 
in the current radix.  (Larger integers will still be displayed in their
entirety.) 

@node Grouping Digits, Float Formats, Radix Modes, Display Modes
@subsection Grouping Digits

@noindent
@kindex d g
@pindex calc-group-digits
@cindex Grouping digits
@cindex Digit grouping
Long numbers can be hard to read if they have too many digits.  For
example, the factorial of 30 is 33 digits long!  Press @kbd{d g}
(@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
are displayed in clumps of 3 or 4 (depending on the current radix)
separated by commas.

The @kbd{d g} command toggles grouping on and off.
With a numeric prefix of 0, this command displays the current state of
the grouping flag; with an argument of minus one it disables grouping;
with a positive argument @expr{N} it enables grouping on every @expr{N}
digits.  For floating-point numbers, grouping normally occurs only
before the decimal point.  A negative prefix argument @expr{-N} enables
grouping every @expr{N} digits both before and after the decimal point.

@kindex d ,
@pindex calc-group-char
The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
character as the grouping separator.  The default is the comma character.
If you find it difficult to read vectors of large integers grouped with
commas, you may wish to use spaces or some other character instead.
This command takes the next character you type, whatever it is, and
uses it as the digit separator.  As a special case, @kbd{d , \} selects
@samp{\,} (@TeX{}'s thin-space symbol) as the digit separator.

Please note that grouped numbers will not generally be parsed correctly
if re-read in textual form, say by the use of @kbd{C-x * y} and @kbd{C-x * g}.
(@xref{Kill and Yank}, for details on these commands.)  One exception is
the @samp{\,} separator, which doesn't interfere with parsing because it
is ignored by @TeX{} language mode.

@node Float Formats, Complex Formats, Grouping Digits, Display Modes
@subsection Float Formats

@noindent
Floating-point quantities are normally displayed in standard decimal
form, with scientific notation used if the exponent is especially high
or low.  All significant digits are normally displayed.  The commands
in this section allow you to choose among several alternative display
formats for floats.

@kindex d n
@pindex calc-normal-notation
The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
display format.  All significant figures in a number are displayed.
With a positive numeric prefix, numbers are rounded if necessary to
that number of significant digits.  With a negative numerix prefix,
the specified number of significant digits less than the current
precision is used.  (Thus @kbd{C-u -2 d n} displays 10 digits if the
current precision is 12.)

@kindex d f
@pindex calc-fix-notation
The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
notation.  The numeric argument is the number of digits after the
decimal point, zero or more.  This format will relax into scientific
notation if a nonzero number would otherwise have been rounded all the
way to zero.  Specifying a negative number of digits is the same as
for a positive number, except that small nonzero numbers will be rounded
to zero rather than switching to scientific notation.

@kindex d s
@pindex calc-sci-notation
@cindex Scientific notation, display of
The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
notation.  A positive argument sets the number of significant figures
displayed, of which one will be before and the rest after the decimal
point.  A negative argument works the same as for @kbd{d n} format.
The default is to display all significant digits.

@kindex d e
@pindex calc-eng-notation
@cindex Engineering notation, display of
The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
notation.  This is similar to scientific notation except that the
exponent is rounded down to a multiple of three, with from one to three
digits before the decimal point.  An optional numeric prefix sets the
number of significant digits to display, as for @kbd{d s}.

It is important to distinguish between the current @emph{precision} and
the current @emph{display format}.  After the commands @kbd{C-u 10 p}
and @kbd{C-u 6 d n} the Calculator computes all results to ten
significant figures but displays only six.  (In fact, intermediate
calculations are often carried to one or two more significant figures,
but values placed on the stack will be rounded down to ten figures.)
Numbers are never actually rounded to the display precision for storage,
except by commands like @kbd{C-k} and @kbd{C-x * y} which operate on the
actual displayed text in the Calculator buffer.

@kindex d .
@pindex calc-point-char
The @kbd{d .} (@code{calc-point-char}) command selects the character used
as a decimal point.  Normally this is a period; users in some countries
may wish to change this to a comma.  Note that this is only a display
style; on entry, periods must always be used to denote floating-point
numbers, and commas to separate elements in a list.

@node Complex Formats, Fraction Formats, Float Formats, Display Modes
@subsection Complex Formats

@noindent
@kindex d c
@pindex calc-complex-notation
There are three supported notations for complex numbers in rectangular
form.  The default is as a pair of real numbers enclosed in parentheses
and separated by a comma: @samp{(a,b)}.  The @kbd{d c}
(@code{calc-complex-notation}) command selects this style.

@kindex d i
@pindex calc-i-notation
@kindex d j
@pindex calc-j-notation
The other notations are @kbd{d i} (@code{calc-i-notation}), in which
numbers are displayed in @samp{a+bi} form, and @kbd{d j}
(@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
in some disciplines.

@cindex @code{i} variable
@vindex i
Complex numbers are normally entered in @samp{(a,b)} format.
If you enter @samp{2+3i} as an algebraic formula, it will be stored as
the formula @samp{2 + 3 * i}.  However, if you use @kbd{=} to evaluate
this formula and you have not changed the variable @samp{i}, the @samp{i}
will be interpreted as @samp{(0,1)} and the formula will be simplified
to @samp{(2,3)}.  Other commands (like @code{calc-sin}) will @emph{not}
interpret the formula @samp{2 + 3 * i} as a complex number.
@xref{Variables}, under ``special constants.''

@node Fraction Formats, HMS Formats, Complex Formats, Display Modes
@subsection Fraction Formats

@noindent
@kindex d o
@pindex calc-over-notation
Display of fractional numbers is controlled by the @kbd{d o}
(@code{calc-over-notation}) command.  By default, a number like
eight thirds is displayed in the form @samp{8:3}.  The @kbd{d o} command
prompts for a one- or two-character format.  If you give one character,
that character is used as the fraction separator.  Common separators are
@samp{:} and @samp{/}.  (During input of numbers, the @kbd{:} key must be
used regardless of the display format; in particular, the @kbd{/} is used
for RPN-style division, @emph{not} for entering fractions.)

If you give two characters, fractions use ``integer-plus-fractional-part''
notation.  For example, the format @samp{+/} would display eight thirds
as @samp{2+2/3}.  If two colons are present in a number being entered,
the number is interpreted in this form (so that the entries @kbd{2:2:3}
and @kbd{8:3} are equivalent).

It is also possible to follow the one- or two-character format with
a number.  For example:  @samp{:10} or @samp{+/3}.  In this case,
Calc adjusts all fractions that are displayed to have the specified
denominator, if possible.  Otherwise it adjusts the denominator to
be a multiple of the specified value.  For example, in @samp{:6} mode
the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
and @expr{1:8} will be displayed as @expr{3:24}.  Integers are also
affected by this mode:  3 is displayed as @expr{18:6}.  Note that the
format @samp{:1} writes fractions the same as @samp{:}, but it writes
integers as @expr{n:1}.

The fraction format does not affect the way fractions or integers are
stored, only the way they appear on the screen.  The fraction format
never affects floats.

@node HMS Formats, Date Formats, Fraction Formats, Display Modes
@subsection HMS Formats

@noindent
@kindex d h
@pindex calc-hms-notation
The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
HMS (hours-minutes-seconds) forms.  It prompts for a string which
consists basically of an ``hours'' marker, optional punctuation, a
``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
Punctuation is zero or more spaces, commas, or semicolons.  The hours
marker is one or more non-punctuation characters.  The minutes and
seconds markers must be single non-punctuation characters.

The default HMS format is @samp{@@ ' "}, producing HMS values of the form
@samp{23@@ 30' 15.75"}.  The format @samp{deg, ms} would display this same
value as @samp{23deg, 30m15.75s}.  During numeric entry, the @kbd{h} or @kbd{o}
keys are recognized as synonyms for @kbd{@@} regardless of display format.
The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
@kbd{"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o}) has
already been typed; otherwise, they have their usual meanings
(@kbd{m-} prefix and @kbd{s-} prefix).  Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
@kbd{0 h 5 s} are some of the ways to enter the quantity ``five seconds.''
The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
@kbd{o}) has already been pressed; otherwise it means to switch to algebraic
entry.

@node Date Formats, Truncating the Stack, HMS Formats, Display Modes
@subsection Date Formats

@noindent
@kindex d d
@pindex calc-date-notation
The @kbd{d d} (@code{calc-date-notation}) command controls the display
of date forms (@pxref{Date Forms}).  It prompts for a string which
contains letters that represent the various parts of a date and time.
To show which parts should be omitted when the form represents a pure
date with no time, parts of the string can be enclosed in @samp{< >}
marks.  If you don't include @samp{< >} markers in the format, Calc
guesses at which parts, if any, should be omitted when formatting
pure dates.

The default format is:  @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
If you enter a blank format string, this default format is
reestablished.

Calc uses @samp{< >} notation for nameless functions as well as for
dates.  @xref{Specifying Operators}.  To avoid confusion with nameless
functions, your date formats should avoid using the @samp{#} character.

@menu
* Date Formatting Codes::
* Free-Form Dates::
* Standard Date Formats::
@end menu

@node Date Formatting Codes, Free-Form Dates, Date Formats, Date Formats
@subsubsection Date Formatting Codes

@noindent
When displaying a date, the current date format is used.  All
characters except for letters and @samp{<} and @samp{>} are
copied literally when dates are formatted.  The portion between
@samp{< >} markers is omitted for pure dates, or included for
date/time forms.  Letters are interpreted according to the table
below.

When dates are read in during algebraic entry, Calc first tries to
match the input string to the current format either with or without
the time part.  The punctuation characters (including spaces) must
match exactly; letter fields must correspond to suitable text in
the input.  If this doesn't work, Calc checks if the input is a
simple number; if so, the number is interpreted as a number of days
since Jan 1, 1 AD.  Otherwise, Calc tries a much more relaxed and
flexible algorithm which is described in the next section.

Weekday names are ignored during reading.

Two-digit year numbers are interpreted as lying in the range
from 1941 to 2039.  Years outside that range are always
entered and displayed in full.  Year numbers with a leading
@samp{+} sign are always interpreted exactly, allowing the
entry and display of the years 1 through 99 AD.

Here is a complete list of the formatting codes for dates:

@table @asis
@item Y
Year:  ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
@item YY
Year:  ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
@item BY
Year:  ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
@item YYY
Year:  ``1991'' for 1991, ``23'' for 23 AD.
@item YYYY
Year:  ``1991'' for 1991, ``+23'' for 23 AD.
@item aa
Year:  ``ad'' or blank.
@item AA
Year:  ``AD'' or blank.
@item aaa
Year:  ``ad '' or blank.  (Note trailing space.)
@item AAA
Year:  ``AD '' or blank.
@item aaaa
Year:  ``a.d.'' or blank.
@item AAAA
Year:  ``A.D.'' or blank.
@item bb
Year:  ``bc'' or blank.
@item BB
Year:  ``BC'' or blank.
@item bbb
Year:  `` bc'' or blank.  (Note leading space.)
@item BBB
Year:  `` BC'' or blank.
@item bbbb
Year:  ``b.c.'' or blank.
@item BBBB
Year:  ``B.C.'' or blank.
@item M
Month:  ``8'' for August.
@item MM
Month:  ``08'' for August.
@item BM
Month:  `` 8'' for August.
@item MMM
Month:  ``AUG'' for August.
@item Mmm
Month:  ``Aug'' for August.
@item mmm
Month:  ``aug'' for August.
@item MMMM
Month:  ``AUGUST'' for August.
@item Mmmm
Month:  ``August'' for August.
@item D
Day:  ``7'' for 7th day of month.
@item DD
Day:  ``07'' for 7th day of month.
@item BD
Day:  `` 7'' for 7th day of month.
@item W
Weekday:  ``0'' for Sunday, ``6'' for Saturday.
@item WWW
Weekday:  ``SUN'' for Sunday.
@item Www
Weekday:  ``Sun'' for Sunday.
@item www
Weekday:  ``sun'' for Sunday.
@item WWWW
Weekday:  ``SUNDAY'' for Sunday.
@item Wwww
Weekday:  ``Sunday'' for Sunday.
@item d
Day of year:  ``34'' for Feb. 3.
@item ddd
Day of year:  ``034'' for Feb. 3.
@item bdd
Day of year:  `` 34'' for Feb. 3.
@item h
Hour:  ``5'' for 5 AM; ``17'' for 5 PM.
@item hh
Hour:  ``05'' for 5 AM; ``17'' for 5 PM.
@item bh
Hour:  `` 5'' for 5 AM; ``17'' for 5 PM.
@item H
Hour:  ``5'' for 5 AM and 5 PM.
@item HH
Hour:  ``05'' for 5 AM and 5 PM.
@item BH
Hour:  `` 5'' for 5 AM and 5 PM.
@item p
AM/PM:  ``a'' or ``p''.
@item P
AM/PM:  ``A'' or ``P''.
@item pp
AM/PM:  ``am'' or ``pm''.
@item PP
AM/PM:  ``AM'' or ``PM''.
@item pppp
AM/PM:  ``a.m.'' or ``p.m.''.
@item PPPP
AM/PM:  ``A.M.'' or ``P.M.''.
@item m
Minutes:  ``7'' for 7.
@item mm
Minutes:  ``07'' for 7.
@item bm
Minutes:  `` 7'' for 7.
@item s
Seconds:  ``7'' for 7;  ``7.23'' for 7.23.
@item ss
Seconds:  ``07'' for 7;  ``07.23'' for 7.23.
@item bs
Seconds:  `` 7'' for 7;  `` 7.23'' for 7.23.
@item SS
Optional seconds:  ``07'' for 7;  blank for 0.
@item BS
Optional seconds:  `` 7'' for 7;  blank for 0.
@item N
Numeric date/time:  ``726842.25'' for 6:00am Wed Jan 9, 1991.
@item n
Numeric date:  ``726842'' for any time on Wed Jan 9, 1991.
@item J
Julian date/time:  ``2448265.75'' for 6:00am Wed Jan 9, 1991.
@item j
Julian date:  ``2448266'' for any time on Wed Jan 9, 1991.
@item U
Unix time:  ``663400800'' for 6:00am Wed Jan 9, 1991.
@item X
Brackets suppression.  An ``X'' at the front of the format
causes the surrounding @w{@samp{< >}} delimiters to be omitted
when formatting dates.  Note that the brackets are still
required for algebraic entry.
@end table

If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
colon is also omitted if the seconds part is zero.

If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
appear in the format, then negative year numbers are displayed
without a minus sign.  Note that ``aa'' and ``bb'' are mutually
exclusive.  Some typical usages would be @samp{YYYY AABB};
@samp{AAAYYYYBBB}; @samp{YYYYBBB}.

The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
reading unless several of these codes are strung together with no
punctuation in between, in which case the input must have exactly as
many digits as there are letters in the format.

The ``j,'' ``J,'' and ``U'' formats do not make any time zone
adjustment.  They effectively use @samp{julian(x,0)} and
@samp{unixtime(x,0)} to make the conversion; @pxref{Date Arithmetic}.

@node Free-Form Dates, Standard Date Formats, Date Formatting Codes, Date Formats
@subsubsection Free-Form Dates

@noindent
When reading a date form during algebraic entry, Calc falls back
on the algorithm described here if the input does not exactly
match the current date format.  This algorithm generally
``does the right thing'' and you don't have to worry about it,
but it is described here in full detail for the curious.

Calc does not distinguish between upper- and lower-case letters
while interpreting dates.

First, the time portion, if present, is located somewhere in the
text and then removed.  The remaining text is then interpreted as
the date.

A time is of the form @samp{hh:mm:ss}, possibly with the seconds
part omitted and possibly with an AM/PM indicator added to indicate
12-hour time.  If the AM/PM is present, the minutes may also be
omitted.  The AM/PM part may be any of the words @samp{am},
@samp{pm}, @samp{noon}, or @samp{midnight}; each of these may be
abbreviated to one letter, and the alternate forms @samp{a.m.},
@samp{p.m.}, and @samp{mid} are also understood.  Obviously
@samp{noon} and @samp{midnight} are allowed only on 12:00:00.
The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
recognized with no number attached.

If there is no AM/PM indicator, the time is interpreted in 24-hour
format.

To read the date portion, all words and numbers are isolated
from the string; other characters are ignored.  All words must
be either month names or day-of-week names (the latter of which
are ignored).  Names can be written in full or as three-letter
abbreviations.

Large numbers, or numbers with @samp{+} or @samp{-} signs,
are interpreted as years.  If one of the other numbers is
greater than 12, then that must be the day and the remaining
number in the input is therefore the month.  Otherwise, Calc
assumes the month, day and year are in the same order that they
appear in the current date format.  If the year is omitted, the
current year is taken from the system clock.

If there are too many or too few numbers, or any unrecognizable
words, then the input is rejected.

If there are any large numbers (of five digits or more) other than
the year, they are ignored on the assumption that they are something
like Julian dates that were included along with the traditional
date components when the date was formatted.

One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
may optionally be used; the latter two are equivalent to a
minus sign on the year value.

If you always enter a four-digit year, and use a name instead
of a number for the month, there is no danger of ambiguity.

@node Standard Date Formats,  , Free-Form Dates, Date Formats
@subsubsection Standard Date Formats

@noindent
There are actually ten standard date formats, numbered 0 through 9.
Entering a blank line at the @kbd{d d} command's prompt gives
you format number 1, Calc's usual format.  You can enter any digit
to select the other formats.

To create your own standard date formats, give a numeric prefix
argument from 0 to 9 to the @w{@kbd{d d}} command.  The format you
enter will be recorded as the new standard format of that
number, as well as becoming the new current date format.
You can save your formats permanently with the @w{@kbd{m m}}
command (@pxref{Mode Settings}).

@table @asis
@item 0
@samp{N}  (Numerical format)
@item 1
@samp{<H:mm:SSpp >Www Mmm D, YYYY}  (American format)
@item 2
@samp{D Mmm YYYY<, h:mm:SS>}  (European format)
@item 3
@samp{Www Mmm BD< hh:mm:ss> YYYY}  (Unix written date format)
@item 4
@samp{M/D/Y< H:mm:SSpp>}  (American slashed format)
@item 5
@samp{D.M.Y< h:mm:SS>}  (European dotted format)
@item 6
@samp{M-D-Y< H:mm:SSpp>}  (American dashed format)
@item 7
@samp{D-M-Y< h:mm:SS>}  (European dashed format)
@item 8
@samp{j<, h:mm:ss>}  (Julian day plus time)
@item 9
@samp{YYddd< hh:mm:ss>}  (Year-day format)
@end table

@node Truncating the Stack, Justification, Date Formats, Display Modes
@subsection Truncating the Stack

@noindent
@kindex d t
@pindex calc-truncate-stack
@cindex Truncating the stack
@cindex Narrowing the stack
The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
line that marks the top-of-stack up or down in the Calculator buffer.
The number right above that line is considered to the be at the top of
the stack.  Any numbers below that line are ``hidden'' from all stack
operations (although still visible to the user).  This is similar to the
Emacs ``narrowing'' feature, except that the values below the @samp{.}
are @emph{visible}, just temporarily frozen.  This feature allows you to
keep several independent calculations running at once in different parts
of the stack, or to apply a certain command to an element buried deep in
the stack.

Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
is on.  Thus, this line and all those below it become hidden.  To un-hide
these lines, move down to the end of the buffer and press @w{@kbd{d t}}.
With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
bottom @expr{n} values in the buffer.  With a negative argument, it hides
all but the top @expr{n} values.  With an argument of zero, it hides zero
values, i.e., moves the @samp{.} all the way down to the bottom.

@kindex d [
@pindex calc-truncate-up
@kindex d ]
@pindex calc-truncate-down
The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
(@code{calc-truncate-down}) commands move the @samp{.} up or down one
line at a time (or several lines with a prefix argument).

@node Justification, Labels, Truncating the Stack, Display Modes
@subsection Justification

@noindent
@kindex d <
@pindex calc-left-justify
@kindex d =
@pindex calc-center-justify
@kindex d >
@pindex calc-right-justify
Values on the stack are normally left-justified in the window.  You can
control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
@kbd{d >} (@code{calc-right-justify}), or @kbd{d =}
(@code{calc-center-justify}).  For example, in Right-Justification mode,
stack entries are displayed flush-right against the right edge of the
window.

If you change the width of the Calculator window you may have to type
@kbd{d @key{SPC}} (@code{calc-refresh}) to re-align right-justified or centered
text.

Right-justification is especially useful together with fixed-point
notation (see @code{d f}; @code{calc-fix-notation}).  With these modes
together, the decimal points on numbers will always line up.

With a numeric prefix argument, the justification commands give you
a little extra control over the display.  The argument specifies the
horizontal ``origin'' of a display line.  It is also possible to
specify a maximum line width using the @kbd{d b} command (@pxref{Normal
Language Modes}).  For reference, the precise rules for formatting and
breaking lines are given below.  Notice that the interaction between
origin and line width is slightly different in each justification
mode.

In Left-Justified mode, the line is indented by a number of spaces
given by the origin (default zero).  If the result is longer than the
maximum line width, if given, or too wide to fit in the Calc window
otherwise, then it is broken into lines which will fit; each broken
line is indented to the origin.

In Right-Justified mode, lines are shifted right so that the rightmost
character is just before the origin, or just before the current
window width if no origin was specified.  If the line is too long
for this, then it is broken; the current line width is used, if
specified, or else the origin is used as a width if that is
specified, or else the line is broken to fit in the window.

In Centering mode, the origin is the column number of the center of
each stack entry.  If a line width is specified, lines will not be
allowed to go past that width; Calc will either indent less or
break the lines if necessary.  If no origin is specified, half the
line width or Calc window width is used.

Note that, in each case, if line numbering is enabled the display
is indented an additional four spaces to make room for the line
number.  The width of the line number is taken into account when
positioning according to the current Calc window width, but not
when positioning by explicit origins and widths.  In the latter
case, the display is formatted as specified, and then uniformly
shifted over four spaces to fit the line numbers.

@node Labels,  , Justification, Display Modes
@subsection Labels

@noindent
@kindex d @{
@pindex calc-left-label
The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
then displays that string to the left of every stack entry.  If the
entries are left-justified (@pxref{Justification}), then they will
appear immediately after the label (unless you specified an origin
greater than the length of the label).  If the entries are centered
or right-justified, the label appears on the far left and does not
affect the horizontal position of the stack entry.

Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.

@kindex d @}
@pindex calc-right-label
The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
label on the righthand side.  It does not affect positioning of
the stack entries unless they are right-justified.  Also, if both
a line width and an origin are given in Right-Justified mode, the
stack entry is justified to the origin and the righthand label is
justified to the line width.

One application of labels would be to add equation numbers to
formulas you are manipulating in Calc and then copying into a
document (possibly using Embedded mode).  The equations would
typically be centered, and the equation numbers would be on the
left or right as you prefer.

@node Language Modes, Modes Variable, Display Modes, Mode Settings
@section Language Modes

@noindent
The commands in this section change Calc to use a different notation for
entry and display of formulas, corresponding to the conventions of some
other common language such as Pascal or La@TeX{}.  Objects displayed on the
stack or yanked from the Calculator to an editing buffer will be formatted
in the current language; objects entered in algebraic entry or yanked from
another buffer will be interpreted according to the current language.

The current language has no effect on things written to or read from the
trail buffer, nor does it affect numeric entry.  Only algebraic entry is
affected.  You can make even algebraic entry ignore the current language
and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.

For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
program; elsewhere in the program you need the derivatives of this formula
with respect to @samp{a[1]} and @samp{a[2]}.  First, type @kbd{d C}
to switch to C notation.  Now use @code{C-u C-x * g} to grab the formula
into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
to the first variable, and @kbd{C-x * y} to yank the formula for the derivative
back into your C program.  Press @kbd{U} to undo the differentiation and
repeat with @kbd{a d a[2] @key{RET}} for the other derivative.

Without being switched into C mode first, Calc would have misinterpreted
the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
@code{atan} was equivalent to Calc's built-in @code{arctan} function,
and would have written the formula back with notations (like implicit
multiplication) which would not have been valid for a C program.

As another example, suppose you are maintaining a C program and a La@TeX{}
document, each of which needs a copy of the same formula.  You can grab the
formula from the program in C mode, switch to La@TeX{} mode, and yank the
formula into the document in La@TeX{} math-mode format.

Language modes are selected by typing the letter @kbd{d} followed by a
shifted letter key.

@menu
* Normal Language Modes::
* C FORTRAN Pascal::
* TeX and LaTeX Language Modes::
* Eqn Language Mode::
* Mathematica Language Mode::
* Maple Language Mode::
* Compositions::
* Syntax Tables::
@end menu

@node Normal Language Modes, C FORTRAN Pascal, Language Modes, Language Modes
@subsection Normal Language Modes

@noindent
@kindex d N
@pindex calc-normal-language
The @kbd{d N} (@code{calc-normal-language}) command selects the usual
notation for Calc formulas, as described in the rest of this manual.
Matrices are displayed in a multi-line tabular format, but all other
objects are written in linear form, as they would be typed from the
keyboard.

@kindex d O
@pindex calc-flat-language
@cindex Matrix display
The @kbd{d O} (@code{calc-flat-language}) command selects a language
identical with the normal one, except that matrices are written in
one-line form along with everything else.  In some applications this
form may be more suitable for yanking data into other buffers.

@kindex d b
@pindex calc-line-breaking
@cindex Line breaking
@cindex Breaking up long lines
Even in one-line mode, long formulas or vectors will still be split
across multiple lines if they exceed the width of the Calculator window.
The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
feature on and off.  (It works independently of the current language.)
If you give a numeric prefix argument of five or greater to the @kbd{d b}
command, that argument will specify the line width used when breaking
long lines.

@kindex d B
@pindex calc-big-language
The @kbd{d B} (@code{calc-big-language}) command selects a language
which uses textual approximations to various mathematical notations,
such as powers, quotients, and square roots:

@example
  ____________
 | a + 1    2
 | ----- + c
\|   b
@end example

@noindent
in place of @samp{sqrt((a+1)/b + c^2)}.

Subscripts like @samp{a_i} are displayed as actual subscripts in Big
mode.  Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
are displayed as @samp{a} with subscripts separated by commas:
@samp{i, j}.  They must still be entered in the usual underscore
notation.

One slight ambiguity of Big notation is that

@example
  3
- -
  4
@end example

@noindent
can represent either the negative rational number @expr{-3:4}, or the
actual expression @samp{-(3/4)}; but the latter formula would normally
never be displayed because it would immediately be evaluated to
@expr{-3:4} or @expr{-0.75}, so this ambiguity is not a problem in
typical use.

Non-decimal numbers are displayed with subscripts.  Thus there is no
way to tell the difference between @samp{16#C2} and @samp{C2_16},
though generally you will know which interpretation is correct.
Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
in Big mode.

In Big mode, stack entries often take up several lines.  To aid
readability, stack entries are separated by a blank line in this mode.
You may find it useful to expand the Calc window's height using
@kbd{C-x ^} (@code{enlarge-window}) or to make the Calc window the only
one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).

Long lines are currently not rearranged to fit the window width in
Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
to scroll across a wide formula.  For really big formulas, you may
even need to use @kbd{@{} and @kbd{@}} to scroll up and down.

@kindex d U
@pindex calc-unformatted-language
The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
the use of operator notation in formulas.  In this mode, the formula
shown above would be displayed:

@example
sqrt(add(div(add(a, 1), b), pow(c, 2)))
@end example

These four modes differ only in display format, not in the format
expected for algebraic entry.  The standard Calc operators work in
all four modes, and unformatted notation works in any language mode
(except that Mathematica mode expects square brackets instead of
parentheses).

@node C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language Modes, Language Modes
@subsection C, FORTRAN, and Pascal Modes

@noindent
@kindex d C
@pindex calc-c-language
@cindex C language
The @kbd{d C} (@code{calc-c-language}) command selects the conventions
of the C language for display and entry of formulas.  This differs from
the normal language mode in a variety of (mostly minor) ways.  In
particular, C language operators and operator precedences are used in
place of Calc's usual ones.  For example, @samp{a^b} means @samp{xor(a,b)}
in C mode; a value raised to a power is written as a function call,
@samp{pow(a,b)}.

In C mode, vectors and matrices use curly braces instead of brackets.
Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
rather than using the @samp{#} symbol.  Array subscripting is
translated into @code{subscr} calls, so that @samp{a[i]} in C
mode is the same as @samp{a_i} in Normal mode.  Assignments
turn into the @code{assign} function, which Calc normally displays
using the @samp{:=} symbol.

The variables @code{pi} and @code{e} would be displayed @samp{pi}
and @samp{e} in Normal mode, but in C mode they are displayed as
@samp{M_PI} and @samp{M_E}, corresponding to the names of constants
typically provided in the @file{<math.h>} header.  Functions whose
names are different in C are translated automatically for entry and
display purposes.  For example, entering @samp{asin(x)} will push the
formula @samp{arcsin(x)} onto the stack; this formula will be displayed
as @samp{asin(x)} as long as C mode is in effect.

@kindex d P
@pindex calc-pascal-language
@cindex Pascal language
The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
conventions.  Like C mode, Pascal mode interprets array brackets and uses
a different table of operators.  Hexadecimal numbers are entered and
displayed with a preceding dollar sign.  (Thus the regular meaning of
@kbd{$2} during algebraic entry does not work in Pascal mode, though
@kbd{$} (and @kbd{$$}, etc.) not followed by digits works the same as
always.)  No special provisions are made for other non-decimal numbers,
vectors, and so on, since there is no universally accepted standard way
of handling these in Pascal.

@kindex d F
@pindex calc-fortran-language
@cindex FORTRAN language
The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
conventions.  Various function names are transformed into FORTRAN
equivalents.  Vectors are written as @samp{/1, 2, 3/}, and may be
entered this way or using square brackets.  Since FORTRAN uses round
parentheses for both function calls and array subscripts, Calc displays
both in the same way; @samp{a(i)} is interpreted as a function call
upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
Also, if the variable @code{a} has been declared to have type
@code{vector} or @code{matrix} then @samp{a(i)} will be parsed as a
subscript.  (@xref{Declarations}.)  Usually it doesn't matter, though;
if you enter the subscript expression @samp{a(i)} and Calc interprets
it as a function call, you'll never know the difference unless you
switch to another language mode or replace @code{a} with an actual
vector (or unless @code{a} happens to be the name of a built-in
function!).

Underscores are allowed in variable and function names in all of these
language modes.  The underscore here is equivalent to the @samp{#} in
Normal mode, or to hyphens in the underlying Emacs Lisp variable names.

FORTRAN and Pascal modes normally do not adjust the case of letters in
formulas.  Most built-in Calc names use lower-case letters.  If you use a
positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
modes will use upper-case letters exclusively for display, and will
convert to lower-case on input.  With a negative prefix, these modes
convert to lower-case for display and input.

@node TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN Pascal, Language Modes
@subsection @TeX{} and La@TeX{} Language Modes

@noindent
@kindex d T
@pindex calc-tex-language
@cindex TeX language
@kindex d L
@pindex calc-latex-language
@cindex LaTeX language
The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
and the @kbd{d L} (@code{calc-latex-language}) command selects the
conventions of ``math mode'' in La@TeX{}, a typesetting language that
uses @TeX{} as its formatting engine.  Calc's La@TeX{} language mode can
read any formula that the @TeX{} language mode can, although La@TeX{}
mode may display it differently.

Formulas are entered and displayed in the appropriate notation;
@texline @math{\sin(a/b)}
@infoline @expr{sin(a/b)}
will appear as @samp{\sin\left( a \over b \right)} in @TeX{} mode and
@samp{\sin\left(\frac@{a@}@{b@}\right)} in La@TeX{} mode.
Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
La@TeX{}; these should be omitted when interfacing with Calc.  To Calc,
the @samp{$} sign has the same meaning it always does in algebraic
formulas (a reference to an existing entry on the stack).

Complex numbers are displayed as in @samp{3 + 4i}.  Fractions and
quotients are written using @code{\over} in @TeX{} mode (as in 
@code{@{a \over b@}}) and @code{\frac} in La@TeX{} mode (as in
@code{\frac@{a@}@{b@}});  binomial coefficients are written with
@code{\choose} in @TeX{} mode (as in @code{@{a \choose b@}}) and
@code{\binom} in La@TeX{} mode (as in @code{\binom@{a@}@{b@}}).
Interval forms are written with @code{\ldots}, and error forms are
written with @code{\pm}. Absolute values are written as in 
@samp{|x + 1|}, and the floor and ceiling functions are written with
@code{\lfloor}, @code{\rfloor}, etc. The words @code{\left} and
@code{\right} are ignored when reading formulas in @TeX{} and La@TeX{}
modes.  Both @code{inf} and @code{uinf} are written as @code{\infty};
when read, @code{\infty} always translates to @code{inf}.

Function calls are written the usual way, with the function name followed
by the arguments in parentheses.  However, functions for which @TeX{}
and La@TeX{} have special names (like @code{\sin}) will use curly braces
instead of parentheses for very simple arguments.  During input, curly
braces and parentheses work equally well for grouping, but when the
document is formatted the curly braces will be invisible.  Thus the
printed result is 
@texline @math{\sin{2 x}}
@infoline @expr{sin 2x} 
but 
@texline @math{\sin(2 + x)}.
@infoline @expr{sin(2 + x)}.

Function and variable names not treated specially by @TeX{} and La@TeX{}
are simply written out as-is, which will cause them to come out in
italic letters in the printed document.  If you invoke @kbd{d T} or
@kbd{d L} with a positive numeric prefix argument, names of more than
one character will instead be enclosed in a protective commands that
will prevent them from being typeset in the math italics; they will be
written @samp{\hbox@{@var{name}@}} in @TeX{} mode and 
@samp{\text@{@var{name}@}} in La@TeX{} mode.  The
@samp{\hbox@{ @}} and @samp{\text@{ @}} notations are ignored during
reading.  If you use a negative prefix argument, such function names are
written @samp{\@var{name}}, and function names that begin with @code{\} during
reading have the @code{\} removed.  (Note that in this mode, long
variable names are still written with @code{\hbox} or @code{\text}.
However, you can always make an actual variable name like @code{\bar} in
any @TeX{} mode.)

During reading, text of the form @samp{\matrix@{ ...@: @}} is replaced
by @samp{[ ...@: ]}.  The same also applies to @code{\pmatrix} and
@code{\bmatrix}.  In La@TeX{} mode this also applies to 
@samp{\begin@{matrix@} ... \end@{matrix@}},
@samp{\begin@{bmatrix@} ... \end@{bmatrix@}},
@samp{\begin@{pmatrix@} ... \end@{pmatrix@}}, as well as
@samp{\begin@{smallmatrix@} ... \end@{smallmatrix@}}.
The symbol @samp{&} is interpreted as a comma,
and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
During output, matrices are displayed in @samp{\matrix@{ a & b \\ c & d@}}
format in @TeX{} mode and in 
@samp{\begin@{pmatrix@} a & b \\ c & d \end@{pmatrix@}} format in
La@TeX{} mode; you may need to edit this afterwards to change to your
preferred matrix form.  If you invoke @kbd{d T} or @kbd{d L} with an
argument of 2 or -2, then matrices will be displayed in two-dimensional
form, such as 

@example
\begin@{pmatrix@}
a & b \\
c & d
\end@{pmatrix@}
@end example

@noindent
This may be convenient for isolated matrices, but could lead to
expressions being displayed like

@example
\begin@{pmatrix@} \times x
a & b \\
c & d
\end@{pmatrix@}
@end example

@noindent
While this wouldn't bother Calc, it is incorrect La@TeX{}.
(Similarly for @TeX{}.)

Accents like @code{\tilde} and @code{\bar} translate into function
calls internally (@samp{tilde(x)}, @samp{bar(x)}).  The @code{\underline}
sequence is treated as an accent.  The @code{\vec} accent corresponds
to the function name @code{Vec}, because @code{vec} is the name of
a built-in Calc function.  The following table shows the accents
in Calc, @TeX{}, La@TeX{} and @dfn{eqn} (described in the next section):

@iftex
@begingroup
@let@calcindexershow=@calcindexernoshow  @c Suppress marginal notes
@let@calcindexersh=@calcindexernoshow
@end iftex
@ignore
@starindex
@end ignore
@tindex acute
@ignore
@starindex
@end ignore
@tindex Acute
@ignore
@starindex
@end ignore
@tindex bar
@ignore
@starindex
@end ignore
@tindex Bar
@ignore
@starindex
@end ignore
@tindex breve
@ignore
@starindex
@end ignore
@tindex Breve
@ignore
@starindex
@end ignore
@tindex check
@ignore
@starindex
@end ignore
@tindex Check
@ignore
@starindex
@end ignore
@tindex dddot
@ignore
@starindex
@end ignore
@tindex ddddot
@ignore
@starindex
@end ignore
@tindex dot
@ignore
@starindex
@end ignore
@tindex Dot
@ignore
@starindex
@end ignore
@tindex dotdot
@ignore
@starindex
@end ignore
@tindex DotDot
@ignore
@starindex
@end ignore
@tindex dyad
@ignore
@starindex
@end ignore
@tindex grave
@ignore
@starindex
@end ignore
@tindex Grave
@ignore
@starindex
@end ignore
@tindex hat
@ignore
@starindex
@end ignore
@tindex Hat
@ignore
@starindex
@end ignore
@tindex Prime
@ignore
@starindex
@end ignore
@tindex tilde
@ignore
@starindex
@end ignore
@tindex Tilde
@ignore
@starindex
@end ignore
@tindex under
@ignore
@starindex
@end ignore
@tindex Vec
@ignore
@starindex
@end ignore
@tindex VEC
@iftex
@endgroup
@end iftex
@example
Calc      TeX           LaTeX         eqn
----      ---           -----         ---
acute     \acute        \acute        
Acute                   \Acute        
bar       \bar          \bar          bar
Bar                     \Bar
breve     \breve        \breve        
Breve                   \Breve        
check     \check        \check        
Check                   \Check        
dddot                   \dddot
ddddot                  \ddddot
dot       \dot          \dot          dot
Dot                     \Dot
dotdot    \ddot         \ddot         dotdot
DotDot                  \Ddot         
dyad                                  dyad
grave     \grave        \grave        
Grave                   \Grave        
hat       \hat          \hat          hat
Hat                     \Hat          
Prime                                 prime
tilde     \tilde        \tilde        tilde
Tilde                   \Tilde
under     \underline    \underline    under
Vec       \vec          \vec          vec
VEC                     \Vec
@end example

The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
@samp{@{@var{a} \to @var{b}@}}.  @TeX{} defines @code{\to} as an
alias for @code{\rightarrow}.  However, if the @samp{=>} is the
top-level expression being formatted, a slightly different notation
is used:  @samp{\evalto @var{a} \to @var{b}}.  The @code{\evalto}
word is ignored by Calc's input routines, and is undefined in @TeX{}.
You will typically want to include one of the following definitions
at the top of a @TeX{} file that uses @code{\evalto}:

@example
\def\evalto@{@}
\def\evalto#1\to@{@}
@end example

The first definition formats evaluates-to operators in the usual
way.  The second causes only the @var{b} part to appear in the
printed document; the @var{a} part and the arrow are hidden.
Another definition you may wish to use is @samp{\let\to=\Rightarrow}
which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
@xref{Evaluates-To Operator}, for a discussion of @code{evalto}.

The complete set of @TeX{} control sequences that are ignored during
reading is:

@example
\hbox  \mbox  \text  \left  \right
\,  \>  \:  \;  \!  \quad  \qquad  \hfil  \hfill
\displaystyle  \textstyle  \dsize  \tsize
\scriptstyle  \scriptscriptstyle  \ssize  \ssize
\rm  \bf  \it  \sl  \roman  \bold  \italic  \slanted
\cal  \mit  \Cal  \Bbb  \frak  \goth
\evalto
@end example

Note that, because these symbols are ignored, reading a @TeX{} or
La@TeX{} formula into Calc and writing it back out may lose spacing and
font information. 

Also, the ``discretionary multiplication sign'' @samp{\*} is read
the same as @samp{*}.

@ifinfo
The @TeX{} version of this manual includes some printed examples at the
end of this section.
@end ifinfo
@iftex
Here are some examples of how various Calc formulas are formatted in @TeX{}:

@example
@group
sin(a^2 / b_i)
\sin\left( {a^2 \over b_i} \right)
@end group
@end example
@tex
$$ \sin\left( a^2 \over b_i \right) $$
@end tex
@sp 1

@example
@group
[(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
[3 + 4i, @{3 \over 4@}, 3 \pm 4, [3 \ldots \infty)]
@end group
@end example
@tex
\turnoffactive
$$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
@end tex
@sp 1

@example
@group
[abs(a), abs(a / b), floor(a), ceil(a / b)]
[|a|, \left| a \over b \right|,
 \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
@end group
@end example
@tex
$$ [|a|, \left| a \over b \right|,
    \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
@end tex
@sp 1

@example
@group
[sin(a), sin(2 a), sin(2 + a), sin(a / b)]
[\sin@{a@}, \sin@{2 a@}, \sin(2 + a),
 \sin\left( @{a \over b@} \right)]
@end group
@end example
@tex
\turnoffactive
$$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
@end tex
@sp 2

First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
@kbd{C-u - d T} (using the example definition
@samp{\def\foo#1@{\tilde F(#1)@}}:

@example
@group
[f(a), foo(bar), sin(pi)]
[f(a), foo(bar), \sin{\pi}]
[f(a), \hbox@{foo@}(\hbox@{bar@}), \sin@{\pi@}]
[f(a), \foo@{\hbox@{bar@}@}, \sin@{\pi@}]
@end group
@end example
@tex
$$ [f(a), foo(bar), \sin{\pi}] $$
$$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
$$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
@end tex
@sp 2

First with @samp{\def\evalto@{@}}, then with @samp{\def\evalto#1\to@{@}}:

@example
@group
2 + 3 => 5
\evalto 2 + 3 \to 5
@end group
@end example
@tex
\turnoffactive
$$ 2 + 3 \to 5 $$
$$ 5 $$
@end tex
@sp 2

First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:

@example
@group
[2 + 3 => 5, a / 2 => (b + c) / 2]
[@{2 + 3 \to 5@}, @{@{a \over 2@} \to @{b + c \over 2@}@}]
@end group
@end example
@tex
\turnoffactive
$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
{\let\to\Rightarrow
$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
@end tex
@sp 2

Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:

@example
@group
[ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
\matrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
\pmatrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
@end group
@end example
@tex
\turnoffactive
$$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
$$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
@end tex
@sp 2
@end iftex

@node Eqn Language Mode, Mathematica Language Mode, TeX and LaTeX Language Modes, Language Modes
@subsection Eqn Language Mode

@noindent
@kindex d E
@pindex calc-eqn-language
@dfn{Eqn} is another popular formatter for math formulas.  It is
designed for use with the TROFF text formatter, and comes standard
with many versions of Unix.  The @kbd{d E} (@code{calc-eqn-language})
command selects @dfn{eqn} notation.

The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
a significant part in the parsing of the language.  For example,
@samp{sqrt x+1 + y} treats @samp{x+1} as the argument of the
@code{sqrt} operator.  @dfn{Eqn} also understands more conventional
grouping using curly braces:  @samp{sqrt@{x+1@} + y}.  Braces are
required only when the argument contains spaces.

In Calc's @dfn{eqn} mode, however, curly braces are required to
delimit arguments of operators like @code{sqrt}.  The first of the
above examples would treat only the @samp{x} as the argument of
@code{sqrt}, and in fact @samp{sin x+1} would be interpreted as
@samp{sin * x + 1}, because @code{sin} is not a special operator
in the @dfn{eqn} language.  If you always surround the argument
with curly braces, Calc will never misunderstand.

Calc also understands parentheses as grouping characters.  Another
peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
words with spaces from any surrounding characters that aren't curly
braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
(The spaces around @code{sin} are important to make @dfn{eqn}
recognize that @code{sin} should be typeset in a roman font, and
the spaces around @code{x} and @code{y} are a good idea just in
case the @dfn{eqn} document has defined special meanings for these
names, too.)

Powers and subscripts are written with the @code{sub} and @code{sup}
operators, respectively.  Note that the caret symbol @samp{^} is
treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
symbol (these are used to introduce spaces of various widths into
the typeset output of @dfn{eqn}).

As in La@TeX{} mode, Calc's formatter omits parentheses around the
arguments of functions like @code{ln} and @code{sin} if they are
``simple-looking''; in this case Calc surrounds the argument with
braces, separated by a @samp{~} from the function name: @samp{sin~@{x@}}.

Font change codes (like @samp{roman @var{x}}) and positioning codes
(like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
@dfn{eqn} reader.  Also ignored are the words @code{left}, @code{right},
@code{mark}, and @code{lineup}.  Quotation marks in @dfn{eqn} mode input
are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
@samp{sqrt @{1+x@}}; this is only an approximation to the true meaning
of quotes in @dfn{eqn}, but it is good enough for most uses.

Accent codes (@samp{@var{x} dot}) are handled by treating them as
function calls (@samp{dot(@var{x})}) internally.  
@xref{TeX and LaTeX Language Modes}, for a table of these accent
functions.  The @code{prime} accent is treated specially if it occurs on
a variable or function name: @samp{f prime prime @w{( x prime )}} is
stored internally as @samp{f'@w{'}(x')}.  For example, taking the
derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.

Assignments are written with the @samp{<-} (left-arrow) symbol,
and @code{evalto} operators are written with @samp{->} or
@samp{evalto ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
of this).  The regular Calc symbols @samp{:=} and @samp{=>} are also
recognized for these operators during reading.

Vectors in @dfn{eqn} mode use regular Calc square brackets, but
matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
The words @code{lcol} and @code{rcol} are recognized as synonyms
for @code{ccol} during input, and are generated instead of @code{ccol}
if the matrix justification mode so specifies.

@node Mathematica Language Mode, Maple Language Mode, Eqn Language Mode, Language Modes
@subsection Mathematica Language Mode

@noindent
@kindex d M
@pindex calc-mathematica-language
@cindex Mathematica language
The @kbd{d M} (@code{calc-mathematica-language}) command selects the
conventions of Mathematica.  Notable differences in Mathematica mode
are that the names of built-in functions are capitalized, and function
calls use square brackets instead of parentheses.  Thus the Calc
formula @samp{sin(2 x)} is entered and displayed @w{@samp{Sin[2 x]}} in
Mathematica mode.

Vectors and matrices use curly braces in Mathematica.  Complex numbers
are written @samp{3 + 4 I}.  The standard special constants in Calc are
written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
@code{Infinity}, @code{ComplexInfinity}, and @code{Indeterminate} in
Mathematica mode.
Non-decimal numbers are written, e.g., @samp{16^^7fff}.  Floating-point
numbers in scientific notation are written @samp{1.23*10.^3}.
Subscripts use double square brackets: @samp{a[[i]]}.

@node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes
@subsection Maple Language Mode

@noindent
@kindex d W
@pindex calc-maple-language
@cindex Maple language
The @kbd{d W} (@code{calc-maple-language}) command selects the
conventions of Maple.

Maple's language is much like C.  Underscores are allowed in symbol
names; square brackets are used for subscripts; explicit @samp{*}s for
multiplications are required.  Use either @samp{^} or @samp{**} to
denote powers.

Maple uses square brackets for lists and curly braces for sets.  Calc
interprets both notations as vectors, and displays vectors with square
brackets.  This means Maple sets will be converted to lists when they
pass through Calc.  As a special case, matrices are written as calls
to the function @code{matrix}, given a list of lists as the argument,
and can be read in this form or with all-capitals @code{MATRIX}.

The Maple interval notation @samp{2 .. 3} has no surrounding brackets;
Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]}, and
writes any kind of interval as @samp{2 .. 3}.  This means you cannot
see the difference between an open and a closed interval while in
Maple display mode.

Maple writes complex numbers as @samp{3 + 4*I}.  Its special constants
are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
@code{inf}, @code{uinf}, and @code{nan} display as @code{infinity}).
Floating-point numbers are written @samp{1.23*10.^3}.

Among things not currently handled by Calc's Maple mode are the
various quote symbols, procedures and functional operators, and
inert (@samp{&}) operators.

@node Compositions, Syntax Tables, Maple Language Mode, Language Modes
@subsection Compositions

@noindent
@cindex Compositions
There are several @dfn{composition functions} which allow you to get
displays in a variety of formats similar to those in Big language
mode.  Most of these functions do not evaluate to anything; they are
placeholders which are left in symbolic form by Calc's evaluator but
are recognized by Calc's display formatting routines.

Two of these, @code{string} and @code{bstring}, are described elsewhere.
@xref{Strings}.  For example, @samp{string("ABC")} is displayed as
@samp{ABC}.  When viewed on the stack it will be indistinguishable from
the variable @code{ABC}, but internally it will be stored as
@samp{string([65, 66, 67])} and can still be manipulated this way; for
example, the selection and vector commands @kbd{j 1 v v j u} would
select the vector portion of this object and reverse the elements, then
deselect to reveal a string whose characters had been reversed.

The composition functions do the same thing in all language modes
(although their components will of course be formatted in the current
language mode).  The one exception is Unformatted mode (@kbd{d U}),
which does not give the composition functions any special treatment.
The functions are discussed here because of their relationship to
the language modes.

@menu
* Composition Basics::
* Horizontal Compositions::
* Vertical Compositions::
* Other Compositions::
* Information about Compositions::
* User-Defined Compositions::
@end menu

@node Composition Basics, Horizontal Compositions, Compositions, Compositions
@subsubsection Composition Basics

@noindent
Compositions are generally formed by stacking formulas together
horizontally or vertically in various ways.  Those formulas are
themselves compositions.  @TeX{} users will find this analogous
to @TeX{}'s ``boxes.''  Each multi-line composition has a
@dfn{baseline}; horizontal compositions use the baselines to
decide how formulas should be positioned relative to one another.
For example, in the Big mode formula

@example
@group
          2
     a + b
17 + ------
       c
@end group
@end example

@noindent
the second term of the sum is four lines tall and has line three as
its baseline.  Thus when the term is combined with 17, line three
is placed on the same level as the baseline of 17.

@tex
\bigskip
@end tex

Another important composition concept is @dfn{precedence}.  This is
an integer that represents the binding strength of various operators.
For example, @samp{*} has higher precedence (195) than @samp{+} (180),
which means that @samp{(a * b) + c} will be formatted without the
parentheses, but @samp{a * (b + c)} will keep the parentheses.

The operator table used by normal and Big language modes has the
following precedences:

@example
_     1200   @r{(subscripts)}
%     1100   @r{(as in n}%@r{)}
-     1000   @r{(as in }-@r{n)}
!     1000   @r{(as in }!@r{n)}
mod    400
+/-    300
!!     210    @r{(as in n}!!@r{)}
!      210    @r{(as in n}!@r{)}
^      200
*      195    @r{(or implicit multiplication)}
/ % \  190
+ -    180    @r{(as in a}+@r{b)}
|      170
< =    160    @r{(and other relations)}
&&     110
||     100
? :     90
!!!     85
&&&     80
|||     75
:=      50
::      45
=>      40
@end example

The general rule is that if an operator with precedence @expr{n}
occurs as an argument to an operator with precedence @expr{m}, then
the argument is enclosed in parentheses if @expr{n < m}.  Top-level
expressions and expressions which are function arguments, vector
components, etc., are formatted with precedence zero (so that they
normally never get additional parentheses).

For binary left-associative operators like @samp{+}, the righthand
argument is actually formatted with one-higher precedence than shown
in the table.  This makes sure @samp{(a + b) + c} omits the parentheses,
but the unnatural form @samp{a + (b + c)} keeps its parentheses.
Right-associative operators like @samp{^} format the lefthand argument
with one-higher precedence.

@ignore
@starindex
@end ignore
@tindex cprec
The @code{cprec} function formats an expression with an arbitrary
precedence.  For example, @samp{cprec(abc, 185)} will combine into
sums and products as follows:  @samp{7 + abc}, @samp{7 (abc)} (because
this @code{cprec} form has higher precedence than addition, but lower
precedence than multiplication).

@tex
\bigskip
@end tex

A final composition issue is @dfn{line breaking}.  Calc uses two
different strategies for ``flat'' and ``non-flat'' compositions.
A non-flat composition is anything that appears on multiple lines
(not counting line breaking).  Examples would be matrices and Big
mode powers and quotients.  Non-flat compositions are displayed
exactly as specified.  If they come out wider than the current
window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
view them.

Flat compositions, on the other hand, will be broken across several
lines if they are too wide to fit the window.  Certain points in a
composition are noted internally as @dfn{break points}.  Calc's
general strategy is to fill each line as much as possible, then to
move down to the next line starting at the first break point that
didn't fit.  However, the line breaker understands the hierarchical
structure of formulas.  It will not break an ``inner'' formula if
it can use an earlier break point from an ``outer'' formula instead.
For example, a vector of sums might be formatted as:

@example
@group
[ a + b + c, d + e + f,
  g + h + i, j + k + l, m ]
@end group
@end example

@noindent
If the @samp{m} can fit, then so, it seems, could the @samp{g}.
But Calc prefers to break at the comma since the comma is part
of a ``more outer'' formula.  Calc would break at a plus sign
only if it had to, say, if the very first sum in the vector had
itself been too large to fit.

Of the composition functions described below, only @code{choriz}
generates break points.  The @code{bstring} function (@pxref{Strings})
also generates breakable items:  A break point is added after every
space (or group of spaces) except for spaces at the very beginning or
end of the string.

Composition functions themselves count as levels in the formula
hierarchy, so a @code{choriz} that is a component of a larger
@code{choriz} will be less likely to be broken.  As a special case,
if a @code{bstring} occurs as a component of a @code{choriz} or
@code{choriz}-like object (such as a vector or a list of arguments
in a function call), then the break points in that @code{bstring}
will be on the same level as the break points of the surrounding
object.

@node Horizontal Compositions, Vertical Compositions, Composition Basics, Compositions
@subsubsection Horizontal Compositions

@noindent
@ignore
@starindex
@end ignore
@tindex choriz
The @code{choriz} function takes a vector of objects and composes
them horizontally.  For example, @samp{choriz([17, a b/c, d])} formats
as @w{@samp{17a b / cd}} in Normal language mode, or as

@example
@group
  a b
17---d
   c
@end group
@end example

@noindent
in Big language mode.  This is actually one case of the general
function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
either or both of @var{sep} and @var{prec} may be omitted.
@var{Prec} gives the @dfn{precedence} to use when formatting
each of the components of @var{vec}.  The default precedence is
the precedence from the surrounding environment.

@var{Sep} is a string (i.e., a vector of character codes as might
be entered with @code{" "} notation) which should separate components
of the composition.  Also, if @var{sep} is given, the line breaker
will allow lines to be broken after each occurrence of @var{sep}.
If @var{sep} is omitted, the composition will not be breakable
(unless any of its component compositions are breakable).

For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
formatted as @samp{2 a + b c + (d = e)}.  To get the @code{choriz}
to have precedence 180 ``outwards'' as well as ``inwards,''
enclose it in a @code{cprec} form:  @samp{2 cprec(choriz(...), 180)}
formats as @samp{2 (a + b c + (d = e))}.

The baseline of a horizontal composition is the same as the
baselines of the component compositions, which are all aligned.

@node Vertical Compositions, Other Compositions, Horizontal Compositions, Compositions
@subsubsection Vertical Compositions

@noindent
@ignore
@starindex
@end ignore
@tindex cvert
The @code{cvert} function makes a vertical composition.  Each
component of the vector is centered in a column.  The baseline of
the result is by default the top line of the resulting composition.
For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
formats in Big mode as

@example
@group
f( a ,  2    )
  bb   a  + 1
  ccc     2
         b
@end group
@end example

@ignore
@starindex
@end ignore
@tindex cbase
There are several special composition functions that work only as
components of a vertical composition.  The @code{cbase} function
controls the baseline of the vertical composition; the baseline
will be the same as the baseline of whatever component is enclosed
in @code{cbase}.  Thus @samp{f(cvert([a, cbase(bb), ccc]),
cvert([a^2 + 1, cbase(b^2)]))} displays as

@example
@group
        2
       a  + 1
   a      2
f(bb ,   b   )
  ccc
@end group
@end example

@ignore
@starindex
@end ignore
@tindex ctbase
@ignore
@starindex
@end ignore
@tindex cbbase
There are also @code{ctbase} and @code{cbbase} functions which
make the baseline of the vertical composition equal to the top
or bottom line (rather than the baseline) of that component.
Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
cvert([cbbase(a / b)])} gives

@example
@group
        a
a       -
- + a + b
b   -
    b
@end group
@end example

There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
function in a given vertical composition.  These functions can also
be written with no arguments:  @samp{ctbase()} is a zero-height object
which means the baseline is the top line of the following item, and
@samp{cbbase()} means the baseline is the bottom line of the preceding
item.

@ignore
@starindex
@end ignore
@tindex crule
The @code{crule} function builds a ``rule,'' or horizontal line,
across a vertical composition.  By itself @samp{crule()} uses @samp{-}
characters to build the rule.  You can specify any other character,
e.g., @samp{crule("=")}.  The argument must be a character code or
vector of exactly one character code.  It is repeated to match the
width of the widest item in the stack.  For example, a quotient
with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:

@example
@group
a + 1
=====
  2
 b
@end group
@end example

@ignore
@starindex
@end ignore
@tindex clvert
@ignore
@starindex
@end ignore
@tindex crvert
Finally, the functions @code{clvert} and @code{crvert} act exactly
like @code{cvert} except that the items are left- or right-justified
in the stack.  Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
gives:

@example
@group
a   +   a
bb     bb
ccc   ccc
@end group
@end example

Like @code{choriz}, the vertical compositions accept a second argument
which gives the precedence to use when formatting the components.
Vertical compositions do not support separator strings.

@node Other Compositions, Information about Compositions, Vertical Compositions, Compositions
@subsubsection Other Compositions

@noindent
@ignore
@starindex
@end ignore
@tindex csup
The @code{csup} function builds a superscripted expression.  For
example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
language mode.  This is essentially a horizontal composition of
@samp{a} and @samp{b}, where @samp{b} is shifted up so that its
bottom line is one above the baseline.

@ignore
@starindex
@end ignore
@tindex csub
Likewise, the @code{csub} function builds a subscripted expression.
This shifts @samp{b} down so that its top line is one below the
bottom line of @samp{a} (note that this is not quite analogous to
@code{csup}).  Other arrangements can be obtained by using
@code{choriz} and @code{cvert} directly.

@ignore
@starindex
@end ignore
@tindex cflat
The @code{cflat} function formats its argument in ``flat'' mode,
as obtained by @samp{d O}, if the current language mode is normal
or Big.  It has no effect in other language modes.  For example,
@samp{a^(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
to improve its readability.

@ignore
@starindex
@end ignore
@tindex cspace
The @code{cspace} function creates horizontal space.  For example,
@samp{cspace(4)} is effectively the same as @samp{string("    ")}.
A second string (i.e., vector of characters) argument is repeated
instead of the space character.  For example, @samp{cspace(4, "ab")}
looks like @samp{abababab}.  If the second argument is not a string,
it is formatted in the normal way and then several copies of that
are composed together:  @samp{cspace(4, a^2)} yields

@example
@group
 2 2 2 2
a a a a
@end group
@end example

@noindent
If the number argument is zero, this is a zero-width object.

@ignore
@starindex
@end ignore
@tindex cvspace
The @code{cvspace} function creates vertical space, or a vertical
stack of copies of a certain string or formatted object.  The
baseline is the center line of the resulting stack.  A numerical
argument of zero will produce an object which contributes zero
height if used in a vertical composition.

@ignore
@starindex
@end ignore
@tindex ctspace
@ignore
@starindex
@end ignore
@tindex cbspace
There are also @code{ctspace} and @code{cbspace} functions which
create vertical space with the baseline the same as the baseline
of the top or bottom copy, respectively, of the second argument.
Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
displays as:

@example
@group
        a
        -
a       b
-   a   a
b + - + -
a   b   b
-   a
b   -
    b
@end group
@end example

@node Information about Compositions, User-Defined Compositions, Other Compositions, Compositions
@subsubsection Information about Compositions

@noindent
The functions in this section are actual functions; they compose their
arguments according to the current language and other display modes,
then return a certain measurement of the composition as an integer.

@ignore
@starindex
@end ignore
@tindex cwidth
The @code{cwidth} function measures the width, in characters, of a
composition.  For example, @samp{cwidth(a + b)} is 5, and
@samp{cwidth(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
@TeX{} mode (for @samp{@{a \over b@}}).  The argument may involve
the composition functions described in this section.

@ignore
@starindex
@end ignore
@tindex cheight
The @code{cheight} function measures the height of a composition.
This is the total number of lines in the argument's printed form.

@ignore
@starindex
@end ignore
@tindex cascent
@ignore
@starindex
@end ignore
@tindex cdescent
The functions @code{cascent} and @code{cdescent} measure the amount
of the height that is above (and including) the baseline, or below
the baseline, respectively.  Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
always equals @samp{cheight(@var{x})}.  For a one-line formula like
@samp{a + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
returns 1.  The only formula for which @code{cascent} will return zero
is @samp{cvspace(0)} or equivalents.

@node User-Defined Compositions,  , Information about Compositions, Compositions
@subsubsection User-Defined Compositions

@noindent
@kindex Z C
@pindex calc-user-define-composition
The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
define the display format for any algebraic function.  You provide a
formula containing a certain number of argument variables on the stack.
Any time Calc formats a call to the specified function in the current
language mode and with that number of arguments, Calc effectively
replaces the function call with that formula with the arguments
replaced.

Calc builds the default argument list by sorting all the variable names
that appear in the formula into alphabetical order.  You can edit this
argument list before pressing @key{RET} if you wish.  Any variables in
the formula that do not appear in the argument list will be displayed
literally; any arguments that do not appear in the formula will not
affect the display at all.

You can define formats for built-in functions, for functions you have
defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
which have no definitions but are being used as purely syntactic objects.
You can define different formats for each language mode, and for each
number of arguments, using a succession of @kbd{Z C} commands.  When
Calc formats a function call, it first searches for a format defined
for the current language mode (and number of arguments); if there is
none, it uses the format defined for the Normal language mode.  If
neither format exists, Calc uses its built-in standard format for that
function (usually just @samp{@var{func}(@var{args})}).

If you execute @kbd{Z C} with the number 0 on the stack instead of a
formula, any defined formats for the function in the current language
mode will be removed.  The function will revert to its standard format.

For example, the default format for the binomial coefficient function
@samp{choose(n, m)} in the Big language mode is

@example
@group
 n
( )
 m
@end group
@end example

@noindent
You might prefer the notation,

@example
@group
 C
n m
@end group
@end example

@noindent
To define this notation, first make sure you are in Big mode,
then put the formula

@smallexample
choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
@end smallexample

@noindent
on the stack and type @kbd{Z C}.  Answer the first prompt with
@code{choose}.  The second prompt will be the default argument list
of @samp{(C m n)}.  Edit this list to be @samp{(n m)} and press
@key{RET}.  Now, try it out:  For example, turn simplification
off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
as an algebraic entry.

@example
@group
 C  +  C
a b   7 3
@end group
@end example

As another example, let's define the usual notation for Stirling
numbers of the first kind, @samp{stir1(n, m)}.  This is just like
the regular format for binomial coefficients but with square brackets
instead of parentheses.

@smallexample
choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
@end smallexample

Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
@samp{(n m)}, and type @key{RET}.

The formula provided to @kbd{Z C} usually will involve composition
functions, but it doesn't have to.  Putting the formula @samp{a + b + c}
onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
This ``sum'' will act exactly like a real sum for all formatting
purposes (it will be parenthesized the same, and so on).  However
it will be computationally unrelated to a sum.  For example, the
formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
Operator precedences have caused the ``sum'' to be written in
parentheses, but the arguments have not actually been summed.
(Generally a display format like this would be undesirable, since
it can easily be confused with a real sum.)

The special function @code{eval} can be used inside a @kbd{Z C}
composition formula to cause all or part of the formula to be
evaluated at display time.  For example, if the formula is
@samp{a + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
as @samp{1 + 5}.  Evaluation will use the default simplifications,
regardless of the current simplification mode.  There are also
@code{evalsimp} and @code{evalextsimp} which simplify as if by
@kbd{a s} and @kbd{a e} (respectively).  Note that these ``functions''
operate only in the context of composition formulas (and also in
rewrite rules, where they serve a similar purpose; @pxref{Rewrite
Rules}).  On the stack, a call to @code{eval} will be left in
symbolic form.

It is not a good idea to use @code{eval} except as a last resort.
It can cause the display of formulas to be extremely slow.  For
example, while @samp{eval(a + b)} might seem quite fast and simple,
there are several situations where it could be slow.  For example,
@samp{a} and/or @samp{b} could be polar complex numbers, in which
case doing the sum requires trigonometry.  Or, @samp{a} could be
the factorial @samp{fact(100)} which is unevaluated because you
have typed @kbd{m O}; @code{eval} will evaluate it anyway to
produce a large, unwieldy integer.

You can save your display formats permanently using the @kbd{Z P}
command (@pxref{Creating User Keys}).

@node Syntax Tables,  , Compositions, Language Modes
@subsection Syntax Tables

@noindent
@cindex Syntax tables
@cindex Parsing formulas, customized
Syntax tables do for input what compositions do for output:  They
allow you to teach custom notations to Calc's formula parser.
Calc keeps a separate syntax table for each language mode.

(Note that the Calc ``syntax tables'' discussed here are completely
unrelated to the syntax tables described in the Emacs manual.)

@kindex Z S
@pindex calc-edit-user-syntax
The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
syntax table for the current language mode.  If you want your
syntax to work in any language, define it in the Normal language
mode.  Type @kbd{C-c C-c} to finish editing the syntax table, or
@kbd{C-x k} to cancel the edit.  The @kbd{m m} command saves all
the syntax tables along with the other mode settings;
@pxref{General Mode Commands}.

@menu
* Syntax Table Basics::
* Precedence in Syntax Tables::
* Advanced Syntax Patterns::
* Conditional Syntax Rules::
@end menu

@node Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables, Syntax Tables
@subsubsection Syntax Table Basics

@noindent
@dfn{Parsing} is the process of converting a raw string of characters,
such as you would type in during algebraic entry, into a Calc formula.
Calc's parser works in two stages.  First, the input is broken down
into @dfn{tokens}, such as words, numbers, and punctuation symbols
like @samp{+}, @samp{:=}, and @samp{+/-}.  Space between tokens is
ignored (except when it serves to separate adjacent words).  Next,
the parser matches this string of tokens against various built-in
syntactic patterns, such as ``an expression followed by @samp{+}
followed by another expression'' or ``a name followed by @samp{(},
zero or more expressions separated by commas, and @samp{)}.''

A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
which allow you to specify new patterns to define your own
favorite input notations.  Calc's parser always checks the syntax
table for the current language mode, then the table for the Normal
language mode, before it uses its built-in rules to parse an
algebraic formula you have entered.  Each syntax rule should go on
its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
and a Calc formula with an optional @dfn{condition}.  (Syntax rules
resemble algebraic rewrite rules, but the notation for patterns is
completely different.)

A syntax pattern is a list of tokens, separated by spaces.
Except for a few special symbols, tokens in syntax patterns are
matched literally, from left to right.  For example, the rule,

@example
foo ( ) := 2+3
@end example

@noindent
would cause Calc to parse the formula @samp{4+foo()*5} as if it
were @samp{4+(2+3)*5}.  Notice that the parentheses were written
as two separate tokens in the rule.  As a result, the rule works
for both @samp{foo()} and @w{@samp{foo (  )}}.  If we had written
the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
as a single, indivisible token, so that @w{@samp{foo( )}} would
not be recognized by the rule.  (It would be parsed as a regular
zero-argument function call instead.)  In fact, this rule would
also make trouble for the rest of Calc's parser:  An unrelated
formula like @samp{bar()} would now be tokenized into @samp{bar ()}
instead of @samp{bar ( )}, so that the standard parser for function
calls would no longer recognize it!

While it is possible to make a token with a mixture of letters
and punctuation symbols, this is not recommended.  It is better to
break it into several tokens, as we did with @samp{foo()} above.

The symbol @samp{#} in a syntax pattern matches any Calc expression.
On the righthand side, the things that matched the @samp{#}s can
be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
matches the leftmost @samp{#} in the pattern).  For example, these
rules match a user-defined function, prefix operator, infix operator,
and postfix operator, respectively:

@example
foo ( # ) := myfunc(#1)
foo # := myprefix(#1)
# foo # := myinfix(#1,#2)
# foo := mypostfix(#1)
@end example

Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
will parse as @samp{mypostfix(2+3)}.

It is important to write the first two rules in the order shown,
because Calc tries rules in order from first to last.  If the
pattern @samp{foo #} came first, it would match anything that could
match the @samp{foo ( # )} rule, since an expression in parentheses
is itself a valid expression.  Thus the @w{@samp{foo ( # )}} rule would
never get to match anything.  Likewise, the last two rules must be
written in the order shown or else @samp{3 foo 4} will be parsed as
@samp{mypostfix(3) * 4}.  (Of course, the best way to avoid these
ambiguities is not to use the same symbol in more than one way at
the same time!  In case you're not convinced, try the following
exercise:  How will the above rules parse the input @samp{foo(3,4)},
if at all?  Work it out for yourself, then try it in Calc and see.)

Calc is quite flexible about what sorts of patterns are allowed.
The only rule is that every pattern must begin with a literal
token (like @samp{foo} in the first two patterns above), or with
a @samp{#} followed by a literal token (as in the last two
patterns).  After that, any mixture is allowed, although putting
two @samp{#}s in a row will not be very useful since two
expressions with nothing between them will be parsed as one
expression that uses implicit multiplication.

As a more practical example, Maple uses the notation
@samp{sum(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
recognize at present.  To handle this syntax, we simply add the
rule,

@example
sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
@end example

@noindent
to the Maple mode syntax table.  As another example, C mode can't
read assignment operators like @samp{++} and @samp{*=}.  We can
define these operators quite easily:

@example
# *= # := muleq(#1,#2)
# ++ := postinc(#1)
++ # := preinc(#1)
@end example

@noindent
To complete the job, we would use corresponding composition functions
and @kbd{Z C} to cause these functions to display in their respective
Maple and C notations.  (Note that the C example ignores issues of
operator precedence, which are discussed in the next section.)

You can enclose any token in quotes to prevent its usual
interpretation in syntax patterns:

@example
# ":=" # := becomes(#1,#2)
@end example

Quotes also allow you to include spaces in a token, although once
again it is generally better to use two tokens than one token with
an embedded space.  To include an actual quotation mark in a quoted
token, precede it with a backslash.  (This also works to include
backslashes in tokens.)

@example
# "bad token" # "/\"\\" # := silly(#1,#2,#3)
@end example

@noindent
This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.

The token @kbd{#} has a predefined meaning in Calc's formula parser;
it is not valid to use @samp{"#"} in a syntax rule.  However, longer
tokens that include the @samp{#} character are allowed.  Also, while
@samp{"$"} and @samp{"\""} are allowed as tokens, their presence in
the syntax table will prevent those characters from working in their
usual ways (referring to stack entries and quoting strings,
respectively).

Finally, the notation @samp{%%} anywhere in a syntax table causes
the rest of the line to be ignored as a comment.

@node Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax Table Basics, Syntax Tables
@subsubsection Precedence

@noindent
Different operators are generally assigned different @dfn{precedences}.
By default, an operator defined by a rule like

@example
# foo # := foo(#1,#2)
@end example

@noindent
will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
will be parsed as @samp{(2*3+4) foo (5 == 6)}.  To change the
precedence of an operator, use the notation @samp{#/@var{p}} in
place of @samp{#}, where @var{p} is an integer precedence level.
For example, 185 lies between the precedences for @samp{+} and
@samp{*}, so if we change this rule to

@example
#/185 foo #/186 := foo(#1,#2)
@end example

@noindent
then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
Also, because we've given the righthand expression slightly higher
precedence, our new operator will be left-associative:
@samp{1 foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
By raising the precedence of the lefthand expression instead, we
can create a right-associative operator.

@xref{Composition Basics}, for a table of precedences of the
standard Calc operators.  For the precedences of operators in other
language modes, look in the Calc source file @file{calc-lang.el}.

@node Advanced Syntax Patterns, Conditional Syntax Rules, Precedence in Syntax Tables, Syntax Tables
@subsubsection Advanced Syntax Patterns

@noindent
To match a function with a variable number of arguments, you could
write

@example
foo ( # ) := myfunc(#1)
foo ( # , # ) := myfunc(#1,#2)
foo ( # , # , # ) := myfunc(#1,#2,#3)
@end example

@noindent
but this isn't very elegant.  To match variable numbers of items,
Calc uses some notations inspired regular expressions and the
``extended BNF'' style used by some language designers.

@example
foo ( @{ # @}*, ) := apply(myfunc,#1)
@end example

The token @samp{@{} introduces a repeated or optional portion.
One of the three tokens @samp{@}*}, @samp{@}+}, or @samp{@}?}
ends the portion.  These will match zero or more, one or more,
or zero or one copies of the enclosed pattern, respectively.
In addition, @samp{@}*} and @samp{@}+} can be followed by a
separator token (with no space in between, as shown above).
Thus @samp{@{ # @}*,} matches nothing, or one expression, or
several expressions separated by commas.

A complete @samp{@{ ... @}} item matches as a vector of the
items that matched inside it.  For example, the above rule will
match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
The Calc @code{apply} function takes a function name and a vector
of arguments and builds a call to the function with those
arguments, so the net result is the formula @samp{myfunc(1,2,3)}.

If the body of a @samp{@{ ... @}} contains several @samp{#}s
(or nested @samp{@{ ... @}} constructs), then the items will be
strung together into the resulting vector.  If the body
does not contain anything but literal tokens, the result will
always be an empty vector.

@example
foo ( @{ # , # @}+, ) := bar(#1)
foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
@end example

@noindent
will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
@samp{foo(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}.  Also, after
some thought it's easy to see how this pair of rules will parse
@samp{foo(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
rule will only match an even number of arguments.  The rule

@example
foo ( # @{ , # , # @}? ) := bar(#1,#2)
@end example

@noindent
will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
@samp{foo(2)} as @samp{bar(2,[])}.

The notation @samp{@{ ... @}?.} (note the trailing period) works
just the same as regular @samp{@{ ... @}?}, except that it does not
count as an argument; the following two rules are equivalent:

@example
foo ( # , @{ also @}? # ) := bar(#1,#3)
foo ( # , @{ also @}?. # ) := bar(#1,#2)
@end example

@noindent
Note that in the first case the optional text counts as @samp{#2},
which will always be an empty vector, but in the second case no
empty vector is produced.

Another variant is @samp{@{ ... @}?$}, which means the body is
optional only at the end of the input formula.  All built-in syntax
rules in Calc use this for closing delimiters, so that during
algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
the closing parenthesis and bracket.  Calc does this automatically
for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
rules, but you can use @samp{@{ ... @}?$} explicitly to get
this effect with any token (such as @samp{"@}"} or @samp{end}).
Like @samp{@{ ... @}?.}, this notation does not count as an
argument.  Conversely, you can use quotes, as in @samp{")"}, to
prevent a closing-delimiter token from being automatically treated
as optional.

Calc's parser does not have full backtracking, which means some
patterns will not work as you might expect:

@example
foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
@end example

@noindent
Here we are trying to make the first argument optional, so that
@samp{foo(2,3)} parses as @samp{bar([],2,3)}.  Unfortunately, Calc
first tries to match @samp{2,} against the optional part of the
pattern, finds a match, and so goes ahead to match the rest of the
pattern.  Later on it will fail to match the second comma, but it
doesn't know how to go back and try the other alternative at that
point.  One way to get around this would be to use two rules:

@example
foo ( # , # , # ) := bar([#1],#2,#3)
foo ( # , # ) := bar([],#1,#2)
@end example

More precisely, when Calc wants to match an optional or repeated
part of a pattern, it scans forward attempting to match that part.
If it reaches the end of the optional part without failing, it
``finalizes'' its choice and proceeds.  If it fails, though, it
backs up and tries the other alternative.  Thus Calc has ``partial''
backtracking.  A fully backtracking parser would go on to make sure
the rest of the pattern matched before finalizing the choice.

@node Conditional Syntax Rules,  , Advanced Syntax Patterns, Syntax Tables
@subsubsection Conditional Syntax Rules

@noindent
It is possible to attach a @dfn{condition} to a syntax rule.  For
example, the rules

@example
foo ( # ) := ifoo(#1) :: integer(#1)
foo ( # ) := gfoo(#1)
@end example

@noindent
will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
@samp{foo(3.5)} and @samp{foo(x)} as calls to @code{gfoo}.  Any
number of conditions may be attached; all must be true for the
rule to succeed.  A condition is ``true'' if it evaluates to a
nonzero number.  @xref{Logical Operations}, for a list of Calc
functions like @code{integer} that perform logical tests.

The exact sequence of events is as follows:  When Calc tries a
rule, it first matches the pattern as usual.  It then substitutes
@samp{#1}, @samp{#2}, etc., in the conditions, if any.  Next, the
conditions are simplified and evaluated in order from left to right,
as if by the @w{@kbd{a s}} algebra command (@pxref{Simplifying Formulas}).
Each result is true if it is a nonzero number, or an expression
that can be proven to be nonzero (@pxref{Declarations}).  If the
results of all conditions are true, the expression (such as
@samp{ifoo(#1)}) has its @samp{#}s substituted, and that is the
result of the parse.  If the result of any condition is false, Calc
goes on to try the next rule in the syntax table.

Syntax rules also support @code{let} conditions, which operate in
exactly the same way as they do in algebraic rewrite rules.
@xref{Other Features of Rewrite Rules}, for details.  A @code{let}
condition is always true, but as a side effect it defines a
variable which can be used in later conditions, and also in the
expression after the @samp{:=} sign:

@example
foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
@end example

@noindent
The @code{dnumint} function tests if a value is numerically an
integer, i.e., either a true integer or an integer-valued float.
This rule will parse @code{foo} with a half-integer argument,
like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.

The lefthand side of a syntax rule @code{let} must be a simple
variable, not the arbitrary pattern that is allowed in rewrite
rules.

The @code{matches} function is also treated specially in syntax
rule conditions (again, in the same way as in rewrite rules).
@xref{Matching Commands}.  If the matching pattern contains
meta-variables, then those meta-variables may be used in later
conditions and in the result expression.  The arguments to
@code{matches} are not evaluated in this situation.

@example
sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
@end example

@noindent
This is another way to implement the Maple mode @code{sum} notation.
In this approach, we allow @samp{#2} to equal the whole expression
@samp{i=1..10}.  Then, we use @code{matches} to break it apart into
its components.  If the expression turns out not to match the pattern,
the syntax rule will fail.  Note that @kbd{Z S} always uses Calc's
Normal language mode for editing expressions in syntax rules, so we
must use regular Calc notation for the interval @samp{[b..c]} that
will correspond to the Maple mode interval @samp{1..10}.

@node Modes Variable, Calc Mode Line, Language Modes, Mode Settings
@section The @code{Modes} Variable

@noindent
@kindex m g
@pindex calc-get-modes
The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
a vector of numbers that describes the various mode settings that
are in effect.  With a numeric prefix argument, it pushes only the
@var{n}th mode, i.e., the @var{n}th element of this vector.  Keyboard
macros can use the @kbd{m g} command to modify their behavior based
on the current mode settings.

@cindex @code{Modes} variable
@vindex Modes
The modes vector is also available in the special variable
@code{Modes}.  In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
It will not work to store into this variable; in fact, if you do,
@code{Modes} will cease to track the current modes.  (The @kbd{m g}
command will continue to work, however.)

In general, each number in this vector is suitable as a numeric
prefix argument to the associated mode-setting command.  (Recall
that the @kbd{~} key takes a number from the stack and gives it as
a numeric prefix to the next command.)

The elements of the modes vector are as follows:

@enumerate
@item
Current precision.  Default is 12; associated command is @kbd{p}.

@item
Binary word size.  Default is 32; associated command is @kbd{b w}.

@item
Stack size (not counting the value about to be pushed by @kbd{m g}).
This is zero if @kbd{m g} is executed with an empty stack.

@item
Number radix.  Default is 10; command is @kbd{d r}.

@item
Floating-point format.  This is the number of digits, plus the
constant 0 for normal notation, 10000 for scientific notation,
20000 for engineering notation, or 30000 for fixed-point notation.
These codes are acceptable as prefix arguments to the @kbd{d n}
command, but note that this may lose information:  For example,
@kbd{d s} and @kbd{C-u 12 d s} have similar (but not quite
identical) effects if the current precision is 12, but they both
produce a code of 10012, which will be treated by @kbd{d n} as
@kbd{C-u 12 d s}.  If the precision then changes, the float format
will still be frozen at 12 significant figures.

@item
Angular mode.  Default is 1 (degrees).  Other values are 2 (radians)
and 3 (HMS).  The @kbd{m d} command accepts these prefixes.

@item
Symbolic mode.  Value is 0 or 1; default is 0.  Command is @kbd{m s}.

@item
Fraction mode.  Value is 0 or 1; default is 0.  Command is @kbd{m f}.

@item
Polar mode.  Value is 0 (rectangular) or 1 (polar); default is 0.
Command is @kbd{m p}.

@item
Matrix/Scalar mode.  Default value is @mathit{-1}.  Value is 0 for Scalar
mode, @mathit{-2} for Matrix mode, @mathit{-3} for square Matrix mode,
or @var{N} for  
@texline @math{N\times N}
@infoline @var{N}x@var{N} 
Matrix mode.  Command is @kbd{m v}.

@item
Simplification mode.  Default is 1.  Value is @mathit{-1} for off (@kbd{m O}),
0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
or 5 for @w{@kbd{m U}}.  The @kbd{m D} command accepts these prefixes.

@item
Infinite mode.  Default is @mathit{-1} (off).  Value is 1 if the mode is on,
or 0 if the mode is on with positive zeros.  Command is @kbd{m i}.
@end enumerate

For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
precision by two, leaving a copy of the old precision on the stack.
Later, @kbd{~ p} will restore the original precision using that
stack value.  (This sequence might be especially useful inside a
keyboard macro.)

As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
oldest (bottommost) stack entry.

Yet another example:  The HP-48 ``round'' command rounds a number
to the current displayed precision.  You could roughly emulate this
in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}.  (This
would not work for fixed-point mode, but it wouldn't be hard to
do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
programming commands.  @xref{Conditionals in Macros}.)

@node Calc Mode Line,  , Modes Variable, Mode Settings
@section The Calc Mode Line

@noindent
@cindex Mode line indicators
This section is a summary of all symbols that can appear on the
Calc mode line, the highlighted bar that appears under the Calc
stack window (or under an editing window in Embedded mode).

The basic mode line format is:

@example
--%%-Calc: 12 Deg @var{other modes}       (Calculator)
@end example

The @samp{%%} is the Emacs symbol for ``read-only''; it shows that
regular Emacs commands are not allowed to edit the stack buffer
as if it were text.

The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
is enabled.  The words after this describe the various Calc modes
that are in effect.

The first mode is always the current precision, an integer.
The second mode is always the angular mode, either @code{Deg},
@code{Rad}, or @code{Hms}.

Here is a complete list of the remaining symbols that can appear
on the mode line:

@table @code
@item Alg
Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).

@item Alg[(
Incomplete algebraic mode (@kbd{C-u m a}).

@item Alg*
Total algebraic mode (@kbd{m t}).

@item Symb
Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).

@item Matrix
Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).

@item Matrix@var{n}
Dimensioned Matrix mode (@kbd{C-u @var{n} m v}; @pxref{Matrix Mode}).

@item SqMatrix
Square Matrix mode (@kbd{C-u m v}; @pxref{Matrix Mode}).

@item Scalar
Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).

@item Polar
Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).

@item Frac
Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).

@item Inf
Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).

@item +Inf
Positive Infinite mode (@kbd{C-u 0 m i}).

@item NoSimp
Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).

@item NumSimp
Default simplifications for numeric arguments only (@kbd{m N}).

@item BinSimp@var{w}
Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).

@item AlgSimp
Algebraic simplification mode (@kbd{m A}).

@item ExtSimp
Extended algebraic simplification mode (@kbd{m E}).

@item UnitSimp
Units simplification mode (@kbd{m U}).

@item Bin
Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).

@item Oct
Current radix is 8 (@kbd{d 8}).

@item Hex
Current radix is 16 (@kbd{d 6}).

@item Radix@var{n}
Current radix is @var{n} (@kbd{d r}).

@item Zero
Leading zeros (@kbd{d z}; @pxref{Radix Modes}).

@item Big
Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).

@item Flat
One-line normal language mode (@kbd{d O}).

@item Unform
Unformatted language mode (@kbd{d U}).

@item C
C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).

@item Pascal
Pascal language mode (@kbd{d P}).

@item Fortran
FORTRAN language mode (@kbd{d F}).

@item TeX
@TeX{} language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).

@item LaTeX
La@TeX{} language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).

@item Eqn
@dfn{Eqn} language mode (@kbd{d E}; @pxref{Eqn Language Mode}).

@item Math
Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).

@item Maple
Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).

@item Norm@var{n}
Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).

@item Fix@var{n}
Fixed point mode with @var{n} digits after the point (@kbd{d f}).

@item Sci
Scientific notation mode (@kbd{d s}).

@item Sci@var{n}
Scientific notation with @var{n} digits (@kbd{d s}).

@item Eng
Engineering notation mode (@kbd{d e}).

@item Eng@var{n}
Engineering notation with @var{n} digits (@kbd{d e}).

@item Left@var{n}
Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).

@item Right
Right-justified display (@kbd{d >}).

@item Right@var{n}
Right-justified display with width @var{n} (@kbd{d >}).

@item Center
Centered display (@kbd{d =}).

@item Center@var{n}
Centered display with center column @var{n} (@kbd{d =}).

@item Wid@var{n}
Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).

@item Wide
No line breaking (@kbd{d b}).

@item Break
Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).

@item Save
Record modes in @file{~/.calc.el} (@kbd{m R}; @pxref{General Mode Commands}).

@item Local
Record modes in Embedded buffer (@kbd{m R}).

@item LocEdit
Record modes as editing-only in Embedded buffer (@kbd{m R}).

@item LocPerm
Record modes as permanent-only in Embedded buffer (@kbd{m R}).

@item Global
Record modes as global in Embedded buffer (@kbd{m R}).

@item Manual
Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
Recomputation}).

@item Graph
GNUPLOT process is alive in background (@pxref{Graphics}).

@item Sel
Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).

@item Dirty
The stack display may not be up-to-date (@pxref{Display Modes}).

@item Inv
``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).

@item Hyp
``Hyperbolic'' prefix was pressed (@kbd{H}).

@item Keep
``Keep-arguments'' prefix was pressed (@kbd{K}).

@item Narrow
Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
@end table

In addition, the symbols @code{Active} and @code{~Active} can appear
as minor modes on an Embedded buffer's mode line.  @xref{Embedded Mode}.

@node Arithmetic, Scientific Functions, Mode Settings, Top
@chapter Arithmetic Functions

@noindent
This chapter describes the Calc commands for doing simple calculations
on numbers, such as addition, absolute value, and square roots.  These
commands work by removing the top one or two values from the stack,
performing the desired operation, and pushing the result back onto the
stack.  If the operation cannot be performed, the result pushed is a
formula instead of a number, such as @samp{2/0} (because division by zero
is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).

Most of the commands described here can be invoked by a single keystroke.
Some of the more obscure ones are two-letter sequences beginning with
the @kbd{f} (``functions'') prefix key.

@xref{Prefix Arguments}, for a discussion of the effect of numeric
prefix arguments on commands in this chapter which do not otherwise
interpret a prefix argument.

@menu
* Basic Arithmetic::
* Integer Truncation::
* Complex Number Functions::
* Conversions::
* Date Arithmetic::
* Financial Functions::
* Binary Functions::
@end menu

@node Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
@section Basic Arithmetic

@noindent
@kindex +
@pindex calc-plus
@ignore
@mindex @null
@end ignore
@tindex +
The @kbd{+} (@code{calc-plus}) command adds two numbers.  The numbers may
be any of the standard Calc data types.  The resulting sum is pushed back
onto the stack.

If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
the result is a vector or matrix sum.  If one argument is a vector and the
other a scalar (i.e., a non-vector), the scalar is added to each of the
elements of the vector to form a new vector.  If the scalar is not a
number, the operation is left in symbolic form:  Suppose you added @samp{x}
to the vector @samp{[1,2]}.  You may want the result @samp{[1+x,2+x]}, or
you may plan to substitute a 2-vector for @samp{x} in the future.  Since
the Calculator can't tell which interpretation you want, it makes the
safest assumption.  @xref{Reducing and Mapping}, for a way to add @samp{x}
to every element of a vector.

If either argument of @kbd{+} is a complex number, the result will in general
be complex.  If one argument is in rectangular form and the other polar,
the current Polar mode determines the form of the result.  If Symbolic
mode is enabled, the sum may be left as a formula if the necessary
conversions for polar addition are non-trivial.

If both arguments of @kbd{+} are HMS forms, the forms are added according to
the usual conventions of hours-minutes-seconds notation.  If one argument
is an HMS form and the other is a number, that number is converted from
degrees or radians (depending on the current Angular mode) to HMS format
and then the two HMS forms are added.

If one argument of @kbd{+} is a date form, the other can be either a
real number, which advances the date by a certain number of days, or
an HMS form, which advances the date by a certain amount of time.
Subtracting two date forms yields the number of days between them.
Adding two date forms is meaningless, but Calc interprets it as the
subtraction of one date form and the negative of the other.  (The
negative of a date form can be understood by remembering that dates
are stored as the number of days before or after Jan 1, 1 AD.)

If both arguments of @kbd{+} are error forms, the result is an error form
with an appropriately computed standard deviation.  If one argument is an
error form and the other is a number, the number is taken to have zero error.
Error forms may have symbolic formulas as their mean and/or error parts;
adding these will produce a symbolic error form result.  However, adding an
error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
work, for the same reasons just mentioned for vectors.  Instead you must
write @samp{(a +/- b) + (c +/- 0)}.

If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
or if one argument is a modulo form and the other a plain number, the
result is a modulo form which represents the sum, modulo @expr{M}, of
the two values.

If both arguments of @kbd{+} are intervals, the result is an interval
which describes all possible sums of the possible input values.  If
one argument is a plain number, it is treated as the interval
@w{@samp{[x ..@: x]}}.

If one argument of @kbd{+} is an infinity and the other is not, the
result is that same infinity.  If both arguments are infinite and in
the same direction, the result is the same infinity, but if they are
infinite in different directions the result is @code{nan}.

@kindex -
@pindex calc-minus
@ignore
@mindex @null
@end ignore
@tindex -
The @kbd{-} (@code{calc-minus}) command subtracts two values.  The top
number on the stack is subtracted from the one behind it, so that the
computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}.  All options
available for @kbd{+} are available for @kbd{-} as well.

@kindex *
@pindex calc-times
@ignore
@mindex @null
@end ignore
@tindex *
The @kbd{*} (@code{calc-times}) command multiplies two numbers.  If one
argument is a vector and the other a scalar, the scalar is multiplied by
the elements of the vector to produce a new vector.  If both arguments
are vectors, the interpretation depends on the dimensions of the
vectors:  If both arguments are matrices, a matrix multiplication is
done.  If one argument is a matrix and the other a plain vector, the
vector is interpreted as a row vector or column vector, whichever is
dimensionally correct.  If both arguments are plain vectors, the result
is a single scalar number which is the dot product of the two vectors.

If one argument of @kbd{*} is an HMS form and the other a number, the
HMS form is multiplied by that amount.  It is an error to multiply two
HMS forms together, or to attempt any multiplication involving date
forms.  Error forms, modulo forms, and intervals can be multiplied;
see the comments for addition of those forms.  When two error forms
or intervals are multiplied they are considered to be statistically
independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
whereas @w{@samp{[-2 ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.

@kindex /
@pindex calc-divide
@ignore
@mindex @null
@end ignore
@tindex /
The @kbd{/} (@code{calc-divide}) command divides two numbers.  Note that
when using algebraic entry, @samp{/} has lower precedence than @samp{*}, 
so that @samp{a/b*c} is interpreted as @samp{a/(b*c)}.

When dividing a scalar @expr{B} by a square matrix @expr{A}, the
computation performed is @expr{B} times the inverse of @expr{A}.  This
also occurs if @expr{B} is itself a vector or matrix, in which case the
effect is to solve the set of linear equations represented by @expr{B}.
If @expr{B} is a matrix with the same number of rows as @expr{A}, or a
plain vector (which is interpreted here as a column vector), then the
equation @expr{A X = B} is solved for the vector or matrix @expr{X}.
Otherwise, if @expr{B} is a non-square matrix with the same number of
@emph{columns} as @expr{A}, the equation @expr{X A = B} is solved.  If
you wish a vector @expr{B} to be interpreted as a row vector to be
solved as @expr{X A = B}, make it into a one-row matrix with @kbd{C-u 1
v p} first.  To force a left-handed solution with a square matrix
@expr{B}, transpose @expr{A} and @expr{B} before dividing, then
transpose the result.

HMS forms can be divided by real numbers or by other HMS forms.  Error
forms can be divided in any combination of ways.  Modulo forms where both
values and the modulo are integers can be divided to get an integer modulo
form result.  Intervals can be divided; dividing by an interval that
encompasses zero or has zero as a limit will result in an infinite
interval.

@kindex ^
@pindex calc-power
@ignore
@mindex @null
@end ignore
@tindex ^
The @kbd{^} (@code{calc-power}) command raises a number to a power.  If
the power is an integer, an exact result is computed using repeated
multiplications.  For non-integer powers, Calc uses Newton's method or
logarithms and exponentials.  Square matrices can be raised to integer
powers.  If either argument is an error (or interval or modulo) form,
the result is also an error (or interval or modulo) form.

@kindex I ^
@tindex nroot
If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
computes an Nth root:  @kbd{125 @key{RET} 3 I ^} computes the number 5.
(This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)

@kindex \
@pindex calc-idiv
@tindex idiv
@ignore
@mindex @null
@end ignore
@tindex \
The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
to produce an integer result.  It is equivalent to dividing with
@key{/}, then rounding down with @kbd{F} (@code{calc-floor}), only a bit
more convenient and efficient.  Also, since it is an all-integer
operation when the arguments are integers, it avoids problems that
@kbd{/ F} would have with floating-point roundoff.

@kindex %
@pindex calc-mod
@ignore
@mindex @null
@end ignore
@tindex %
The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
operation.  Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
for all real numbers @expr{a} and @expr{b} (except @expr{b=0}).  For
positive @expr{b}, the result will always be between 0 (inclusive) and
@expr{b} (exclusive).  Modulo does not work for HMS forms and error forms.
If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
must be positive real number.

@kindex :
@pindex calc-fdiv
@tindex fdiv
The @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command
divides the two integers on the top of the stack to produce a fractional
result.  This is a convenient shorthand for enabling Fraction mode (with
@kbd{m f}) temporarily and using @samp{/}.  Note that during numeric entry
the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
you would have to type @kbd{8 @key{RET} 6 @key{RET} :}.  (Of course, in
this case, it would be much easier simply to enter the fraction directly
as @kbd{8:6 @key{RET}}!)

@kindex n
@pindex calc-change-sign
The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
of the stack.  It works on numbers, vectors and matrices, HMS forms, date
forms, error forms, intervals, and modulo forms.

@kindex A
@pindex calc-abs
@tindex abs
The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the absolute
value of a number.  The result of @code{abs} is always a nonnegative
real number:  With a complex argument, it computes the complex magnitude.
With a vector or matrix argument, it computes the Frobenius norm, i.e.,
the square root of the sum of the squares of the absolute values of the
elements.  The absolute value of an error form is defined by replacing
the mean part with its absolute value and leaving the error part the same.
The absolute value of a modulo form is undefined.  The absolute value of
an interval is defined in the obvious way.

@kindex f A
@pindex calc-abssqr
@tindex abssqr
The @kbd{f A} (@code{calc-abssqr}) [@code{abssqr}] command computes the
absolute value squared of a number, vector or matrix, or error form.

@kindex f s
@pindex calc-sign
@tindex sign
The @kbd{f s} (@code{calc-sign}) [@code{sign}] command returns 1 if its
argument is positive, @mathit{-1} if its argument is negative, or 0 if its
argument is zero.  In algebraic form, you can also write @samp{sign(a,x)}
which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
zero depending on the sign of @samp{a}.

@kindex &
@pindex calc-inv
@tindex inv
@cindex Reciprocal
The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
reciprocal of a number, i.e., @expr{1 / x}.  Operating on a square
matrix, it computes the inverse of that matrix.

@kindex Q
@pindex calc-sqrt
@tindex sqrt
The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] command computes the square
root of a number.  For a negative real argument, the result will be a
complex number whose form is determined by the current Polar mode.

@kindex f h
@pindex calc-hypot
@tindex hypot
The @kbd{f h} (@code{calc-hypot}) [@code{hypot}] command computes the square
root of the sum of the squares of two numbers.  That is, @samp{hypot(a,b)}
is the length of the hypotenuse of a right triangle with sides @expr{a}
and @expr{b}.  If the arguments are complex numbers, their squared
magnitudes are used.

@kindex f Q
@pindex calc-isqrt
@tindex isqrt
The @kbd{f Q} (@code{calc-isqrt}) [@code{isqrt}] command computes the
integer square root of an integer.  This is the true square root of the
number, rounded down to an integer.  For example, @samp{isqrt(10)}
produces 3.  Note that, like @kbd{\} [@code{idiv}], this uses exact
integer arithmetic throughout to avoid roundoff problems.  If the input
is a floating-point number or other non-integer value, this is exactly
the same as @samp{floor(sqrt(x))}.

@kindex f n
@kindex f x
@pindex calc-min
@tindex min
@pindex calc-max
@tindex max
The @kbd{f n} (@code{calc-min}) [@code{min}] and @kbd{f x} (@code{calc-max})
[@code{max}] commands take the minimum or maximum of two real numbers,
respectively.  These commands also work on HMS forms, date forms,
intervals, and infinities.  (In algebraic expressions, these functions
take any number of arguments and return the maximum or minimum among
all the arguments.)

@kindex f M
@kindex f X
@pindex calc-mant-part
@tindex mant
@pindex calc-xpon-part
@tindex xpon
The @kbd{f M} (@code{calc-mant-part}) [@code{mant}] function extracts
the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
(@code{calc-xpon-part}) [@code{xpon}] extracts the ``exponent'' part
@expr{e}.  The original number is equal to 
@texline @math{m \times 10^e},
@infoline @expr{m * 10^e},
where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
@expr{m=e=0} if the original number is zero.  For integers
and fractions, @code{mant} returns the number unchanged and @code{xpon}
returns zero.  The @kbd{v u} (@code{calc-unpack}) command can also be
used to ``unpack'' a floating-point number; this produces an integer
mantissa and exponent, with the constraint that the mantissa is not
a multiple of ten (again except for the @expr{m=e=0} case).

@kindex f S
@pindex calc-scale-float
@tindex scf
The @kbd{f S} (@code{calc-scale-float}) [@code{scf}] function scales a number
by a given power of ten.  Thus, @samp{scf(mant(x), xpon(x)) = x} for any
real @samp{x}.  The second argument must be an integer, but the first
may actually be any numeric value.  For example, @samp{scf(5,-2) = 0.05}
or @samp{1:20} depending on the current Fraction mode.

@kindex f [
@kindex f ]
@pindex calc-decrement
@pindex calc-increment
@tindex decr
@tindex incr
The @kbd{f [} (@code{calc-decrement}) [@code{decr}] and @kbd{f ]}
(@code{calc-increment}) [@code{incr}] functions decrease or increase
a number by one unit.  For integers, the effect is obvious.  For
floating-point numbers, the change is by one unit in the last place.
For example, incrementing @samp{12.3456} when the current precision
is 6 digits yields @samp{12.3457}.  If the current precision had been
8 digits, the result would have been @samp{12.345601}.  Incrementing
@samp{0.0} produces 
@texline @math{10^{-p}},
@infoline @expr{10^-p}, 
where @expr{p} is the current
precision.  These operations are defined only on integers and floats.
With numeric prefix arguments, they change the number by @expr{n} units.

Note that incrementing followed by decrementing, or vice-versa, will
almost but not quite always cancel out.  Suppose the precision is
6 digits and the number @samp{9.99999} is on the stack.  Incrementing
will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
One digit has been dropped.  This is an unavoidable consequence of the
way floating-point numbers work.

Incrementing a date/time form adjusts it by a certain number of seconds.
Incrementing a pure date form adjusts it by a certain number of days.

@node Integer Truncation, Complex Number Functions, Basic Arithmetic, Arithmetic
@section Integer Truncation

@noindent
There are four commands for truncating a real number to an integer,
differing mainly in their treatment of negative numbers.  All of these
commands have the property that if the argument is an integer, the result
is the same integer.  An integer-valued floating-point argument is converted
to integer form.

If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
expressed as an integer-valued floating-point number.

@cindex Integer part of a number
@kindex F
@pindex calc-floor
@tindex floor
@tindex ffloor
@ignore
@mindex @null
@end ignore
@kindex H F
The @kbd{F} (@code{calc-floor}) [@code{floor} or @code{ffloor}] command
truncates a real number to the next lower integer, i.e., toward minus
infinity.  Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
@mathit{-4}.

@kindex I F
@pindex calc-ceiling
@tindex ceil
@tindex fceil
@ignore
@mindex @null
@end ignore
@kindex H I F
The @kbd{I F} (@code{calc-ceiling}) [@code{ceil} or @code{fceil}]
command truncates toward positive infinity.  Thus @kbd{3.6 I F} produces
4, and @kbd{_3.6 I F} produces @mathit{-3}.

@kindex R
@pindex calc-round
@tindex round
@tindex fround
@ignore
@mindex @null
@end ignore
@kindex H R
The @kbd{R} (@code{calc-round}) [@code{round} or @code{fround}] command
rounds to the nearest integer.  When the fractional part is .5 exactly,
this command rounds away from zero.  (All other rounding in the
Calculator uses this convention as well.)  Thus @kbd{3.5 R} produces 4
but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.

@kindex I R
@pindex calc-trunc
@tindex trunc
@tindex ftrunc
@ignore
@mindex @null
@end ignore
@kindex H I R
The @kbd{I R} (@code{calc-trunc}) [@code{trunc} or @code{ftrunc}]
command truncates toward zero.  In other words, it ``chops off''
everything after the decimal point.  Thus @kbd{3.6 I R} produces 3 and
@kbd{_3.6 I R} produces @mathit{-3}.

These functions may not be applied meaningfully to error forms, but they
do work for intervals.  As a convenience, applying @code{floor} to a
modulo form floors the value part of the form.  Applied to a vector,
these functions operate on all elements of the vector one by one.
Applied to a date form, they operate on the internal numerical
representation of dates, converting a date/time form into a pure date.

@ignore
@starindex
@end ignore
@tindex rounde
@ignore
@starindex
@end ignore
@tindex roundu
@ignore
@starindex
@end ignore
@tindex frounde
@ignore
@starindex
@end ignore
@tindex froundu
There are two more rounding functions which can only be entered in
algebraic notation.  The @code{roundu} function is like @code{round}
except that it rounds up, toward plus infinity, when the fractional
part is .5.  This distinction matters only for negative arguments.
Also, @code{rounde} rounds to an even number in the case of a tie,
rounding up or down as necessary.  For example, @samp{rounde(3.5)} and
@samp{rounde(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
The advantage of round-to-even is that the net error due to rounding
after a long calculation tends to cancel out to zero.  An important
subtle point here is that the number being fed to @code{rounde} will
already have been rounded to the current precision before @code{rounde}
begins.  For example, @samp{rounde(2.500001)} with a current precision
of 6 will incorrectly, or at least surprisingly, yield 2 because the
argument will first have been rounded down to @expr{2.5} (which
@code{rounde} sees as an exact tie between 2 and 3).

Each of these functions, when written in algebraic formulas, allows
a second argument which specifies the number of digits after the
decimal point to keep.  For example, @samp{round(123.4567, 2)} will
produce the answer 123.46, and @samp{round(123.4567, -1)} will
produce 120 (i.e., the cutoff is one digit to the @emph{left} of
the decimal point).  A second argument of zero is equivalent to
no second argument at all.

@cindex Fractional part of a number
To compute the fractional part of a number (i.e., the amount which, when
added to `@tfn{floor(}@var{n}@tfn{)}', will produce @var{n}) just take @var{n}
modulo 1 using the @code{%} command.

Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
and @kbd{f Q} (integer square root) commands, which are analogous to
@kbd{/}, @kbd{B}, and @kbd{Q}, respectively, except that they take integer
arguments and return the result rounded down to an integer.

@node Complex Number Functions, Conversions, Integer Truncation, Arithmetic
@section Complex Number Functions

@noindent
@kindex J
@pindex calc-conj
@tindex conj
The @kbd{J} (@code{calc-conj}) [@code{conj}] command computes the
complex conjugate of a number.  For complex number @expr{a+bi}, the
complex conjugate is @expr{a-bi}.  If the argument is a real number,
this command leaves it the same.  If the argument is a vector or matrix,
this command replaces each element by its complex conjugate.

@kindex G
@pindex calc-argument
@tindex arg
The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the
``argument'' or polar angle of a complex number.  For a number in polar
notation, this is simply the second component of the pair
@texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'.
@infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'.
The result is expressed according to the current angular mode and will
be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
(inclusive), or the equivalent range in radians.

@pindex calc-imaginary
The @code{calc-imaginary} command multiplies the number on the
top of the stack by the imaginary number @expr{i = (0,1)}.  This
command is not normally bound to a key in Calc, but it is available
on the @key{IMAG} button in Keypad mode.

@kindex f r
@pindex calc-re
@tindex re
The @kbd{f r} (@code{calc-re}) [@code{re}] command replaces a complex number
by its real part.  This command has no effect on real numbers.  (As an
added convenience, @code{re} applied to a modulo form extracts
the value part.)

@kindex f i
@pindex calc-im
@tindex im
The @kbd{f i} (@code{calc-im}) [@code{im}] command replaces a complex number
by its imaginary part; real numbers are converted to zero.  With a vector
or matrix argument, these functions operate element-wise.

@ignore
@mindex v p
@end ignore
@kindex v p (complex)
@pindex calc-pack
The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
the stack into a composite object such as a complex number.  With
a prefix argument of @mathit{-1}, it produces a rectangular complex number;
with an argument of @mathit{-2}, it produces a polar complex number.
(Also, @pxref{Building Vectors}.)

@ignore
@mindex v u
@end ignore
@kindex v u (complex)
@pindex calc-unpack
The @kbd{v u} (@code{calc-unpack}) command takes the complex number
(or other composite object) on the top of the stack and unpacks it
into its separate components.

@node Conversions, Date Arithmetic, Complex Number Functions, Arithmetic
@section Conversions

@noindent
The commands described in this section convert numbers from one form
to another; they are two-key sequences beginning with the letter @kbd{c}.

@kindex c f
@pindex calc-float
@tindex pfloat
The @kbd{c f} (@code{calc-float}) [@code{pfloat}] command converts the
number on the top of the stack to floating-point form.  For example,
@expr{23} is converted to @expr{23.0}, @expr{3:2} is converted to
@expr{1.5}, and @expr{2.3} is left the same.  If the value is a composite
object such as a complex number or vector, each of the components is
converted to floating-point.  If the value is a formula, all numbers
in the formula are converted to floating-point.  Note that depending
on the current floating-point precision, conversion to floating-point
format may lose information.

As a special exception, integers which appear as powers or subscripts
are not floated by @kbd{c f}.  If you really want to float a power,
you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
Because @kbd{c f} cannot examine the formula outside of the selection,
it does not notice that the thing being floated is a power.
@xref{Selecting Subformulas}.

The normal @kbd{c f} command is ``pervasive'' in the sense that it
applies to all numbers throughout the formula.  The @code{pfloat}
algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
changes to @samp{a + 1.0} as soon as it is evaluated.

@kindex H c f
@tindex float
With the Hyperbolic flag, @kbd{H c f} [@code{float}] operates
only on the number or vector of numbers at the top level of its
argument.  Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
is left unevaluated because its argument is not a number.

You should use @kbd{H c f} if you wish to guarantee that the final
value, once all the variables have been assigned, is a float; you
would use @kbd{c f} if you wish to do the conversion on the numbers
that appear right now.

@kindex c F
@pindex calc-fraction
@tindex pfrac
The @kbd{c F} (@code{calc-fraction}) [@code{pfrac}] command converts a
floating-point number into a fractional approximation.  By default, it
produces a fraction whose decimal representation is the same as the
input number, to within the current precision.  You can also give a
numeric prefix argument to specify a tolerance, either directly, or,
if the prefix argument is zero, by using the number on top of the stack
as the tolerance.  If the tolerance is a positive integer, the fraction
is correct to within that many significant figures.  If the tolerance is
a non-positive integer, it specifies how many digits fewer than the current
precision to use.  If the tolerance is a floating-point number, the
fraction is correct to within that absolute amount.

@kindex H c F
@tindex frac
The @code{pfrac} function is pervasive, like @code{pfloat}.
There is also a non-pervasive version, @kbd{H c F} [@code{frac}],
which is analogous to @kbd{H c f} discussed above.

@kindex c d
@pindex calc-to-degrees
@tindex deg
The @kbd{c d} (@code{calc-to-degrees}) [@code{deg}] command converts a
number into degrees form.  The value on the top of the stack may be an
HMS form (interpreted as degrees-minutes-seconds), or a real number which
will be interpreted in radians regardless of the current angular mode.

@kindex c r
@pindex calc-to-radians
@tindex rad
The @kbd{c r} (@code{calc-to-radians}) [@code{rad}] command converts an
HMS form or angle in degrees into an angle in radians.

@kindex c h
@pindex calc-to-hms
@tindex hms
The @kbd{c h} (@code{calc-to-hms}) [@code{hms}] command converts a real
number, interpreted according to the current angular mode, to an HMS
form describing the same angle.  In algebraic notation, the @code{hms}
function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
(The three-argument version is independent of the current angular mode.)

@pindex calc-from-hms
The @code{calc-from-hms} command converts the HMS form on the top of the
stack into a real number according to the current angular mode.

@kindex c p
@kindex I c p
@pindex calc-polar
@tindex polar
@tindex rect
The @kbd{c p} (@code{calc-polar}) command converts the complex number on
the top of the stack from polar to rectangular form, or from rectangular
to polar form, whichever is appropriate.  Real numbers are left the same.
This command is equivalent to the @code{rect} or @code{polar}
functions in algebraic formulas, depending on the direction of
conversion.  (It uses @code{polar}, except that if the argument is
already a polar complex number, it uses @code{rect} instead.  The
@kbd{I c p} command always uses @code{rect}.)

@kindex c c
@pindex calc-clean
@tindex pclean
The @kbd{c c} (@code{calc-clean}) [@code{pclean}] command ``cleans'' the
number on the top of the stack.  Floating point numbers are re-rounded
according to the current precision.  Polar numbers whose angular
components have strayed from the @mathit{-180} to @mathit{+180} degree range
are normalized.  (Note that results will be undesirable if the current
angular mode is different from the one under which the number was
produced!)  Integers and fractions are generally unaffected by this
operation.  Vectors and formulas are cleaned by cleaning each component
number (i.e., pervasively).

If the simplification mode is set below the default level, it is raised
to the default level for the purposes of this command.  Thus, @kbd{c c}
applies the default simplifications even if their automatic application
is disabled.  @xref{Simplification Modes}.

@cindex Roundoff errors, correcting
A numeric prefix argument to @kbd{c c} sets the floating-point precision
to that value for the duration of the command.  A positive prefix (of at
least 3) sets the precision to the specified value; a negative or zero
prefix decreases the precision by the specified amount.

@kindex c 0-9
@pindex calc-clean-num
The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
to @kbd{c c} with the corresponding negative prefix argument.  If roundoff
errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
decimal place often conveniently does the trick.

The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
through @kbd{c 9} commands, also ``clip'' very small floating-point
numbers to zero.  If the exponent is less than or equal to the negative
of the specified precision, the number is changed to 0.0.  For example,
if the current precision is 12, then @kbd{c 2} changes the vector
@samp{[1e-8, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
Numbers this small generally arise from roundoff noise.

If the numbers you are using really are legitimately this small,
you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
(The plain @kbd{c c} command rounds to the current precision but
does not clip small numbers.)

One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
a prefix argument, is that integer-valued floats are converted to
plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
produces @samp{[1, 1.5, 2, 2.5, 3]}.  This is not done for huge
numbers (@samp{1e100} is technically an integer-valued float, but
you wouldn't want it automatically converted to a 100-digit integer).

@kindex H c 0-9
@kindex H c c
@tindex clean
With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
operate non-pervasively [@code{clean}].

@node Date Arithmetic, Financial Functions, Conversions, Arithmetic
@section Date Arithmetic

@noindent
@cindex Date arithmetic, additional functions
The commands described in this section perform various conversions
and calculations involving date forms (@pxref{Date Forms}).  They
use the @kbd{t} (for time/date) prefix key followed by shifted
letters.

The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
commands.  In particular, adding a number to a date form advances the
date form by a certain number of days; adding an HMS form to a date
form advances the date by a certain amount of time; and subtracting two
date forms produces a difference measured in days.  The commands
described here provide additional, more specialized operations on dates.

Many of these commands accept a numeric prefix argument; if you give
plain @kbd{C-u} as the prefix, these commands will instead take the
additional argument from the top of the stack.

@menu
* Date Conversions::
* Date Functions::
* Time Zones::
* Business Days::
@end menu

@node Date Conversions, Date Functions, Date Arithmetic, Date Arithmetic
@subsection Date Conversions

@noindent
@kindex t D
@pindex calc-date
@tindex date
The @kbd{t D} (@code{calc-date}) [@code{date}] command converts a
date form into a number, measured in days since Jan 1, 1 AD.  The
result will be an integer if @var{date} is a pure date form, or a
fraction or float if @var{date} is a date/time form.  Or, if its
argument is a number, it converts this number into a date form.

With a numeric prefix argument, @kbd{t D} takes that many objects
(up to six) from the top of the stack and interprets them in one
of the following ways:

The @samp{date(@var{year}, @var{month}, @var{day})} function
builds a pure date form out of the specified year, month, and
day, which must all be integers.  @var{Year} is a year number,
such as 1991 (@emph{not} the same as 91!).  @var{Month} must be
an integer in the range 1 to 12; @var{day} must be in the range
1 to 31.  If the specified month has fewer than 31 days and
@var{day} is too large, the equivalent day in the following
month will be used.

The @samp{date(@var{month}, @var{day})} function builds a
pure date form using the current year, as determined by the
real-time clock.

The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
function builds a date/time form using an @var{hms} form.

The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
@var{minute}, @var{second})} function builds a date/time form.
@var{hour} should be an integer in the range 0 to 23;
@var{minute} should be an integer in the range 0 to 59;
@var{second} should be any real number in the range @samp{[0 .. 60)}.
The last two arguments default to zero if omitted.

@kindex t J
@pindex calc-julian
@tindex julian
@cindex Julian day counts, conversions
The @kbd{t J} (@code{calc-julian}) [@code{julian}] command converts
a date form into a Julian day count, which is the number of days
since noon on Jan 1, 4713 BC.  A pure date is converted to an integer
Julian count representing noon of that day.  A date/time form is
converted to an exact floating-point Julian count, adjusted to
interpret the date form in the current time zone but the Julian
day count in Greenwich Mean Time.  A numeric prefix argument allows
you to specify the time zone; @pxref{Time Zones}.  Use a prefix of
zero to suppress the time zone adjustment.  Note that pure date forms
are never time-zone adjusted.

This command can also do the opposite conversion, from a Julian day
count (either an integer day, or a floating-point day and time in
the GMT zone), into a pure date form or a date/time form in the
current or specified time zone.

@kindex t U
@pindex calc-unix-time
@tindex unixtime
@cindex Unix time format, conversions
The @kbd{t U} (@code{calc-unix-time}) [@code{unixtime}] command
converts a date form into a Unix time value, which is the number of
seconds since midnight on Jan 1, 1970, or vice-versa.  The numeric result
will be an integer if the current precision is 12 or less; for higher
precisions, the result may be a float with (@var{precision}@minus{}12)
digits after the decimal.  Just as for @kbd{t J}, the numeric time
is interpreted in the GMT time zone and the date form is interpreted
in the current or specified zone.  Some systems use Unix-like
numbering but with the local time zone; give a prefix of zero to
suppress the adjustment if so.

@kindex t C
@pindex calc-convert-time-zones
@tindex tzconv
@cindex Time Zones, converting between
The @kbd{t C} (@code{calc-convert-time-zones}) [@code{tzconv}]
command converts a date form from one time zone to another.  You
are prompted for each time zone name in turn; you can answer with
any suitable Calc time zone expression (@pxref{Time Zones}).
If you answer either prompt with a blank line, the local time
zone is used for that prompt.  You can also answer the first
prompt with @kbd{$} to take the two time zone names from the
stack (and the date to be converted from the third stack level).

@node Date Functions, Business Days, Date Conversions, Date Arithmetic
@subsection Date Functions

@noindent
@kindex t N
@pindex calc-now
@tindex now
The @kbd{t N} (@code{calc-now}) [@code{now}] command pushes the
current date and time on the stack as a date form.  The time is
reported in terms of the specified time zone; with no numeric prefix
argument, @kbd{t N} reports for the current time zone.

@kindex t P
@pindex calc-date-part
The @kbd{t P} (@code{calc-date-part}) command extracts one part
of a date form.  The prefix argument specifies the part; with no
argument, this command prompts for a part code from 1 to 9.
The various part codes are described in the following paragraphs.

@tindex year
The @kbd{M-1 t P} [@code{year}] function extracts the year number
from a date form as an integer, e.g., 1991.  This and the
following functions will also accept a real number for an
argument, which is interpreted as a standard Calc day number.
Note that this function will never return zero, since the year
1 BC immediately precedes the year 1 AD.

@tindex month
The @kbd{M-2 t P} [@code{month}] function extracts the month number
from a date form as an integer in the range 1 to 12.

@tindex day
The @kbd{M-3 t P} [@code{day}] function extracts the day number
from a date form as an integer in the range 1 to 31.

@tindex hour
The @kbd{M-4 t P} [@code{hour}] function extracts the hour from
a date form as an integer in the range 0 (midnight) to 23.  Note
that 24-hour time is always used.  This returns zero for a pure
date form.  This function (and the following two) also accept
HMS forms as input.

@tindex minute
The @kbd{M-5 t P} [@code{minute}] function extracts the minute
from a date form as an integer in the range 0 to 59.

@tindex second
The @kbd{M-6 t P} [@code{second}] function extracts the second
from a date form.  If the current precision is 12 or less,
the result is an integer in the range 0 to 59.  For higher
precisions, the result may instead be a floating-point number.

@tindex weekday
The @kbd{M-7 t P} [@code{weekday}] function extracts the weekday
number from a date form as an integer in the range 0 (Sunday)
to 6 (Saturday).

@tindex yearday
The @kbd{M-8 t P} [@code{yearday}] function extracts the day-of-year
number from a date form as an integer in the range 1 (January 1)
to 366 (December 31 of a leap year).

@tindex time
The @kbd{M-9 t P} [@code{time}] function extracts the time portion
of a date form as an HMS form.  This returns @samp{0@@ 0' 0"}
for a pure date form.

@kindex t M
@pindex calc-new-month
@tindex newmonth
The @kbd{t M} (@code{calc-new-month}) [@code{newmonth}] command
computes a new date form that represents the first day of the month
specified by the input date.  The result is always a pure date
form; only the year and month numbers of the input are retained.
With a numeric prefix argument @var{n} in the range from 1 to 31,
@kbd{t M} computes the @var{n}th day of the month.  (If @var{n}
is greater than the actual number of days in the month, or if
@var{n} is zero, the last day of the month is used.)

@kindex t Y
@pindex calc-new-year
@tindex newyear
The @kbd{t Y} (@code{calc-new-year}) [@code{newyear}] command
computes a new pure date form that represents the first day of
the year specified by the input.  The month, day, and time
of the input date form are lost.  With a numeric prefix argument
@var{n} in the range from 1 to 366, @kbd{t Y} computes the
@var{n}th day of the year (366 is treated as 365 in non-leap
years).  A prefix argument of 0 computes the last day of the
year (December 31).  A negative prefix argument from @mathit{-1} to
@mathit{-12} computes the first day of the @var{n}th month of the year.

@kindex t W
@pindex calc-new-week
@tindex newweek
The @kbd{t W} (@code{calc-new-week}) [@code{newweek}] command
computes a new pure date form that represents the Sunday on or before
the input date.  With a numeric prefix argument, it can be made to
use any day of the week as the starting day; the argument must be in
the range from 0 (Sunday) to 6 (Saturday).  This function always
subtracts between 0 and 6 days from the input date.

Here's an example use of @code{newweek}:  Find the date of the next
Wednesday after a given date.  Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
will give you the following Wednesday.  A further look at the definition
of @code{newweek} shows that if the input date is itself a Wednesday,
this formula will return the Wednesday one week in the future.  An
exercise for the reader is to modify this formula to yield the same day
if the input is already a Wednesday.  Another interesting exercise is
to preserve the time-of-day portion of the input (@code{newweek} resets
the time to midnight; hint:@: how can @code{newweek} be defined in terms
of the @code{weekday} function?).

@ignore
@starindex
@end ignore
@tindex pwday
The @samp{pwday(@var{date})} function (not on any key) computes the
day-of-month number of the Sunday on or before @var{date}.  With
two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
number of the Sunday on or before day number @var{day} of the month
specified by @var{date}.  The @var{day} must be in the range from
7 to 31; if the day number is greater than the actual number of days
in the month, the true number of days is used instead.  Thus
@samp{pwday(@var{date}, 7)} finds the first Sunday of the month, and
@samp{pwday(@var{date}, 31)} finds the last Sunday of the month.
With a third @var{weekday} argument, @code{pwday} can be made to look
for any day of the week instead of Sunday.

@kindex t I
@pindex calc-inc-month
@tindex incmonth
The @kbd{t I} (@code{calc-inc-month}) [@code{incmonth}] command
increases a date form by one month, or by an arbitrary number of
months specified by a numeric prefix argument.  The time portion,
if any, of the date form stays the same.  The day also stays the
same, except that if the new month has fewer days the day
number may be reduced to lie in the valid range.  For example,
@samp{incmonth(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
in this case).

@ignore
@starindex
@end ignore
@tindex incyear
The @samp{incyear(@var{date}, @var{step})} function increases
a date form by the specified number of years, which may be
any positive or negative integer.  Note that @samp{incyear(d, n)}
is equivalent to @w{@samp{incmonth(d, 12*n)}}, but these do not have
simple equivalents in terms of day arithmetic because
months and years have varying lengths.  If the @var{step}
argument is omitted, 1 year is assumed.  There is no keyboard
command for this function; use @kbd{C-u 12 t I} instead.

There is no @code{newday} function at all because @kbd{F} [@code{floor}]
serves this purpose.  Similarly, instead of @code{incday} and
@code{incweek} simply use @expr{d + n} or @expr{d + 7 n}.

@xref{Basic Arithmetic}, for the @kbd{f ]} [@code{incr}] command
which can adjust a date/time form by a certain number of seconds.

@node Business Days, Time Zones, Date Functions, Date Arithmetic
@subsection Business Days

@noindent
Often time is measured in ``business days'' or ``working days,''
where weekends and holidays are skipped.  Calc's normal date
arithmetic functions use calendar days, so that subtracting two
consecutive Mondays will yield a difference of 7 days.  By contrast,
subtracting two consecutive Mondays would yield 5 business days
(assuming two-day weekends and the absence of holidays).

@kindex t +
@kindex t -
@tindex badd
@tindex bsub
@pindex calc-business-days-plus
@pindex calc-business-days-minus
The @kbd{t +} (@code{calc-business-days-plus}) [@code{badd}]
and @kbd{t -} (@code{calc-business-days-minus}) [@code{bsub}]
commands perform arithmetic using business days.  For @kbd{t +},
one argument must be a date form and the other must be a real
number (positive or negative).  If the number is not an integer,
then a certain amount of time is added as well as a number of
days; for example, adding 0.5 business days to a time in Friday
evening will produce a time in Monday morning.  It is also
possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
half a business day.  For @kbd{t -}, the arguments are either a
date form and a number or HMS form, or two date forms, in which
case the result is the number of business days between the two
dates.

@cindex @code{Holidays} variable
@vindex Holidays
By default, Calc considers any day that is not a Saturday or
Sunday to be a business day.  You can define any number of
additional holidays by editing the variable @code{Holidays}.
(There is an @w{@kbd{s H}} convenience command for editing this
variable.)  Initially, @code{Holidays} contains the vector
@samp{[sat, sun]}.  Entries in the @code{Holidays} vector may
be any of the following kinds of objects:

@itemize @bullet
@item
Date forms (pure dates, not date/time forms).  These specify
particular days which are to be treated as holidays.

@item
Intervals of date forms.  These specify a range of days, all of
which are holidays (e.g., Christmas week).  @xref{Interval Forms}.

@item
Nested vectors of date forms.  Each date form in the vector is
considered to be a holiday.

@item
Any Calc formula which evaluates to one of the above three things.
If the formula involves the variable @expr{y}, it stands for a
yearly repeating holiday; @expr{y} will take on various year
numbers like 1992.  For example, @samp{date(y, 12, 25)} specifies
Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
Thanksgiving (which is held on the fourth Thursday of November).
If the formula involves the variable @expr{m}, that variable
takes on month numbers from 1 to 12:  @samp{date(y, m, 15)} is
a holiday that takes place on the 15th of every month.

@item
A weekday name, such as @code{sat} or @code{sun}.  This is really
a variable whose name is a three-letter, lower-case day name.

@item
An interval of year numbers (integers).  This specifies the span of
years over which this holiday list is to be considered valid.  Any
business-day arithmetic that goes outside this range will result
in an error message.  Use this if you are including an explicit
list of holidays, rather than a formula to generate them, and you
want to make sure you don't accidentally go beyond the last point
where the holidays you entered are complete.  If there is no
limiting interval in the @code{Holidays} vector, the default
@samp{[1 .. 2737]} is used.  (This is the absolute range of years
for which Calc's business-day algorithms will operate.)

@item
An interval of HMS forms.  This specifies the span of hours that
are to be considered one business day.  For example, if this
range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
the business day is only eight hours long, so that @kbd{1.5 t +}
on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
Likewise, @kbd{t -} will now express differences in time as
fractions of an eight-hour day.  Times before 9am will be treated
as 9am by business date arithmetic, and times at or after 5pm will
be treated as 4:59:59pm.  If there is no HMS interval in @code{Holidays},
the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
(Regardless of the type of bounds you specify, the interval is
treated as inclusive on the low end and exclusive on the high end,
so that the work day goes from 9am up to, but not including, 5pm.)
@end itemize

If the @code{Holidays} vector is empty, then @kbd{t +} and
@kbd{t -} will act just like @kbd{+} and @kbd{-} because there will
then be no difference between business days and calendar days.

Calc expands the intervals and formulas you give into a complete
list of holidays for internal use.  This is done mainly to make
sure it can detect multiple holidays.  (For example,
@samp{<Jan 1, 1989>} is both New Year's Day and a Sunday, but
Calc's algorithms take care to count it only once when figuring
the number of holidays between two dates.)

Since the complete list of holidays for all the years from 1 to
2737 would be huge, Calc actually computes only the part of the
list between the smallest and largest years that have been involved
in business-day calculations so far.  Normally, you won't have to
worry about this.  Keep in mind, however, that if you do one
calculation for 1992, and another for 1792, even if both involve
only a small range of years, Calc will still work out all the
holidays that fall in that 200-year span.

If you add a (positive) number of days to a date form that falls on a
weekend or holiday, the date form is treated as if it were the most
recent business day.  (Thus adding one business day to a Friday,
Saturday, or Sunday will all yield the following Monday.)  If you
subtract a number of days from a weekend or holiday, the date is
effectively on the following business day.  (So subtracting one business
day from Saturday, Sunday, or Monday yields the preceding Friday.)  The
difference between two dates one or both of which fall on holidays
equals the number of actual business days between them.  These
conventions are consistent in the sense that, if you add @var{n}
business days to any date, the difference between the result and the
original date will come out to @var{n} business days.  (It can't be
completely consistent though; a subtraction followed by an addition
might come out a bit differently, since @kbd{t +} is incapable of
producing a date that falls on a weekend or holiday.)

@ignore
@starindex
@end ignore
@tindex holiday
There is a @code{holiday} function, not on any keys, that takes
any date form and returns 1 if that date falls on a weekend or
holiday, as defined in @code{Holidays}, or 0 if the date is a
business day.

@node Time Zones,  , Business Days, Date Arithmetic
@subsection Time Zones

@noindent
@cindex Time zones
@cindex Daylight saving time
Time zones and daylight saving time are a complicated business.
The conversions to and from Julian and Unix-style dates automatically
compute the correct time zone and daylight saving adjustment to use,
provided they can figure out this information.  This section describes
Calc's time zone adjustment algorithm in detail, in case you want to
do conversions in different time zones or in case Calc's algorithms
can't determine the right correction to use.

Adjustments for time zones and daylight saving time are done by
@kbd{t U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
commands.  In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
to exactly 30 days even though there is a daylight-saving
transition in between.  This is also true for Julian pure dates:
@samp{julian(<may 1 1991>) - julian(<apr 1 1991>)}.  But Julian
and Unix date/times will adjust for daylight saving time:  using Calc's
default daylight saving time rule (see the explanation below),
@samp{julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
evaluates to @samp{29.95833} (that's 29 days and 23 hours)
because one hour was lost when daylight saving commenced on
April 7, 1991.

In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
computes the actual number of 24-hour periods between two dates, whereas
@samp{@var{date1} - @var{date2}} computes the number of calendar
days between two dates without taking daylight saving into account.

@pindex calc-time-zone
@ignore
@starindex
@end ignore
@tindex tzone
The @code{calc-time-zone} [@code{tzone}] command converts the time
zone specified by its numeric prefix argument into a number of
seconds difference from Greenwich mean time (GMT).  If the argument
is a number, the result is simply that value multiplied by 3600.
Typical arguments for North America are 5 (Eastern) or 8 (Pacific).  If
Daylight Saving time is in effect, one hour should be subtracted from
the normal difference.

If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
date arithmetic commands that include a time zone argument) takes the
zone argument from the top of the stack.  (In the case of @kbd{t J}
and @kbd{t U}, the normal argument is then taken from the second-to-top
stack position.)  This allows you to give a non-integer time zone
adjustment.  The time-zone argument can also be an HMS form, or
it can be a variable which is a time zone name in upper- or lower-case.
For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
(for Pacific standard and daylight saving times, respectively).

North American and European time zone names are defined as follows;
note that for each time zone there is one name for standard time,
another for daylight saving time, and a third for ``generalized'' time
in which the daylight saving adjustment is computed from context.

@smallexample
@group
YST  PST  MST  CST  EST  AST    NST    GMT   WET     MET    MEZ
 9    8    7    6    5    4     3.5     0     -1      -2     -2

YDT  PDT  MDT  CDT  EDT  ADT    NDT    BST  WETDST  METDST  MESZ
 8    7    6    5    4    3     2.5     -1    -2      -3     -3

YGT  PGT  MGT  CGT  EGT  AGT    NGT    BGT   WEGT    MEGT   MEGZ
9/8  8/7  7/6  6/5  5/4  4/3  3.5/2.5  0/-1 -1/-2   -2/-3  -2/-3
@end group
@end smallexample

@vindex math-tzone-names
To define time zone names that do not appear in the above table,
you must modify the Lisp variable @code{math-tzone-names}.  This
is a list of lists describing the different time zone names; its
structure is best explained by an example.  The three entries for
Pacific Time look like this:

@smallexample
@group
( ( "PST" 8 0 )    ; Name as an upper-case string, then standard
  ( "PDT" 8 -1 )   ; adjustment, then daylight saving adjustment.
  ( "PGT" 8 "PST" "PDT" ) )   ; Generalized time zone.
@end group
@end smallexample

@cindex @code{TimeZone} variable
@vindex TimeZone
With no arguments, @code{calc-time-zone} or @samp{tzone()} obtains an
argument from the Calc variable @code{TimeZone} if a value has been
stored for that variable.  If not, Calc runs the Unix @samp{date}
command and looks for one of the above time zone names in the output;
if this does not succeed, @samp{tzone()} leaves itself unevaluated.
The time zone name in the @samp{date} output may be followed by a signed
adjustment, e.g., @samp{GMT+5} or @samp{GMT+0500} which specifies a
number of hours and minutes to be added to the base time zone.
Calc stores the time zone it finds into @code{TimeZone} to speed
later calls to @samp{tzone()}.

The special time zone name @code{local} is equivalent to no argument,
i.e., it uses the local time zone as obtained from the @code{date}
command.

If the time zone name found is one of the standard or daylight
saving zone names from the above table, and Calc's internal
daylight saving algorithm says that time and zone are consistent
(e.g., @code{PDT} accompanies a date that Calc's algorithm would also
consider to be daylight saving, or @code{PST} accompanies a date
that Calc would consider to be standard time), then Calc substitutes
the corresponding generalized time zone (like @code{PGT}).

If your system does not have a suitable @samp{date} command, you
may wish to put a @samp{(setq var-TimeZone ...)} in your Emacs
initialization file to set the time zone.  (Since you are interacting
with the variable @code{TimeZone} directly from Emacs Lisp, the
@code{var-} prefix needs to be present.)  The easiest way to do
this is to edit the @code{TimeZone} variable using Calc's @kbd{s T}
command, then use the @kbd{s p} (@code{calc-permanent-variable})
command to save the value of @code{TimeZone} permanently.

The @kbd{t J} and @code{t U} commands with no numeric prefix
arguments do the same thing as @samp{tzone()}.  If the current
time zone is a generalized time zone, e.g., @code{EGT}, Calc
examines the date being converted to tell whether to use standard
or daylight saving time.  But if the current time zone is explicit,
e.g., @code{EST} or @code{EDT}, then that adjustment is used exactly
and Calc's daylight saving algorithm is not consulted.

Some places don't follow the usual rules for daylight saving time.
The state of Arizona, for example, does not observe daylight saving
time.  If you run Calc during the winter season in Arizona, the
Unix @code{date} command will report @code{MST} time zone, which
Calc will change to @code{MGT}.  If you then convert a time that
lies in the summer months, Calc will apply an incorrect daylight
saving time adjustment.  To avoid this, set your @code{TimeZone}
variable explicitly to @code{MST} to force the use of standard,
non-daylight-saving time.

@vindex math-daylight-savings-hook
@findex math-std-daylight-savings
By default Calc always considers daylight saving time to begin at
2 a.m.@: on the second Sunday of March (for years from 2007 on) or on
the last Sunday in April (for years before 2007), and to end at 2 a.m.@:
on the first Sunday of November. (for years from 2007 on) or the last
Sunday in October (for years before 2007).  These are the rules that have
been in effect in much of North America since 1966 and takes into
account the rule change that began in 2007.  If you are in a
country that uses different rules for computing daylight saving time,
you have two choices:  Write your own daylight saving hook, or control
time zones explicitly by setting the @code{TimeZone} variable and/or
always giving a time-zone argument for the conversion functions.

The Lisp variable @code{math-daylight-savings-hook} holds the
name of a function that is used to compute the daylight saving
adjustment for a given date.  The default is
@code{math-std-daylight-savings}, which computes an adjustment
(either 0 or @mathit{-1}) using the North American rules given above.

The daylight saving hook function is called with four arguments:
The date, as a floating-point number in standard Calc format;
a six-element list of the date decomposed into year, month, day,
hour, minute, and second, respectively; a string which contains
the generalized time zone name in upper-case, e.g., @code{"WEGT"};
and a special adjustment to be applied to the hour value when
converting into a generalized time zone (see below).

@findex math-prev-weekday-in-month
The Lisp function @code{math-prev-weekday-in-month} is useful for
daylight saving computations.  This is an internal version of
the user-level @code{pwday} function described in the previous
section. It takes four arguments:  The floating-point date value,
the corresponding six-element date list, the day-of-month number,
and the weekday number (0-6).

The default daylight saving hook ignores the time zone name, but a
more sophisticated hook could use different algorithms for different
time zones.  It would also be possible to use different algorithms
depending on the year number, but the default hook always uses the
algorithm for 1987 and later.  Here is a listing of the default
daylight saving hook:

@smallexample
(defun math-std-daylight-savings (date dt zone bump)
  (cond ((< (nth 1 dt) 4) 0)
        ((= (nth 1 dt) 4)
         (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
           (cond ((< (nth 2 dt) sunday) 0)
                 ((= (nth 2 dt) sunday)
                  (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
                 (t -1))))
        ((< (nth 1 dt) 10) -1)
        ((= (nth 1 dt) 10)
         (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
           (cond ((< (nth 2 dt) sunday) -1)
                 ((= (nth 2 dt) sunday)
                  (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
                 (t 0))))
        (t 0))
)
@end smallexample

@noindent
The @code{bump} parameter is equal to zero when Calc is converting
from a date form in a generalized time zone into a GMT date value.
It is @mathit{-1} when Calc is converting in the other direction.  The
adjustments shown above ensure that the conversion behaves correctly
and reasonably around the 2 a.m.@: transition in each direction.

There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
beginning of daylight saving time; converting a date/time form that
falls in this hour results in a time value for the following hour,
from 3 a.m.@: to 4 a.m.  At the end of daylight saving time, the
hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
form that falls in this hour results in a time value for the first
manifestation of that time (@emph{not} the one that occurs one hour later).

If @code{math-daylight-savings-hook} is @code{nil}, then the
daylight saving adjustment is always taken to be zero.

In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
computes the time zone adjustment for a given zone name at a
given date.  The @var{date} is ignored unless @var{zone} is a
generalized time zone.  If @var{date} is a date form, the
daylight saving computation is applied to it as it appears.
If @var{date} is a numeric date value, it is adjusted for the
daylight-saving version of @var{zone} before being given to
the daylight saving hook.  This odd-sounding rule ensures
that the daylight-saving computation is always done in
local time, not in the GMT time that a numeric @var{date}
is typically represented in.

@ignore
@starindex
@end ignore
@tindex dsadj
The @samp{dsadj(@var{date}, @var{zone})} function computes the
daylight saving adjustment that is appropriate for @var{date} in
time zone @var{zone}.  If @var{zone} is explicitly in or not in
daylight saving time (e.g., @code{PDT} or @code{PST}) the
@var{date} is ignored.  If @var{zone} is a generalized time zone,
the algorithms described above are used.  If @var{zone} is omitted,
the computation is done for the current time zone.

@xref{Reporting Bugs}, for the address of Calc's author, if you
should wish to contribute your improved versions of
@code{math-tzone-names} and @code{math-daylight-savings-hook}
to the Calc distribution.

@node Financial Functions, Binary Functions, Date Arithmetic, Arithmetic
@section Financial Functions

@noindent
Calc's financial or business functions use the @kbd{b} prefix
key followed by a shifted letter.  (The @kbd{b} prefix followed by
a lower-case letter is used for operations on binary numbers.)

Note that the rate and the number of intervals given to these
functions must be on the same time scale, e.g., both months or
both years.  Mixing an annual interest rate with a time expressed
in months will give you very wrong answers!

It is wise to compute these functions to a higher precision than
you really need, just to make sure your answer is correct to the
last penny; also, you may wish to check the definitions at the end
of this section to make sure the functions have the meaning you expect.

@menu
* Percentages::
* Future Value::
* Present Value::
* Related Financial Functions::
* Depreciation Functions::
* Definitions of Financial Functions::
@end menu

@node Percentages, Future Value, Financial Functions, Financial Functions
@subsection Percentages

@kindex M-%
@pindex calc-percent
@tindex %
@tindex percent
The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
say 5.4, and converts it to an equivalent actual number.  For example,
@kbd{5.4 M-%} enters 0.054 on the stack.  (That's the @key{META} or
@key{ESC} key combined with @kbd{%}.)

Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
You can enter @samp{5.4%} yourself during algebraic entry.  The
@samp{%} operator simply means, ``the preceding value divided by
100.''  The @samp{%} operator has very high precedence, so that
@samp{1+8%} is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
(The @samp{%} operator is just a postfix notation for the
@code{percent} function, just like @samp{20!} is the notation for
@samp{fact(20)}, or twenty-factorial.)

The formula @samp{5.4%} would normally evaluate immediately to
0.054, but the @kbd{M-%} command suppresses evaluation as it puts
the formula onto the stack.  However, the next Calc command that
uses the formula @samp{5.4%} will evaluate it as its first step.
The net effect is that you get to look at @samp{5.4%} on the stack,
but Calc commands see it as @samp{0.054}, which is what they expect.

In particular, @samp{5.4%} and @samp{0.054} are suitable values
for the @var{rate} arguments of the various financial functions,
but the number @samp{5.4} is probably @emph{not} suitable---it
represents a rate of 540 percent!

The key sequence @kbd{M-% *} effectively means ``percent-of.''
For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
68 (and also 68% of 25, which comes out to the same thing).

@kindex c %
@pindex calc-convert-percent
The @kbd{c %} (@code{calc-convert-percent}) command converts the
value on the top of the stack from numeric to percentage form.
For example, if 0.08 is on the stack, @kbd{c %} converts it to
@samp{8%}.  The quantity is the same, it's just represented
differently.  (Contrast this with @kbd{M-%}, which would convert
this number to @samp{0.08%}.)  The @kbd{=} key is a convenient way
to convert a formula like @samp{8%} back to numeric form, 0.08.

To compute what percentage one quantity is of another quantity,
use @kbd{/ c %}.  For example, @w{@kbd{17 @key{RET} 68 / c %}} displays
@samp{25%}.

@kindex b %
@pindex calc-percent-change
@tindex relch
The @kbd{b %} (@code{calc-percent-change}) [@code{relch}] command
calculates the percentage change from one number to another.
For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
since 50 is 25% larger than 40.  A negative result represents a
decrease:  @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
20% smaller than 50.  (The answers are different in magnitude
because, in the first case, we're increasing by 25% of 40, but
in the second case, we're decreasing by 20% of 50.)  The effect
of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
the answer to percentage form as if by @kbd{c %}.

@node Future Value, Present Value, Percentages, Financial Functions
@subsection Future Value

@noindent
@kindex b F
@pindex calc-fin-fv
@tindex fv
The @kbd{b F} (@code{calc-fin-fv}) [@code{fv}] command computes
the future value of an investment.  It takes three arguments
from the stack:  @samp{fv(@var{rate}, @var{n}, @var{payment})}.
If you give payments of @var{payment} every year for @var{n}
years, and the money you have paid earns interest at @var{rate} per
year, then this function tells you what your investment would be
worth at the end of the period.  (The actual interval doesn't
have to be years, as long as @var{n} and @var{rate} are expressed
in terms of the same intervals.)  This function assumes payments
occur at the @emph{end} of each interval.

@kindex I b F
@tindex fvb
The @kbd{I b F} [@code{fvb}] command does the same computation,
but assuming your payments are at the beginning of each interval.
Suppose you plan to deposit $1000 per year in a savings account
earning 5.4% interest, starting right now.  How much will be
in the account after five years?  @code{fvb(5.4%, 5, 1000) = 5870.73}.
Thus you will have earned $870 worth of interest over the years.
Using the stack, this calculation would have been
@kbd{5.4 M-% 5 @key{RET} 1000 I b F}.  Note that the rate is expressed
as a number between 0 and 1, @emph{not} as a percentage.

@kindex H b F
@tindex fvl
The @kbd{H b F} [@code{fvl}] command computes the future value
of an initial lump sum investment.  Suppose you could deposit
those five thousand dollars in the bank right now; how much would
they be worth in five years?  @code{fvl(5.4%, 5, 5000) = 6503.89}.

The algebraic functions @code{fv} and @code{fvb} accept an optional
fourth argument, which is used as an initial lump sum in the sense
of @code{fvl}.  In other words, @code{fv(@var{rate}, @var{n},
@var{payment}, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
+ fvl(@var{rate}, @var{n}, @var{initial})}.

To illustrate the relationships between these functions, we could
do the @code{fvb} calculation ``by hand'' using @code{fvl}.  The
final balance will be the sum of the contributions of our five
deposits at various times.  The first deposit earns interest for
five years:  @code{fvl(5.4%, 5, 1000) = 1300.78}.  The second
deposit only earns interest for four years:  @code{fvl(5.4%, 4, 1000) =
1234.13}.  And so on down to the last deposit, which earns one
year's interest:  @code{fvl(5.4%, 1, 1000) = 1054.00}.  The sum of
these five values is, sure enough, $5870.73, just as was computed
by @code{fvb} directly.

What does @code{fv(5.4%, 5, 1000) = 5569.96} mean?  The payments
are now at the ends of the periods.  The end of one year is the same
as the beginning of the next, so what this really means is that we've
lost the payment at year zero (which contributed $1300.78), but we're
now counting the payment at year five (which, since it didn't have
a chance to earn interest, counts as $1000).  Indeed, @expr{5569.96 =
5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).

@node Present Value, Related Financial Functions, Future Value, Financial Functions
@subsection Present Value

@noindent
@kindex b P
@pindex calc-fin-pv
@tindex pv
The @kbd{b P} (@code{calc-fin-pv}) [@code{pv}] command computes
the present value of an investment.  Like @code{fv}, it takes
three arguments:  @code{pv(@var{rate}, @var{n}, @var{payment})}.
It computes the present value of a series of regular payments.
Suppose you have the chance to make an investment that will
pay $2000 per year over the next four years; as you receive
these payments you can put them in the bank at 9% interest.
You want to know whether it is better to make the investment, or
to keep the money in the bank where it earns 9% interest right
from the start.  The calculation @code{pv(9%, 4, 2000)} gives the
result 6479.44.  If your initial investment must be less than this,
say, $6000, then the investment is worthwhile.  But if you had to
put up $7000, then it would be better just to leave it in the bank.

Here is the interpretation of the result of @code{pv}:  You are
trying to compare the return from the investment you are
considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
the return from leaving the money in the bank, which is
@code{fvl(9%, 4, @var{x})} where @var{x} is the amount of money
you would have to put up in advance.  The @code{pv} function
finds the break-even point, @expr{x = 6479.44}, at which
@code{fvl(9%, 4, 6479.44)} is also equal to 9146.26.  This is
the largest amount you should be willing to invest.

@kindex I b P
@tindex pvb
The @kbd{I b P} [@code{pvb}] command solves the same problem,
but with payments occurring at the beginning of each interval.
It has the same relationship to @code{fvb} as @code{pv} has
to @code{fv}.  For example @code{pvb(9%, 4, 2000) = 7062.59},
a larger number than @code{pv} produced because we get to start
earning interest on the return from our investment sooner.

@kindex H b P
@tindex pvl
The @kbd{H b P} [@code{pvl}] command computes the present value of
an investment that will pay off in one lump sum at the end of the
period.  For example, if we get our $8000 all at the end of the
four years, @code{pvl(9%, 4, 8000) = 5667.40}.  This is much
less than @code{pv} reported, because we don't earn any interest
on the return from this investment.  Note that @code{pvl} and
@code{fvl} are simple inverses:  @code{fvl(9%, 4, 5667.40) = 8000}.

You can give an optional fourth lump-sum argument to @code{pv}
and @code{pvb}; this is handled in exactly the same way as the
fourth argument for @code{fv} and @code{fvb}.

@kindex b N
@pindex calc-fin-npv
@tindex npv
The @kbd{b N} (@code{calc-fin-npv}) [@code{npv}] command computes
the net present value of a series of irregular investments.
The first argument is the interest rate.  The second argument is
a vector which represents the expected return from the investment
at the end of each interval.  For example, if the rate represents
a yearly interest rate, then the vector elements are the return
from the first year, second year, and so on.

Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
Obviously this function is more interesting when the payments are
not all the same!

The @code{npv} function can actually have two or more arguments.
Multiple arguments are interpreted in the same way as for the
vector statistical functions like @code{vsum}.
@xref{Single-Variable Statistics}.  Basically, if there are several
payment arguments, each either a vector or a plain number, all these
values are collected left-to-right into the complete list of payments.
A numeric prefix argument on the @kbd{b N} command says how many
payment values or vectors to take from the stack.

@kindex I b N
@tindex npvb
The @kbd{I b N} [@code{npvb}] command computes the net present
value where payments occur at the beginning of each interval
rather than at the end.

@node Related Financial Functions, Depreciation Functions, Present Value, Financial Functions
@subsection Related Financial Functions

@noindent
The functions in this section are basically inverses of the
present value functions with respect to the various arguments.

@kindex b M
@pindex calc-fin-pmt
@tindex pmt
The @kbd{b M} (@code{calc-fin-pmt}) [@code{pmt}] command computes
the amount of periodic payment necessary to amortize a loan.
Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
value of @var{payment} such that @code{pv(@var{rate}, @var{n},
@var{payment}) = @var{amount}}.

@kindex I b M
@tindex pmtb
The @kbd{I b M} [@code{pmtb}] command does the same computation
but using @code{pvb} instead of @code{pv}.  Like @code{pv} and
@code{pvb}, these functions can also take a fourth argument which
represents an initial lump-sum investment.

@kindex H b M
The @kbd{H b M} key just invokes the @code{fvl} function, which is
the inverse of @code{pvl}.  There is no explicit @code{pmtl} function.

@kindex b #
@pindex calc-fin-nper
@tindex nper
The @kbd{b #} (@code{calc-fin-nper}) [@code{nper}] command computes
the number of regular payments necessary to amortize a loan.
Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
the value of @var{n} such that @code{pv(@var{rate}, @var{n},
@var{payment}) = @var{amount}}.  If @var{payment} is too small
ever to amortize a loan for @var{amount} at interest rate @var{rate},
the @code{nper} function is left in symbolic form.

@kindex I b #
@tindex nperb
The @kbd{I b #} [@code{nperb}] command does the same computation
but using @code{pvb} instead of @code{pv}.  You can give a fourth
lump-sum argument to these functions, but the computation will be
rather slow in the four-argument case.

@kindex H b #
@tindex nperl
The @kbd{H b #} [@code{nperl}] command does the same computation
using @code{pvl}.  By exchanging @var{payment} and @var{amount} you
can also get the solution for @code{fvl}.  For example,
@code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
bank account earning 8%, it will take nine years to grow to $2000.

@kindex b T
@pindex calc-fin-rate
@tindex rate
The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes
the rate of return on an investment.  This is also an inverse of @code{pv}:
@code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of
@var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
@var{amount}}.  The result is expressed as a formula like @samp{6.3%}.

@kindex I b T
@kindex H b T
@tindex rateb
@tindex ratel
The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}]
commands solve the analogous equations with @code{pvb} or @code{pvl}
in place of @code{pv}.  Also, @code{rate} and @code{rateb} can
accept an optional fourth argument just like @code{pv} and @code{pvb}.
To redo the above example from a different perspective,
@code{ratel(9, 2000, 1000) = 8.00597%}, which says you will need an
interest rate of 8% in order to double your account in nine years.

@kindex b I
@pindex calc-fin-irr
@tindex irr
The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the
analogous function to @code{rate} but for net present value.
Its argument is a vector of payments.  Thus @code{irr(@var{payments})}
computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
this rate is known as the @dfn{internal rate of return}.

@kindex I b I
@tindex irrb
The @kbd{I b I} [@code{irrb}] command computes the internal rate of
return assuming payments occur at the beginning of each period.

@node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions
@subsection Depreciation Functions

@noindent
The functions in this section calculate @dfn{depreciation}, which is
the amount of value that a possession loses over time.  These functions
are characterized by three parameters:  @var{cost}, the original cost
of the asset; @var{salvage}, the value the asset will have at the end
of its expected ``useful life''; and @var{life}, the number of years
(or other periods) of the expected useful life.

There are several methods for calculating depreciation that differ in
the way they spread the depreciation over the lifetime of the asset.

@kindex b S
@pindex calc-fin-sln
@tindex sln
The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the
``straight-line'' depreciation.  In this method, the asset depreciates
by the same amount every year (or period).  For example,
@samp{sln(12000, 2000, 5)} returns 2000.  The asset costs $12000
initially and will be worth $2000 after five years; it loses $2000
per year.

@kindex b Y
@pindex calc-fin-syd
@tindex syd
The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the
accelerated ``sum-of-years'-digits'' depreciation.  Here the depreciation
is higher during the early years of the asset's life.  Since the
depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
parameter which specifies which year is requested, from 1 to @var{life}.
If @var{period} is outside this range, the @code{syd} function will
return zero.

@kindex b D
@pindex calc-fin-ddb
@tindex ddb
The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an
accelerated depreciation using the double-declining balance method.
It also takes a fourth @var{period} parameter.

For symmetry, the @code{sln} function will accept a @var{period}
parameter as well, although it will ignore its value except that the
return value will as usual be zero if @var{period} is out of range.

For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
the three depreciation methods:

@example
@group
[ [ 2000, 3333, 4800 ]
  [ 2000, 2667, 2880 ]
  [ 2000, 2000, 1728 ]
  [ 2000, 1333,  592 ]
  [ 2000,  667,   0  ] ]
@end group
@end example

@noindent
(Values have been rounded to nearest integers in this figure.)
We see that @code{sln} depreciates by the same amount each year,
@kbd{syd} depreciates more at the beginning and less at the end,
and @kbd{ddb} weights the depreciation even more toward the beginning.

Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
the total depreciation in any method is (by definition) the
difference between the cost and the salvage value.

@node Definitions of Financial Functions,  , Depreciation Functions, Financial Functions
@subsection Definitions

@noindent
For your reference, here are the actual formulas used to compute
Calc's financial functions.

Calc will not evaluate a financial function unless the @var{rate} or
@var{n} argument is known.  However, @var{payment} or @var{amount} can
be a variable.  Calc expands these functions according to the
formulas below for symbolic arguments only when you use the @kbd{a "}
(@code{calc-expand-formula}) command, or when taking derivatives or
integrals or solving equations involving the functions.

@ifinfo
These formulas are shown using the conventions of Big display
mode (@kbd{d B}); for example, the formula for @code{fv} written
linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.

@example
                                        n
                              (1 + rate)  - 1
fv(rate, n, pmt) =      pmt * ---------------
                                   rate

                                         n
                              ((1 + rate)  - 1) (1 + rate)
fvb(rate, n, pmt) =     pmt * ----------------------------
                                         rate

                                        n
fvl(rate, n, pmt) =     pmt * (1 + rate)

                                            -n
                              1 - (1 + rate)
pv(rate, n, pmt) =      pmt * ----------------
                                    rate

                                             -n
                              (1 - (1 + rate)  ) (1 + rate)
pvb(rate, n, pmt) =     pmt * -----------------------------
                                         rate

                                        -n
pvl(rate, n, pmt) =     pmt * (1 + rate)

                                    -1               -2               -3
npv(rate, [a, b, c]) =  a*(1 + rate)   + b*(1 + rate)   + c*(1 + rate)

                                        -1               -2
npvb(rate, [a, b, c]) = a + b*(1 + rate)   + c*(1 + rate)

                                             -n
                        (amt - x * (1 + rate)  ) * rate
pmt(rate, n, amt, x) =  -------------------------------
                                             -n
                               1 - (1 + rate)

                                             -n
                        (amt - x * (1 + rate)  ) * rate
pmtb(rate, n, amt, x) = -------------------------------
                                        -n
                         (1 - (1 + rate)  ) (1 + rate)

                                   amt * rate
nper(rate, pmt, amt) =  - log(1 - ------------, 1 + rate)
                                      pmt

                                    amt * rate
nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
                                  pmt * (1 + rate)

                              amt
nperl(rate, pmt, amt) = - log(---, 1 + rate)
                              pmt

                           1/n
                        pmt
ratel(n, pmt, amt) =    ------ - 1
                           1/n
                        amt

                        cost - salv
sln(cost, salv, life) = -----------
                           life

                             (cost - salv) * (life - per + 1)
syd(cost, salv, life, per) = --------------------------------
                                  life * (life + 1) / 2

                             book * 2
ddb(cost, salv, life, per) = --------,  book = cost - depreciation so far
                               life
@end example
@end ifinfo
@tex
\turnoffactive
$$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
$$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
$$ \code{fvl}(r, n, p) = p (1 + r)^n $$
$$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
$$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
$$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
$$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
$$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
$$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
$$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
                               (1 - (1 + r)^{-n}) (1 + r) } $$
$$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
$$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
$$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
$$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
$$ \code{sln}(c, s, l) = { c - s \over l } $$
$$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
$$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
@end tex

@noindent
In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.

These functions accept any numeric objects, including error forms,
intervals, and even (though not very usefully) complex numbers.  The
above formulas specify exactly the behavior of these functions with
all sorts of inputs.

Note that if the first argument to the @code{log} in @code{nper} is
negative, @code{nper} leaves itself in symbolic form rather than
returning a (financially meaningless) complex number.

@samp{rate(num, pmt, amt)} solves the equation
@samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
(@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
for an initial guess.  The @code{rateb} function is the same except
that it uses @code{pvb}.  Note that @code{ratel} can be solved
directly; its formula is shown in the above list.

Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
for @samp{rate}.

If you give a fourth argument to @code{nper} or @code{nperb}, Calc
will also use @kbd{H a R} to solve the equation using an initial
guess interval of @samp{[0 .. 100]}.

A fourth argument to @code{fv} simply sums the two components
calculated from the above formulas for @code{fv} and @code{fvl}.
The same is true of @code{fvb}, @code{pv}, and @code{pvb}.

The @kbd{ddb} function is computed iteratively; the ``book'' value
starts out equal to @var{cost}, and decreases according to the above
formula for the specified number of periods.  If the book value
would decrease below @var{salvage}, it only decreases to @var{salvage}
and the depreciation is zero for all subsequent periods.  The @code{ddb}
function returns the amount the book value decreased in the specified
period.

@node Binary Functions,  , Financial Functions, Arithmetic
@section Binary Number Functions

@noindent
The commands in this chapter all use two-letter sequences beginning with
the @kbd{b} prefix.

@cindex Binary numbers
The ``binary'' operations actually work regardless of the currently
displayed radix, although their results make the most sense in a radix
like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @w{@kbd{d 6}}
commands, respectively).  You may also wish to enable display of leading
zeros with @kbd{d z}.  @xref{Radix Modes}.

@cindex Word size for binary operations
The Calculator maintains a current @dfn{word size} @expr{w}, an
arbitrary positive or negative integer.  For a positive word size, all
of the binary operations described here operate modulo @expr{2^w}.  In
particular, negative arguments are converted to positive integers modulo
@expr{2^w} by all binary functions.

If the word size is negative, binary operations produce 2's complement
integers from 
@texline @math{-2^{-w-1}}
@infoline @expr{-(2^(-w-1))} 
to 
@texline @math{2^{-w-1}-1}
@infoline @expr{2^(-w-1)-1} 
inclusive.  Either mode accepts inputs in any range; the sign of
@expr{w} affects only the results produced.

@kindex b c
@pindex calc-clip
@tindex clip
The @kbd{b c} (@code{calc-clip})
[@code{clip}] command can be used to clip a number by reducing it modulo
@expr{2^w}.  The commands described in this chapter automatically clip
their results to the current word size.  Note that other operations like
addition do not use the current word size, since integer addition
generally is not ``binary.''  (However, @pxref{Simplification Modes},
@code{calc-bin-simplify-mode}.)  For example, with a word size of 8
bits @kbd{b c} converts a number to the range 0 to 255; with a word
size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.

@kindex b w
@pindex calc-word-size
The default word size is 32 bits.  All operations except the shifts and
rotates allow you to specify a different word size for that one
operation by giving a numeric prefix argument:  @kbd{C-u 8 b c} clips the
top of stack to the range 0 to 255 regardless of the current word size.
To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
This command displays a prompt with the current word size; press @key{RET}
immediately to keep this word size, or type a new word size at the prompt.

When the binary operations are written in symbolic form, they take an
optional second (or third) word-size parameter.  When a formula like
@samp{and(a,b)} is finally evaluated, the word size current at that time
will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
@mathit{-8} will always be used.  A symbolic binary function will be left
in symbolic form unless the all of its argument(s) are integers or
integer-valued floats.

If either or both arguments are modulo forms for which @expr{M} is a
power of two, that power of two is taken as the word size unless a
numeric prefix argument overrides it.  The current word size is never
consulted when modulo-power-of-two forms are involved.

@kindex b a
@pindex calc-and
@tindex and
The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise
AND of the two numbers on the top of the stack.  In other words, for each
of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
bit of the result is 1 if and only if both input bits are 1:
@samp{and(2#1100, 2#1010) = 2#1000}.

@kindex b o
@pindex calc-or
@tindex or
The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise
inclusive OR of two numbers.  A bit is 1 if either of the input bits, or
both, are 1:  @samp{or(2#1100, 2#1010) = 2#1110}.

@kindex b x
@pindex calc-xor
@tindex xor
The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise
exclusive OR of two numbers.  A bit is 1 if exactly one of the input bits
is 1:  @samp{xor(2#1100, 2#1010) = 2#0110}.

@kindex b d
@pindex calc-diff
@tindex diff
The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise
difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
so that @samp{diff(2#1100, 2#1010) = 2#0100}.

@kindex b n
@pindex calc-not
@tindex not
The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise
NOT of a number.  A bit is 1 if the input bit is 0 and vice-versa.

@kindex b l
@pindex calc-lshift-binary
@tindex lsh
The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a
number left by one bit, or by the number of bits specified in the numeric
prefix argument.  A negative prefix argument performs a logical right shift,
in which zeros are shifted in on the left.  In symbolic form, @samp{lsh(a)}
is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
Bits shifted ``off the end,'' according to the current word size, are lost.

@kindex H b l
@kindex H b r
@ignore
@mindex @idots
@end ignore
@kindex H b L
@ignore
@mindex @null
@end ignore
@kindex H b R
@ignore
@mindex @null
@end ignore
@kindex H b t
The @kbd{H b l} command also does a left shift, but it takes two arguments
from the stack (the value to shift, and, at top-of-stack, the number of
bits to shift).  This version interprets the prefix argument just like
the regular binary operations, i.e., as a word size.  The Hyperbolic flag
has a similar effect on the rest of the binary shift and rotate commands.

@kindex b r
@pindex calc-rshift-binary
@tindex rsh
The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a
number right by one bit, or by the number of bits specified in the numeric
prefix argument:  @samp{rsh(a,n) = lsh(a,-n)}.

@kindex b L
@pindex calc-lshift-arith
@tindex ash
The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a
number left.  It is analogous to @code{lsh}, except that if the shift
is rightward (the prefix argument is negative), an arithmetic shift
is performed as described below.

@kindex b R
@pindex calc-rshift-arith
@tindex rash
The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs
an ``arithmetic'' shift to the right, in which the leftmost bit (according
to the current word size) is duplicated rather than shifting in zeros.
This corresponds to dividing by a power of two where the input is interpreted
as a signed, twos-complement number.  (The distinction between the @samp{rsh}
and @samp{rash} operations is totally independent from whether the word
size is positive or negative.)  With a negative prefix argument, this
performs a standard left shift.

@kindex b t
@pindex calc-rotate-binary
@tindex rot
The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a
number one bit to the left.  The leftmost bit (according to the current
word size) is dropped off the left and shifted in on the right.  With a
numeric prefix argument, the number is rotated that many bits to the left
or right.

@xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that
pack and unpack binary integers into sets.  (For example, @kbd{b u}
unpacks the number @samp{2#11001} to the set of bit-numbers
@samp{[0, 3, 4]}.)  Type @kbd{b u V #} to count the number of ``1''
bits in a binary integer.

Another interesting use of the set representation of binary integers
is to reverse the bits in, say, a 32-bit integer.  Type @kbd{b u} to
unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
with 31 minus that bit-number; type @kbd{b p} to pack the set back
into a binary integer.

@node Scientific Functions, Matrix Functions, Arithmetic, Top
@chapter Scientific Functions

@noindent
The functions described here perform trigonometric and other transcendental
calculations.  They generally produce floating-point answers correct to the
full current precision.  The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
flag keys must be used to get some of these functions from the keyboard.

@kindex P
@pindex calc-pi
@cindex @code{pi} variable
@vindex pi
@kindex H P
@cindex @code{e} variable
@vindex e
@kindex I P
@cindex @code{gamma} variable
@vindex gamma
@cindex Gamma constant, Euler's
@cindex Euler's gamma constant
@kindex H I P
@cindex @code{phi} variable
@cindex Phi, golden ratio
@cindex Golden ratio
One miscellaneous command is shift-@kbd{P} (@code{calc-pi}), which pushes
the value of @cpi{} (at the current precision) onto the stack.  With the
Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
With the Inverse flag, it pushes Euler's constant 
@texline @math{\gamma}
@infoline @expr{gamma} 
(about 0.5772).  With both Inverse and Hyperbolic, it
pushes the ``golden ratio'' 
@texline @math{\phi}
@infoline @expr{phi} 
(about 1.618).  (At present, Euler's constant is not available
to unlimited precision; Calc knows only the first 100 digits.)
In Symbolic mode, these commands push the
actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
respectively, instead of their values; @pxref{Symbolic Mode}.

@ignore
@mindex Q
@end ignore
@ignore
@mindex I Q
@end ignore
@kindex I Q
@tindex sqr
The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere;
@pxref{Basic Arithmetic}.  With the Inverse flag [@code{sqr}], this command
computes the square of the argument.

@xref{Prefix Arguments}, for a discussion of the effect of numeric
prefix arguments on commands in this chapter which do not otherwise
interpret a prefix argument.

@menu
* Logarithmic Functions::
* Trigonometric and Hyperbolic Functions::
* Advanced Math Functions::
* Branch Cuts::
* Random Numbers::
* Combinatorial Functions::
* Probability Distribution Functions::
@end menu

@node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions
@section Logarithmic Functions

@noindent
@kindex L
@pindex calc-ln
@tindex ln
@ignore
@mindex @null
@end ignore
@kindex I E
The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural
logarithm of the real or complex number on the top of the stack.  With
the Inverse flag it computes the exponential function instead, although
this is redundant with the @kbd{E} command.

@kindex E
@pindex calc-exp
@tindex exp
@ignore
@mindex @null
@end ignore
@kindex I L
The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the
exponential, i.e., @expr{e} raised to the power of the number on the stack.
The meanings of the Inverse and Hyperbolic flags follow from those for
the @code{calc-ln} command.

@kindex H L
@kindex H E
@pindex calc-log10
@tindex log10
@tindex exp10
@ignore
@mindex @null
@end ignore
@kindex H I L
@ignore
@mindex @null
@end ignore
@kindex H I E
The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common
(base-10) logarithm of a number.  (With the Inverse flag [@code{exp10}],
it raises ten to a given power.)  Note that the common logarithm of a
complex number is computed by taking the natural logarithm and dividing
by 
@texline @math{\ln10}.
@infoline @expr{ln(10)}.

@kindex B
@kindex I B
@pindex calc-log
@tindex log
@tindex alog
The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm
to any base.  For example, @kbd{1024 @key{RET} 2 B} produces 10, since
@texline @math{2^{10} = 1024}.
@infoline @expr{2^10 = 1024}.  
In certain cases like @samp{log(3,9)}, the result
will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
mode setting.  With the Inverse flag [@code{alog}], this command is
similar to @kbd{^} except that the order of the arguments is reversed.

@kindex f I
@pindex calc-ilog
@tindex ilog
The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the
integer logarithm of a number to any base.  The number and the base must
themselves be positive integers.  This is the true logarithm, rounded
down to an integer.  Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
range from 1000 to 9999.  If both arguments are positive integers, exact
integer arithmetic is used; otherwise, this is equivalent to
@samp{floor(log(x,b))}.

@kindex f E
@pindex calc-expm1
@tindex expm1
The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes
@texline @math{e^x - 1},
@infoline @expr{exp(x)-1}, 
but using an algorithm that produces a more accurate
answer when the result is close to zero, i.e., when 
@texline @math{e^x}
@infoline @expr{exp(x)} 
is close to one.

@kindex f L
@pindex calc-lnp1
@tindex lnp1
The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes
@texline @math{\ln(x+1)},
@infoline @expr{ln(x+1)}, 
producing a more accurate answer when @expr{x} is close to zero.

@node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions
@section Trigonometric/Hyperbolic Functions

@noindent
@kindex S
@pindex calc-sin
@tindex sin
The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine
of an angle or complex number.  If the input is an HMS form, it is interpreted
as degrees-minutes-seconds; otherwise, the input is interpreted according
to the current angular mode.  It is best to use Radians mode when operating
on complex numbers.

Calc's ``units'' mechanism includes angular units like @code{deg},
@code{rad}, and @code{grad}.  While @samp{sin(45 deg)} is not evaluated
all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
of the current angular mode.  @xref{Basic Operations on Units}.

Also, the symbolic variable @code{pi} is not ordinarily recognized in
arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
the @kbd{a s} (@code{calc-simplify}) command recognizes many such
formulas when the current angular mode is Radians @emph{and} Symbolic
mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
@xref{Symbolic Mode}.  Beware, this simplification occurs even if you
have stored a different value in the variable @samp{pi}; this is one
reason why changing built-in variables is a bad idea.  Arguments of
the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
Calc includes similar formulas for @code{cos} and @code{tan}.

The @kbd{a s} command knows all angles which are integer multiples of
@cpiover{12}, @cpiover{10}, or @cpiover{8} radians.  In Degrees mode,
analogous simplifications occur for integer multiples of 15 or 18
degrees, and for arguments plus multiples of 90 degrees.

@kindex I S
@pindex calc-arcsin
@tindex arcsin
With the Inverse flag, @code{calc-sin} computes an arcsine.  This is also
available as the @code{calc-arcsin} command or @code{arcsin} algebraic
function.  The returned argument is converted to degrees, radians, or HMS
notation depending on the current angular mode.

@kindex H S
@pindex calc-sinh
@tindex sinh
@kindex H I S
@pindex calc-arcsinh
@tindex arcsinh
With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
sine, also available as @code{calc-sinh} [@code{sinh}].  With the
Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
(@code{calc-arcsinh}) [@code{arcsinh}].

@kindex C
@pindex calc-cos
@tindex cos
@ignore
@mindex @idots
@end ignore
@kindex I C
@pindex calc-arccos
@ignore
@mindex @null
@end ignore
@tindex arccos
@ignore
@mindex @null
@end ignore
@kindex H C
@pindex calc-cosh
@ignore
@mindex @null
@end ignore
@tindex cosh
@ignore
@mindex @null
@end ignore
@kindex H I C
@pindex calc-arccosh
@ignore
@mindex @null
@end ignore
@tindex arccosh
@ignore
@mindex @null
@end ignore
@kindex T
@pindex calc-tan
@ignore
@mindex @null
@end ignore
@tindex tan
@ignore
@mindex @null
@end ignore
@kindex I T
@pindex calc-arctan
@ignore
@mindex @null
@end ignore
@tindex arctan
@ignore
@mindex @null
@end ignore
@kindex H T
@pindex calc-tanh
@ignore
@mindex @null
@end ignore
@tindex tanh
@ignore
@mindex @null
@end ignore
@kindex H I T
@pindex calc-arctanh
@ignore
@mindex @null
@end ignore
@tindex arctanh
The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine
of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}]
computes the tangent, along with all the various inverse and hyperbolic
variants of these functions.

@kindex f T
@pindex calc-arctan2
@tindex arctan2
The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two
numbers from the stack and computes the arc tangent of their ratio.  The
result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
(inclusive) degrees, or the analogous range in radians.  A similar
result would be obtained with @kbd{/} followed by @kbd{I T}, but the
value would only be in the range from @mathit{-90} to @mathit{+90} degrees
since the division loses information about the signs of the two
components, and an error might result from an explicit division by zero
which @code{arctan2} would avoid.  By (arbitrary) definition,
@samp{arctan2(0,0)=0}.

@pindex calc-sincos
@ignore
@starindex
@end ignore
@tindex sincos
@ignore
@starindex
@end ignore
@ignore
@mindex arc@idots
@end ignore
@tindex arcsincos
The @code{calc-sincos} [@code{sincos}] command computes the sine and
cosine of a number, returning them as a vector of the form
@samp{[@var{cos}, @var{sin}]}.
With the Inverse flag [@code{arcsincos}], this command takes a two-element
vector as an argument and computes @code{arctan2} of the elements.
(This command does not accept the Hyperbolic flag.)

@pindex calc-sec
@tindex sec
@pindex calc-csc
@tindex csc
@pindex calc-cot
@tindex cot
@pindex calc-sech
@tindex sech
@pindex calc-csch
@tindex csch
@pindex calc-coth
@tindex coth
The remaining trigonometric functions, @code{calc-sec} [@code{sec}],
@code{calc-csc} [@code{csc}] and @code{calc-sec} [@code{sec}], are also
available.  With the Hyperbolic flag, these compute their hyperbolic
counterparts, which are also available separately as @code{calc-sech}
[@code{sech}], @code{calc-csch} [@code{csch}] and @code{calc-sech}
[@code{sech}].  (These commmands do not accept the Inverse flag.)

@node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions
@section Advanced Mathematical Functions

@noindent
Calc can compute a variety of less common functions that arise in
various branches of mathematics.  All of the functions described in
this section allow arbitrary complex arguments and, except as noted,
will work to arbitrarily large precisions.  They can not at present
handle error forms or intervals as arguments.

NOTE:  These functions are still experimental.  In particular, their
accuracy is not guaranteed in all domains.  It is advisable to set the
current precision comfortably higher than you actually need when
using these functions.  Also, these functions may be impractically
slow for some values of the arguments.

@kindex f g
@pindex calc-gamma
@tindex gamma
The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler
gamma function.  For positive integer arguments, this is related to the
factorial function:  @samp{gamma(n+1) = fact(n)}.  For general complex
arguments the gamma function can be defined by the following definite
integral:  
@texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
@infoline @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.  
(The actual implementation uses far more efficient computational methods.)

@kindex f G
@tindex gammaP
@ignore
@mindex @idots
@end ignore
@kindex I f G
@ignore
@mindex @null
@end ignore
@kindex H f G
@ignore
@mindex @null
@end ignore
@kindex H I f G
@pindex calc-inc-gamma
@ignore
@mindex @null
@end ignore
@tindex gammaQ
@ignore
@mindex @null
@end ignore
@tindex gammag
@ignore
@mindex @null
@end ignore
@tindex gammaG
The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes
the incomplete gamma function, denoted @samp{P(a,x)}.  This is defined by
the integral, 
@texline @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)}.
@infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
definition of the normal gamma function).

Several other varieties of incomplete gamma function are defined.
The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command.
You can think of this as taking the other half of the integral, from
@expr{x} to infinity.

@ifinfo
The functions corresponding to the integrals that define @expr{P(a,x)}
and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
(where @expr{g} and @expr{G} represent the lower- and upper-case Greek
letter gamma).  You can obtain these using the @kbd{H f G} [@code{gammag}]
and @kbd{H I f G} [@code{gammaG}] commands.
@end ifinfo
@tex
\turnoffactive
The functions corresponding to the integrals that define $P(a,x)$
and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
You can obtain these using the \kbd{H f G} [\code{gammag}] and
\kbd{I H f G} [\code{gammaG}] commands.
@end tex

@kindex f b
@pindex calc-beta
@tindex beta
The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the
Euler beta function, which is defined in terms of the gamma function as
@texline @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
@infoline @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)}, 
or by
@texline @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
@infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.

@kindex f B
@kindex H f B
@pindex calc-inc-beta
@tindex betaI
@tindex betaB
The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes
the incomplete beta function @expr{I(x,a,b)}.  It is defined by
@texline @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)}.
@infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}.
Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
un-normalized version [@code{betaB}].

@kindex f e
@kindex I f e
@pindex calc-erf
@tindex erf
@tindex erfc
The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the
error function 
@texline @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
@infoline @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}]
is the corresponding integral from @samp{x} to infinity; the sum
@texline @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
@infoline @expr{erf(x) + erfc(x) = 1}.

@kindex f j
@kindex f y
@pindex calc-bessel-J
@pindex calc-bessel-Y
@tindex besJ
@tindex besY
The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y}
(@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel
functions of the first and second kinds, respectively.
In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
@expr{n} is often an integer, but is not required to be one.
Calc's implementation of the Bessel functions currently limits the
precision to 8 digits, and may not be exact even to that precision.
Use with care!

@node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions
@section Branch Cuts and Principal Values

@noindent
@cindex Branch cuts
@cindex Principal values
All of the logarithmic, trigonometric, and other scientific functions are
defined for complex numbers as well as for reals.
This section describes the values
returned in cases where the general result is a family of possible values.
Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
second edition, in these matters.  This section will describe each
function briefly; for a more detailed discussion (including some nifty
diagrams), consult Steele's book.

Note that the branch cuts for @code{arctan} and @code{arctanh} were
changed between the first and second editions of Steele.  Versions of
Calc starting with 2.00 follow the second edition.

The new branch cuts exactly match those of the HP-28/48 calculators.
They also match those of Mathematica 1.2, except that Mathematica's
@code{arctan} cut is always in the right half of the complex plane,
and its @code{arctanh} cut is always in the top half of the plane.
Calc's cuts are continuous with quadrants I and III for @code{arctan},
or II and IV for @code{arctanh}.

Note:  The current implementations of these functions with complex arguments
are designed with proper behavior around the branch cuts in mind, @emph{not}
efficiency or accuracy.  You may need to increase the floating precision
and wait a while to get suitable answers from them.

For @samp{sqrt(a+bi)}:  When @expr{a<0} and @expr{b} is small but positive
or zero, the result is close to the @expr{+i} axis.  For @expr{b} small and
negative, the result is close to the @expr{-i} axis.  The result always lies
in the right half of the complex plane.

For @samp{ln(a+bi)}:  The real part is defined as @samp{ln(abs(a+bi))}.
The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
negative real axis.

The following table describes these branch cuts in another way.
If the real and imaginary parts of @expr{z} are as shown, then
the real and imaginary parts of @expr{f(z)} will be as shown.
Here @code{eps} stands for a small positive value; each
occurrence of @code{eps} may stand for a different small value.

@smallexample
     z           sqrt(z)       ln(z)
----------------------------------------
   +,   0         +,  0       any, 0
   -,   0         0,  +       any, pi
   -, +eps      +eps, +      +eps, +
   -, -eps      +eps, -      +eps, -
@end smallexample

For @samp{z1^z2}:  This is defined by @samp{exp(ln(z1)*z2)}.
One interesting consequence of this is that @samp{(-8)^1:3} does
not evaluate to @mathit{-2} as you might expect, but to the complex
number @expr{(1., 1.732)}.  Both of these are valid cube roots
of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
less-obvious root for the sake of mathematical consistency.

For @samp{arcsin(z)}:  This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.

For @samp{arccos(z)}:  This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
or equivalently by @samp{pi/2 - arcsin(z)}.  The branch cuts are on
the real axis, less than @mathit{-1} and greater than 1.

For @samp{arctan(z)}:  This is defined by
@samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}.  The branch cuts are on the
imaginary axis, below @expr{-i} and above @expr{i}.

For @samp{arcsinh(z)}:  This is defined by @samp{ln(z + sqrt(1+z^2))}.
The branch cuts are on the imaginary axis, below @expr{-i} and
above @expr{i}.

For @samp{arccosh(z)}:  This is defined by
@samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}.  The branch cut is on the
real axis less than 1.

For @samp{arctanh(z)}:  This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.

The following tables for @code{arcsin}, @code{arccos}, and
@code{arctan} assume the current angular mode is Radians.  The
hyperbolic functions operate independently of the angular mode.

@smallexample
       z             arcsin(z)            arccos(z)
-------------------------------------------------------
 (-1..1),  0      (-pi/2..pi/2), 0       (0..pi), 0
 (-1..1), +eps    (-pi/2..pi/2), +eps    (0..pi), -eps
 (-1..1), -eps    (-pi/2..pi/2), -eps    (0..pi), +eps
   <-1,    0          -pi/2,     +         pi,    -
   <-1,  +eps      -pi/2 + eps,  +      pi - eps, -
   <-1,  -eps      -pi/2 + eps,  -      pi - eps, +
    >1,    0           pi/2,     -          0,    +
    >1,  +eps       pi/2 - eps,  +        +eps,   -
    >1,  -eps       pi/2 - eps,  -        +eps,   +
@end smallexample

@smallexample
       z            arccosh(z)         arctanh(z)
-----------------------------------------------------
 (-1..1),  0        0,  (0..pi)       any,     0
 (-1..1), +eps    +eps, (0..pi)       any,    +eps
 (-1..1), -eps    +eps, (-pi..0)      any,    -eps
   <-1,    0        +,    pi           -,     pi/2
   <-1,  +eps       +,  pi - eps       -,  pi/2 - eps
   <-1,  -eps       +, -pi + eps       -, -pi/2 + eps
    >1,    0        +,     0           +,    -pi/2
    >1,  +eps       +,   +eps          +,  pi/2 - eps
    >1,  -eps       +,   -eps          +, -pi/2 + eps
@end smallexample

@smallexample
       z           arcsinh(z)           arctan(z)
-----------------------------------------------------
   0, (-1..1)    0, (-pi/2..pi/2)         0,     any
   0,   <-1      -,    -pi/2            -pi/2,    -
 +eps,  <-1      +, -pi/2 + eps       pi/2 - eps, -
 -eps,  <-1      -, -pi/2 + eps      -pi/2 + eps, -
   0,    >1      +,     pi/2             pi/2,    +
 +eps,   >1      +,  pi/2 - eps       pi/2 - eps, +
 -eps,   >1      -,  pi/2 - eps      -pi/2 + eps, +
@end smallexample

Finally, the following identities help to illustrate the relationship
between the complex trigonometric and hyperbolic functions.  They
are valid everywhere, including on the branch cuts.

@smallexample
sin(i*z)  = i*sinh(z)       arcsin(i*z)  = i*arcsinh(z)
cos(i*z)  =   cosh(z)       arcsinh(i*z) = i*arcsin(z)
tan(i*z)  = i*tanh(z)       arctan(i*z)  = i*arctanh(z)
sinh(i*z) = i*sin(z)        cosh(i*z)    =   cos(z)
@end smallexample

The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
for general complex arguments, but their branch cuts and principal values
are not rigorously specified at present.

@node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions
@section Random Numbers

@noindent
@kindex k r
@pindex calc-random
@tindex random
The @kbd{k r} (@code{calc-random}) [@code{random}] command produces
random numbers of various sorts.

Given a positive numeric prefix argument @expr{M}, it produces a random
integer @expr{N} in the range 
@texline @math{0 \le N < M}.
@infoline @expr{0 <= N < M}.  
Each of the @expr{M} values appears with equal probability.

With no numeric prefix argument, the @kbd{k r} command takes its argument
from the stack instead.  Once again, if this is a positive integer @expr{M}
the result is a random integer less than @expr{M}.  However, note that
while numeric prefix arguments are limited to six digits or so, an @expr{M}
taken from the stack can be arbitrarily large.  If @expr{M} is negative,
the result is a random integer in the range 
@texline @math{M < N \le 0}.
@infoline @expr{M < N <= 0}.

If the value on the stack is a floating-point number @expr{M}, the result
is a random floating-point number @expr{N} in the range 
@texline @math{0 \le N < M}
@infoline @expr{0 <= N < M}
or 
@texline @math{M < N \le 0},
@infoline @expr{M < N <= 0}, 
according to the sign of @expr{M}.

If @expr{M} is zero, the result is a Gaussian-distributed random real
number; the distribution has a mean of zero and a standard deviation
of one.  The algorithm used generates random numbers in pairs; thus,
every other call to this function will be especially fast.

If @expr{M} is an error form 
@texline @math{m} @code{+/-} @math{\sigma}
@infoline @samp{m +/- s} 
where @var{m} and 
@texline @math{\sigma}
@infoline @var{s} 
are both real numbers, the result uses a Gaussian distribution with mean
@var{m} and standard deviation 
@texline @math{\sigma}.
@infoline @var{s}.

If @expr{M} is an interval form, the lower and upper bounds specify the
acceptable limits of the random numbers.  If both bounds are integers,
the result is a random integer in the specified range.  If either bound
is floating-point, the result is a random real number in the specified
range.  If the interval is open at either end, the result will be sure
not to equal that end value.  (This makes a big difference for integer
intervals, but for floating-point intervals it's relatively minor:
with a precision of 6, @samp{random([1.0..2.0))} will return any of one
million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
additionally return 2.00000, but the probability of this happening is
extremely small.)

If @expr{M} is a vector, the result is one element taken at random from
the vector.  All elements of the vector are given equal probabilities.

@vindex RandSeed
The sequence of numbers produced by @kbd{k r} is completely random by
default, i.e., the sequence is seeded each time you start Calc using
the current time and other information.  You can get a reproducible
sequence by storing a particular ``seed value'' in the Calc variable
@code{RandSeed}.  Any integer will do for a seed; integers of from 1
to 12 digits are good.  If you later store a different integer into
@code{RandSeed}, Calc will switch to a different pseudo-random
sequence.  If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
from the current time.  If you store the same integer that you used
before back into @code{RandSeed}, you will get the exact same sequence
of random numbers as before.

@pindex calc-rrandom
The @code{calc-rrandom} command (not on any key) produces a random real
number between zero and one.  It is equivalent to @samp{random(1.0)}.

@kindex k a
@pindex calc-random-again
The @kbd{k a} (@code{calc-random-again}) command produces another random
number, re-using the most recent value of @expr{M}.  With a numeric
prefix argument @var{n}, it produces @var{n} more random numbers using
that value of @expr{M}.

@kindex k h
@pindex calc-shuffle
@tindex shuffle
The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
random values with no duplicates.  The value on the top of the stack
specifies the set from which the random values are drawn, and may be any
of the @expr{M} formats described above.  The numeric prefix argument
gives the length of the desired list.  (If you do not provide a numeric
prefix argument, the length of the list is taken from the top of the
stack, and @expr{M} from second-to-top.)

If @expr{M} is a floating-point number, zero, or an error form (so
that the random values are being drawn from the set of real numbers)
there is little practical difference between using @kbd{k h} and using
@kbd{k r} several times.  But if the set of possible values consists
of just a few integers, or the elements of a vector, then there is
a very real chance that multiple @kbd{k r}'s will produce the same
number more than once.  The @kbd{k h} command produces a vector whose
elements are always distinct.  (Actually, there is a slight exception:
If @expr{M} is a vector, no given vector element will be drawn more
than once, but if several elements of @expr{M} are equal, they may
each make it into the result vector.)

One use of @kbd{k h} is to rearrange a list at random.  This happens
if the prefix argument is equal to the number of values in the list:
@kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
@samp{[2.5, 1, 1.5, 3, 2]}.  As a convenient feature, if the argument
@var{n} is negative it is replaced by the size of the set represented
by @expr{M}.  Naturally, this is allowed only when @expr{M} specifies
a small discrete set of possibilities.

To do the equivalent of @kbd{k h} but with duplications allowed,
given @expr{M} on the stack and with @var{n} just entered as a numeric
prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
@kbd{V M k r} to ``map'' the normal @kbd{k r} function over the
elements of this vector.  @xref{Matrix Functions}.

@menu
* Random Number Generator::     (Complete description of Calc's algorithm)
@end menu

@node Random Number Generator,  , Random Numbers, Random Numbers
@subsection Random Number Generator

Calc's random number generator uses several methods to ensure that
the numbers it produces are highly random.  Knuth's @emph{Art of
Computer Programming}, Volume II, contains a thorough description
of the theory of random number generators and their measurement and
characterization.

If @code{RandSeed} has no stored value, Calc calls Emacs' built-in
@code{random} function to get a stream of random numbers, which it
then treats in various ways to avoid problems inherent in the simple
random number generators that many systems use to implement @code{random}.

When Calc's random number generator is first invoked, it ``seeds''
the low-level random sequence using the time of day, so that the
random number sequence will be different every time you use Calc.

Since Emacs Lisp doesn't specify the range of values that will be
returned by its @code{random} function, Calc exercises the function
several times to estimate the range.  When Calc subsequently uses
the @code{random} function, it takes only 10 bits of the result
near the most-significant end.  (It avoids at least the bottom
four bits, preferably more, and also tries to avoid the top two
bits.)  This strategy works well with the linear congruential
generators that are typically used to implement @code{random}.

If @code{RandSeed} contains an integer, Calc uses this integer to
seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
computing 
@texline @math{X_{n-55} - X_{n-24}}.
@infoline @expr{X_n-55 - X_n-24}).  
This method expands the seed
value into a large table which is maintained internally; the variable
@code{RandSeed} is changed from, e.g., 42 to the vector @expr{[42]}
to indicate that the seed has been absorbed into this table.  When
@code{RandSeed} contains a vector, @kbd{k r} and related commands
continue to use the same internal table as last time.  There is no
way to extract the complete state of the random number generator
so that you can restart it from any point; you can only restart it
from the same initial seed value.  A simple way to restart from the
same seed is to type @kbd{s r RandSeed} to get the seed vector,
@kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed}
to reseed the generator with that number.

Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
of Knuth.  It fills a table with 13 random 10-bit numbers.  Then,
to generate a new random number, it uses the previous number to
index into the table, picks the value it finds there as the new
random number, then replaces that table entry with a new value
obtained from a call to the base random number generator (either
the additive congruential generator or the @code{random} function
supplied by the system).  If there are any flaws in the base
generator, shuffling will tend to even them out.  But if the system
provides an excellent @code{random} function, shuffling will not
damage its randomness.

To create a random integer of a certain number of digits, Calc
builds the integer three decimal digits at a time.  For each group
of three digits, Calc calls its 10-bit shuffling random number generator
(which returns a value from 0 to 1023); if the random value is 1000
or more, Calc throws it out and tries again until it gets a suitable
value.

To create a random floating-point number with precision @var{p}, Calc
simply creates a random @var{p}-digit integer and multiplies by
@texline @math{10^{-p}}.
@infoline @expr{10^-p}.  
The resulting random numbers should be very clean, but note
that relatively small numbers will have few significant random digits.
In other words, with a precision of 12, you will occasionally get
numbers on the order of 
@texline @math{10^{-9}}
@infoline @expr{10^-9} 
or 
@texline @math{10^{-10}},
@infoline @expr{10^-10}, 
but those numbers will only have two or three random digits since they
correspond to small integers times 
@texline @math{10^{-12}}.
@infoline @expr{10^-12}.

To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
counts the digits in @var{m}, creates a random integer with three
additional digits, then reduces modulo @var{m}.  Unless @var{m} is a
power of ten the resulting values will be very slightly biased toward
the lower numbers, but this bias will be less than 0.1%.  (For example,
if @var{m} is 42, Calc will reduce a random integer less than 100000
modulo 42 to get a result less than 42.  It is easy to show that the
numbers 40 and 41 will be only 2380/2381 as likely to result from this
modulo operation as numbers 39 and below.)  If @var{m} is a power of
ten, however, the numbers should be completely unbiased.

The Gaussian random numbers generated by @samp{random(0.0)} use the
``polar'' method described in Knuth section 3.4.1C.  This method
generates a pair of Gaussian random numbers at a time, so only every
other call to @samp{random(0.0)} will require significant calculations.

@node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions
@section Combinatorial Functions

@noindent
Commands relating to combinatorics and number theory begin with the
@kbd{k} key prefix.

@kindex k g
@pindex calc-gcd
@tindex gcd
The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the
Greatest Common Divisor of two integers.  It also accepts fractions;
the GCD of two fractions is defined by taking the GCD of the
numerators, and the LCM of the denominators.  This definition is
consistent with the idea that @samp{a / gcd(a,x)} should yield an
integer for any @samp{a} and @samp{x}.  For other types of arguments,
the operation is left in symbolic form.

@kindex k l
@pindex calc-lcm
@tindex lcm
The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the
Least Common Multiple of two integers or fractions.  The product of
the LCM and GCD of two numbers is equal to the product of the
numbers.

@kindex k E
@pindex calc-extended-gcd
@tindex egcd
The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes
the GCD of two integers @expr{x} and @expr{y} and returns a vector
@expr{[g, a, b]} where 
@texline @math{g = \gcd(x,y) = a x + b y}.
@infoline @expr{g = gcd(x,y) = a x + b y}.

@kindex !
@pindex calc-factorial
@tindex fact
@ignore
@mindex @null
@end ignore
@tindex !
The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the
factorial of the number at the top of the stack.  If the number is an
integer, the result is an exact integer.  If the number is an
integer-valued float, the result is a floating-point approximation.  If
the number is a non-integral real number, the generalized factorial is used,
as defined by the Euler Gamma function.  Please note that computation of
large factorials can be slow; using floating-point format will help
since fewer digits must be maintained.  The same is true of many of
the commands in this section.

@kindex k d
@pindex calc-double-factorial
@tindex dfact
@ignore
@mindex @null
@end ignore
@tindex !!
The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command
computes the ``double factorial'' of an integer.  For an even integer,
this is the product of even integers from 2 to @expr{N}.  For an odd
integer, this is the product of odd integers from 3 to @expr{N}.  If
the argument is an integer-valued float, the result is a floating-point
approximation.  This function is undefined for negative even integers.
The notation @expr{N!!} is also recognized for double factorials.

@kindex k c
@pindex calc-choose
@tindex choose
The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the
binomial coefficient @expr{N}-choose-@expr{M}, where @expr{M} is the number
on the top of the stack and @expr{N} is second-to-top.  If both arguments
are integers, the result is an exact integer.  Otherwise, the result is a
floating-point approximation.  The binomial coefficient is defined for all
real numbers by
@texline @math{N! \over M! (N-M)!\,}.
@infoline @expr{N! / M! (N-M)!}.

@kindex H k c
@pindex calc-perm
@tindex perm
@ifinfo
The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the
number-of-permutations function @expr{N! / (N-M)!}.
@end ifinfo
@tex
The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
number-of-perm\-utations function $N! \over (N-M)!\,$.
@end tex

@kindex k b
@kindex H k b
@pindex calc-bernoulli-number
@tindex bern
The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command
computes a given Bernoulli number.  The value at the top of the stack
is a nonnegative integer @expr{n} that specifies which Bernoulli number
is desired.  The @kbd{H k b} command computes a Bernoulli polynomial,
taking @expr{n} from the second-to-top position and @expr{x} from the
top of the stack.  If @expr{x} is a variable or formula the result is
a polynomial in @expr{x}; if @expr{x} is a number the result is a number.

@kindex k e
@kindex H k e
@pindex calc-euler-number
@tindex euler
The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly
computes an Euler number, and @w{@kbd{H k e}} computes an Euler polynomial.
Bernoulli and Euler numbers occur in the Taylor expansions of several
functions.

@kindex k s
@kindex H k s
@pindex calc-stirling-number
@tindex stir1
@tindex stir2
The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command
computes a Stirling number of the first 
@texline kind@tie{}@math{n \brack m},
@infoline kind,
given two integers @expr{n} and @expr{m} on the stack.  The @kbd{H k s}
[@code{stir2}] command computes a Stirling number of the second 
@texline kind@tie{}@math{n \brace m}.
@infoline kind.
These are the number of @expr{m}-cycle permutations of @expr{n} objects,
and the number of ways to partition @expr{n} objects into @expr{m}
non-empty sets, respectively.

@kindex k p
@pindex calc-prime-test
@cindex Primes
The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
the top of the stack is prime.  For integers less than eight million, the
answer is always exact and reasonably fast.  For larger integers, a
probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
The number is first checked against small prime factors (up to 13).  Then,
any number of iterations of the algorithm are performed.  Each step either
discovers that the number is non-prime, or substantially increases the
certainty that the number is prime.  After a few steps, the chance that
a number was mistakenly described as prime will be less than one percent.
(Indeed, this is a worst-case estimate of the probability; in practice
even a single iteration is quite reliable.)  After the @kbd{k p} command,
the number will be reported as definitely prime or non-prime if possible,
or otherwise ``probably'' prime with a certain probability of error.

@ignore
@starindex
@end ignore
@tindex prime
The normal @kbd{k p} command performs one iteration of the primality
test.  Pressing @kbd{k p} repeatedly for the same integer will perform
additional iterations.  Also, @kbd{k p} with a numeric prefix performs
the specified number of iterations.  There is also an algebraic function
@samp{prime(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
is (probably) prime and 0 if not.

@kindex k f
@pindex calc-prime-factors
@tindex prfac
The @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command
attempts to decompose an integer into its prime factors.  For numbers up
to 25 million, the answer is exact although it may take some time.  The
result is a vector of the prime factors in increasing order.  For larger
inputs, prime factors above 5000 may not be found, in which case the
last number in the vector will be an unfactored integer greater than 25
million (with a warning message).  For negative integers, the first
element of the list will be @mathit{-1}.  For inputs @mathit{-1}, @mathit{0}, and
@mathit{1}, the result is a list of the same number.

@kindex k n
@pindex calc-next-prime
@ignore
@mindex nextpr@idots
@end ignore
@tindex nextprime
The @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds
the next prime above a given number.  Essentially, it searches by calling
@code{calc-prime-test} on successive integers until it finds one that
passes the test.  This is quite fast for integers less than eight million,
but once the probabilistic test comes into play the search may be rather
slow.  Ordinarily this command stops for any prime that passes one iteration
of the primality test.  With a numeric prefix argument, a number must pass
the specified number of iterations before the search stops.  (This only
matters when searching above eight million.)  You can always use additional
@kbd{k p} commands to increase your certainty that the number is indeed
prime.

@kindex I k n
@pindex calc-prev-prime
@ignore
@mindex prevpr@idots
@end ignore
@tindex prevprime
The @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command
analogously finds the next prime less than a given number.

@kindex k t
@pindex calc-totient
@tindex totient
The @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the
Euler ``totient'' 
@texline function@tie{}@math{\phi(n)},
@infoline function,
the number of integers less than @expr{n} which
are relatively prime to @expr{n}.

@kindex k m
@pindex calc-moebius
@tindex moebius
The @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the
@texline M@"obius @math{\mu}
@infoline Moebius ``mu''
function.  If the input number is a product of @expr{k}
distinct factors, this is @expr{(-1)^k}.  If the input number has any
duplicate factors (i.e., can be divided by the same prime more than once),
the result is zero.

@node Probability Distribution Functions,  , Combinatorial Functions, Scientific Functions
@section Probability Distribution Functions

@noindent
The functions in this section compute various probability distributions.
For continuous distributions, this is the integral of the probability
density function from @expr{x} to infinity.  (These are the ``upper
tail'' distribution functions; there are also corresponding ``lower
tail'' functions which integrate from minus infinity to @expr{x}.)
For discrete distributions, the upper tail function gives the sum
from @expr{x} to infinity; the lower tail function gives the sum
from minus infinity up to, but not including,@w{ }@expr{x}.

To integrate from @expr{x} to @expr{y}, just use the distribution
function twice and subtract.  For example, the probability that a
Gaussian random variable with mean 2 and standard deviation 1 will
lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
(``the probability that it is greater than 2.5, but not greater than 2.8''),
or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.

@kindex k B
@kindex I k B
@pindex calc-utpb
@tindex utpb
@tindex ltpb
The @kbd{k B} (@code{calc-utpb}) [@code{utpb}] function uses the
binomial distribution.  Push the parameters @var{n}, @var{p}, and
then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
probability that an event will occur @var{x} or more times out
of @var{n} trials, if its probability of occurring in any given
trial is @var{p}.  The @kbd{I k B} [@code{ltpb}] function is
the probability that the event will occur fewer than @var{x} times.

The other probability distribution functions similarly take the
form @kbd{k @var{X}} (@code{calc-utp@var{x}}) [@code{utp@var{x}}]
and @kbd{I k @var{X}} [@code{ltp@var{x}}], for various letters
@var{x}.  The arguments to the algebraic functions are the value of
the random variable first, then whatever other parameters define the
distribution.  Note these are among the few Calc functions where the
order of the arguments in algebraic form differs from the order of
arguments as found on the stack.  (The random variable comes last on
the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
k N M-@key{RET} @key{DEL} 2.8 k N -}, using @kbd{M-@key{RET} @key{DEL}} to
recover the original arguments but substitute a new value for @expr{x}.)

@kindex k C
@pindex calc-utpc
@tindex utpc
@ignore
@mindex @idots
@end ignore
@kindex I k C
@ignore
@mindex @null
@end ignore
@tindex ltpc
The @samp{utpc(x,v)} function uses the chi-square distribution with
@texline @math{\nu}
@infoline @expr{v} 
degrees of freedom.  It is the probability that a model is
correct if its chi-square statistic is @expr{x}.

@kindex k F
@pindex calc-utpf
@tindex utpf
@ignore
@mindex @idots
@end ignore
@kindex I k F
@ignore
@mindex @null
@end ignore
@tindex ltpf
The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
various statistical tests.  The parameters 
@texline @math{\nu_1}
@infoline @expr{v1} 
and 
@texline @math{\nu_2}
@infoline @expr{v2}
are the degrees of freedom in the numerator and denominator,
respectively, used in computing the statistic @expr{F}.

@kindex k N
@pindex calc-utpn
@tindex utpn
@ignore
@mindex @idots
@end ignore
@kindex I k N
@ignore
@mindex @null
@end ignore
@tindex ltpn
The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
with mean @expr{m} and standard deviation 
@texline @math{\sigma}.
@infoline @expr{s}.  
It is the probability that such a normal-distributed random variable
would exceed @expr{x}.

@kindex k P
@pindex calc-utpp
@tindex utpp
@ignore
@mindex @idots
@end ignore
@kindex I k P
@ignore
@mindex @null
@end ignore
@tindex ltpp
The @samp{utpp(n,x)} function uses a Poisson distribution with
mean @expr{x}.  It is the probability that @expr{n} or more such
Poisson random events will occur.

@kindex k T
@pindex calc-ltpt
@tindex utpt
@ignore
@mindex @idots
@end ignore
@kindex I k T
@ignore
@mindex @null
@end ignore
@tindex ltpt
The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
with 
@texline @math{\nu}
@infoline @expr{v} 
degrees of freedom.  It is the probability that a
t-distributed random variable will be greater than @expr{t}.
(Note:  This computes the distribution function 
@texline @math{A(t|\nu)}
@infoline @expr{A(t|v)}
where 
@texline @math{A(0|\nu) = 1}
@infoline @expr{A(0|v) = 1} 
and 
@texline @math{A(\infty|\nu) \to 0}.
@infoline @expr{A(inf|v) -> 0}.  
The @code{UTPT} operation on the HP-48 uses a different definition which
returns half of Calc's value:  @samp{UTPT(t,v) = .5*utpt(t,v)}.)

While Calc does not provide inverses of the probability distribution
functions, the @kbd{a R} command can be used to solve for the inverse.
Since the distribution functions are monotonic, @kbd{a R} is guaranteed
to be able to find a solution given any initial guess.
@xref{Numerical Solutions}.

@node Matrix Functions, Algebra, Scientific Functions, Top
@chapter Vector/Matrix Functions

@noindent
Many of the commands described here begin with the @kbd{v} prefix.
(For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.)
The commands usually apply to both plain vectors and matrices; some
apply only to matrices or only to square matrices.  If the argument
has the wrong dimensions the operation is left in symbolic form.

Vectors are entered and displayed using @samp{[a,b,c]} notation.
Matrices are vectors of which all elements are vectors of equal length.
(Though none of the standard Calc commands use this concept, a
three-dimensional matrix or rank-3 tensor could be defined as a
vector of matrices, and so on.)

@menu
* Packing and Unpacking::
* Building Vectors::
* Extracting Elements::
* Manipulating Vectors::
* Vector and Matrix Arithmetic::
* Set Operations::
* Statistical Operations::
* Reducing and Mapping::
* Vector and Matrix Formats::
@end menu

@node Packing and Unpacking, Building Vectors, Matrix Functions, Matrix Functions
@section Packing and Unpacking

@noindent
Calc's ``pack'' and ``unpack'' commands collect stack entries to build
composite objects such as vectors and complex numbers.  They are
described in this chapter because they are most often used to build
vectors.

@kindex v p
@pindex calc-pack
The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several
elements from the stack into a matrix, complex number, HMS form, error
form, etc.  It uses a numeric prefix argument to specify the kind of
object to be built; this argument is referred to as the ``packing mode.''
If the packing mode is a nonnegative integer, a vector of that
length is created.  For example, @kbd{C-u 5 v p} will pop the top
five stack elements and push back a single vector of those five
elements.  (@kbd{C-u 0 v p} simply creates an empty vector.)

The same effect can be had by pressing @kbd{[} to push an incomplete
vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
the incomplete object up past a certain number of elements, and
then pressing @kbd{]} to complete the vector.

Negative packing modes create other kinds of composite objects:

@table @cite
@item -1
Two values are collected to build a complex number.  For example,
@kbd{5 @key{RET} 7 C-u -1 v p} creates the complex number
@expr{(5, 7)}.  The result is always a rectangular complex
number.  The two input values must both be real numbers,
i.e., integers, fractions, or floats.  If they are not, Calc
will instead build a formula like @samp{a + (0, 1) b}.  (The
other packing modes also create a symbolic answer if the
components are not suitable.)

@item -2
Two values are collected to build a polar complex number.
The first is the magnitude; the second is the phase expressed
in either degrees or radians according to the current angular
mode.

@item -3
Three values are collected into an HMS form.  The first
two values (hours and minutes) must be integers or
integer-valued floats.  The third value may be any real
number.

@item -4
Two values are collected into an error form.  The inputs
may be real numbers or formulas.

@item -5
Two values are collected into a modulo form.  The inputs
must be real numbers.

@item -6
Two values are collected into the interval @samp{[a .. b]}.
The inputs may be real numbers, HMS or date forms, or formulas.

@item -7
Two values are collected into the interval @samp{[a .. b)}.

@item -8
Two values are collected into the interval @samp{(a .. b]}.

@item -9
Two values are collected into the interval @samp{(a .. b)}.

@item -10
Two integer values are collected into a fraction.

@item -11
Two values are collected into a floating-point number.
The first is the mantissa; the second, which must be an
integer, is the exponent.  The result is the mantissa
times ten to the power of the exponent.

@item -12
This is treated the same as @mathit{-11} by the @kbd{v p} command.
When unpacking, @mathit{-12} specifies that a floating-point mantissa
is desired.

@item -13
A real number is converted into a date form.

@item -14
Three numbers (year, month, day) are packed into a pure date form.

@item -15
Six numbers are packed into a date/time form.
@end table

With any of the two-input negative packing modes, either or both
of the inputs may be vectors.  If both are vectors of the same
length, the result is another vector made by packing corresponding
elements of the input vectors.  If one input is a vector and the
other is a plain number, the number is packed along with each vector
element to produce a new vector.  For example, @kbd{C-u -4 v p}
could be used to convert a vector of numbers and a vector of errors
into a single vector of error forms; @kbd{C-u -5 v p} could convert
a vector of numbers and a single number @var{M} into a vector of
numbers modulo @var{M}.

If you don't give a prefix argument to @kbd{v p}, it takes
the packing mode from the top of the stack.  The elements to
be packed then begin at stack level 2.  Thus
@kbd{1 @key{RET} 2 @key{RET} 4 n v p} is another way to
enter the error form @samp{1 +/- 2}.

If the packing mode taken from the stack is a vector, the result is a
matrix with the dimensions specified by the elements of the vector,
which must each be integers.  For example, if the packing mode is
@samp{[2, 3]}, then six numbers will be taken from the stack and
returned in the form @samp{[@w{[a, b, c]}, [d, e, f]]}.

If any elements of the vector are negative, other kinds of
packing are done at that level as described above.  For
example, @samp{[2, 3, -4]} takes 12 objects and creates a
@texline @math{2\times3}
@infoline 2x3
matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
Also, @samp{[-4, -10]} will convert four integers into an
error form consisting of two fractions:  @samp{a:b +/- c:d}.

@ignore
@starindex
@end ignore
@tindex pack
There is an equivalent algebraic function,
@samp{pack(@var{mode}, @var{items})} where @var{mode} is a
packing mode (an integer or a vector of integers) and @var{items}
is a vector of objects to be packed (re-packed, really) according
to that mode.  For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}.  The function is
left in symbolic form if the packing mode is invalid, or if the
number of data items does not match the number of items required
by the mode.

@kindex v u
@pindex calc-unpack
The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
number, HMS form, or other composite object on the top of the stack and
``unpacks'' it, pushing each of its elements onto the stack as separate
objects.  Thus, it is the ``inverse'' of @kbd{v p}.  If the value
at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
each of the arguments of the top-level operator onto the stack.

You can optionally give a numeric prefix argument to @kbd{v u}
to specify an explicit (un)packing mode.  If the packing mode is
negative and the input is actually a vector or matrix, the result
will be two or more similar vectors or matrices of the elements.
For example, given the vector @samp{[@w{a +/- b}, c^2, d +/- 7]},
the result of @kbd{C-u -4 v u} will be the two vectors
@samp{[a, c^2, d]} and @w{@samp{[b, 0, 7]}}.

Note that the prefix argument can have an effect even when the input is
not a vector.  For example, if the input is the number @mathit{-5}, then
@kbd{c-u -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
number).  Plain @kbd{v u} with this input would complain that the input
is not a composite object.

Unpacking mode @mathit{-11} converts a float into an integer mantissa and
an integer exponent, where the mantissa is not divisible by 10
(except that 0.0 is represented by a mantissa and exponent of 0).
Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
and integer exponent, where the mantissa (for non-zero numbers)
is guaranteed to lie in the range [1 .. 10).  In both cases,
the mantissa is shifted left or right (and the exponent adjusted
to compensate) in order to satisfy these constraints.

Positive unpacking modes are treated differently than for @kbd{v p}.
A mode of 1 is much like plain @kbd{v u} with no prefix argument,
except that in addition to the components of the input object,
a suitable packing mode to re-pack the object is also pushed.
Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
original object.

A mode of 2 unpacks two levels of the object; the resulting
re-packing mode will be a vector of length 2.  This might be used
to unpack a matrix, say, or a vector of error forms.  Higher
unpacking modes unpack the input even more deeply.

@ignore
@starindex
@end ignore
@tindex unpack
There are two algebraic functions analogous to @kbd{v u}.
The @samp{unpack(@var{mode}, @var{item})} function unpacks the
@var{item} using the given @var{mode}, returning the result as
a vector of components.  Here the @var{mode} must be an
integer, not a vector.  For example, @samp{unpack(-4, a +/- b)}
returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.

@ignore
@starindex
@end ignore
@tindex unpackt
The @code{unpackt} function is like @code{unpack} but instead
of returning a simple vector of items, it returns a vector of
two things:  The mode, and the vector of items.  For example,
@samp{unpackt(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
The identity for re-building the original object is
@samp{apply(pack, unpackt(@var{n}, @var{x})) = @var{x}}.  (The
@code{apply} function builds a function call given the function
name and a vector of arguments.)

@cindex Numerator of a fraction, extracting
Subscript notation is a useful way to extract a particular part
of an object.  For example, to get the numerator of a rational
number, you can use @samp{unpack(-10, @var{x})_1}.

@node Building Vectors, Extracting Elements, Packing and Unpacking, Matrix Functions
@section Building Vectors

@noindent
Vectors and matrices can be added,
subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.

@kindex |
@pindex calc-concat
@ignore
@mindex @null
@end ignore
@tindex |
The @kbd{|} (@code{calc-concat}) [@code{vconcat}] command ``concatenates'' two vectors
into one.  For example, after @kbd{@w{[ 1 , 2 ]} [ 3 , 4 ] |}, the stack
will contain the single vector @samp{[1, 2, 3, 4]}.  If the arguments
are matrices, the rows of the first matrix are concatenated with the
rows of the second.  (In other words, two matrices are just two vectors
of row-vectors as far as @kbd{|} is concerned.)

If either argument to @kbd{|} is a scalar (a non-vector), it is treated
like a one-element vector for purposes of concatenation:  @kbd{1 [ 2 , 3 ] |}
produces the vector @samp{[1, 2, 3]}.  Likewise, if one argument is a
matrix and the other is a plain vector, the vector is treated as a
one-row matrix.

@kindex H |
@tindex append
The @kbd{H |} (@code{calc-append}) [@code{append}] command concatenates
two vectors without any special cases.  Both inputs must be vectors.
Whether or not they are matrices is not taken into account.  If either
argument is a scalar, the @code{append} function is left in symbolic form.
See also @code{cons} and @code{rcons} below.

@kindex I |
@kindex H I |
The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
two stack arguments in the opposite order.  Thus @kbd{I |} is equivalent
to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster.

@kindex v d
@pindex calc-diag
@tindex diag
The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
square matrix.  The optional numeric prefix gives the number of rows
and columns in the matrix.  If the value at the top of the stack is a
vector, the elements of the vector are used as the diagonal elements; the
prefix, if specified, must match the size of the vector.  If the value on
the stack is a scalar, it is used for each element on the diagonal, and
the prefix argument is required.

To build a constant square matrix, e.g., a 
@texline @math{3\times3}
@infoline 3x3
matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
matrix first and then add a constant value to that matrix.  (Another
alternative would be to use @kbd{v b} and @kbd{v a}; see below.)

@kindex v i
@pindex calc-ident
@tindex idn
The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity
matrix of the specified size.  It is a convenient form of @kbd{v d}
where the diagonal element is always one.  If no prefix argument is given,
this command prompts for one.

In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
except that @expr{a} is required to be a scalar (non-vector) quantity.
If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
identity matrix of unknown size.  Calc can operate algebraically on
such generic identity matrices, and if one is combined with a matrix
whose size is known, it is converted automatically to an identity
matrix of a suitable matching size.  The @kbd{v i} command with an
argument of zero creates a generic identity matrix, @samp{idn(1)}.
Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
identity matrices are immediately expanded to the current default
dimensions.

@kindex v x
@pindex calc-index
@tindex index
The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
prefix argument.  If you do not provide a prefix argument, you will be
prompted to enter a suitable number.  If @var{n} is negative, the result
is a vector of negative integers from @var{n} to @mathit{-1}.

With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
three values from the stack: @var{n}, @var{start}, and @var{incr} (with
@var{incr} at top-of-stack).  Counting starts at @var{start} and increases
by @var{incr} for successive vector elements.  If @var{start} or @var{n}
is in floating-point format, the resulting vector elements will also be
floats.  Note that @var{start} and @var{incr} may in fact be any kind
of numbers or formulas.

When @var{start} and @var{incr} are specified, a negative @var{n} has a
different interpretation:  It causes a geometric instead of arithmetic
sequence to be generated.  For example, @samp{index(-3, a, b)} produces
@samp{[a, a b, a b^2]}.  If you omit @var{incr} in the algebraic form,
@samp{index(@var{n}, @var{start})}, the default value for @var{incr}
is one for positive @var{n} or two for negative @var{n}.

@kindex v b
@pindex calc-build-vector
@tindex cvec
The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
vector of @var{n} copies of the value on the top of the stack, where @var{n}
is the numeric prefix argument.  In algebraic formulas, @samp{cvec(x,n,m)}
can also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}.
(Interactively, just use @kbd{v b} twice: once to build a row, then again
to build a matrix of copies of that row.)

@kindex v h
@kindex I v h
@pindex calc-head
@pindex calc-tail
@tindex head
@tindex tail
The @kbd{v h} (@code{calc-head}) [@code{head}] function returns the first
element of a vector.  The @kbd{I v h} (@code{calc-tail}) [@code{tail}]
function returns the vector with its first element removed.  In both
cases, the argument must be a non-empty vector.

@kindex v k
@pindex calc-cons
@tindex cons
The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h}
and a vector @var{t} from the stack, and produces the vector whose head is
@var{h} and whose tail is @var{t}.  This is similar to @kbd{|}, except
if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.

@kindex H v h
@tindex rhead
@ignore
@mindex @idots
@end ignore
@kindex H I v h
@ignore
@mindex @null
@end ignore
@kindex H v k
@ignore
@mindex @null
@end ignore
@tindex rtail
@ignore
@mindex @null
@end ignore
@tindex rcons
Each of these three functions also accepts the Hyperbolic flag [@code{rhead},
@code{rtail}, @code{rcons}] in which case @var{t} instead represents
the @emph{last} single element of the vector, with @var{h}
representing the remainder of the vector.  Thus the vector
@samp{[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
@samp{rhead([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.

@node Extracting Elements, Manipulating Vectors, Building Vectors, Matrix Functions
@section Extracting Vector Elements

@noindent
@kindex v r
@pindex calc-mrow
@tindex mrow
The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
the matrix on the top of the stack, or one element of the plain vector on
the top of the stack.  The row or element is specified by the numeric
prefix argument; the default is to prompt for the row or element number.
The matrix or vector is replaced by the specified row or element in the
form of a vector or scalar, respectively.

@cindex Permutations, applying
With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
the element or row from the top of the stack, and the vector or matrix
from the second-to-top position.  If the index is itself a vector of
integers, the result is a vector of the corresponding elements of the
input vector, or a matrix of the corresponding rows of the input matrix.
This command can be used to obtain any permutation of a vector.

With @kbd{C-u}, if the index is an interval form with integer components,
it is interpreted as a range of indices and the corresponding subvector or
submatrix is returned.

@cindex Subscript notation
@kindex a _
@pindex calc-subscript
@tindex subscr
@tindex _
Subscript notation in algebraic formulas (@samp{a_b}) stands for the
Calc function @code{subscr}, which is synonymous with @code{mrow}.
Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
@expr{k} is one, two, or three, respectively.  A double subscript
(@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
access the element at row @expr{i}, column @expr{j} of a matrix.
The @kbd{a _} (@code{calc-subscript}) command creates a subscript
formula @samp{a_b} out of two stack entries.  (It is on the @kbd{a}
``algebra'' prefix because subscripted variables are often used
purely as an algebraic notation.)

@tindex mrrow
Given a negative prefix argument, @kbd{v r} instead deletes one row or
element from the matrix or vector on the top of the stack.  Thus
@kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r}
replaces the matrix with the same matrix with its second row removed.
In algebraic form this function is called @code{mrrow}.

@tindex getdiag
Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
of a square matrix in the form of a vector.  In algebraic form this
function is called @code{getdiag}.

@kindex v c
@pindex calc-mcol
@tindex mcol
@tindex mrcol
The @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is
the analogous operation on columns of a matrix.  Given a plain vector
it extracts (or removes) one element, just like @kbd{v r}.  If the
index in @kbd{C-u v c} is an interval or vector and the argument is a
matrix, the result is a submatrix with only the specified columns
retained (and possibly permuted in the case of a vector index).

To extract a matrix element at a given row and column, use @kbd{v r} to
extract the row as a vector, then @kbd{v c} to extract the column element
from that vector.  In algebraic formulas, it is often more convenient to
use subscript notation:  @samp{m_i_j} gives row @expr{i}, column @expr{j}
of matrix @expr{m}.

@kindex v s
@pindex calc-subvector
@tindex subvec
The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts
a subvector of a vector.  The arguments are the vector, the starting
index, and the ending index, with the ending index in the top-of-stack
position.  The starting index indicates the first element of the vector
to take.  The ending index indicates the first element @emph{past} the
range to be taken.  Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
the subvector @samp{[b, c]}.  You could get the same result using
@samp{mrow([a, b, c, d, e], @w{[2 .. 4)})}.

If either the start or the end index is zero or negative, it is
interpreted as relative to the end of the vector.  Thus
@samp{subvec([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}.  In
the algebraic form, the end index can be omitted in which case it
is taken as zero, i.e., elements from the starting element to the
end of the vector are used.  The infinity symbol, @code{inf}, also
has this effect when used as the ending index.

@kindex I v s
@tindex rsubvec
With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector
from a vector.  The arguments are interpreted the same as for the
normal @kbd{v s} command.  Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
produces @samp{[a, d, e]}.  It is always true that @code{subvec} and
@code{rsubvec} return complementary parts of the input vector.

@xref{Selecting Subformulas}, for an alternative way to operate on
vectors one element at a time.

@node Manipulating Vectors, Vector and Matrix Arithmetic, Extracting Elements, Matrix Functions
@section Manipulating Vectors

@noindent
@kindex v l
@pindex calc-vlength
@tindex vlen
The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
length of a vector.  The length of a non-vector is considered to be zero.
Note that matrices are just vectors of vectors for the purposes of this
command.

@kindex H v l
@tindex mdims
With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector
of the dimensions of a vector, matrix, or higher-order object.  For
example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
its argument is a 
@texline @math{2\times3}
@infoline 2x3
matrix.

@kindex v f
@pindex calc-vector-find
@tindex find
The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches
along a vector for the first element equal to a given target.  The target
is on the top of the stack; the vector is in the second-to-top position.
If a match is found, the result is the index of the matching element.
Otherwise, the result is zero.  The numeric prefix argument, if given,
allows you to select any starting index for the search.

@kindex v a
@pindex calc-arrange-vector
@tindex arrange
@cindex Arranging a matrix
@cindex Reshaping a matrix
@cindex Flattening a matrix
The @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] command
rearranges a vector to have a certain number of columns and rows.  The
numeric prefix argument specifies the number of columns; if you do not
provide an argument, you will be prompted for the number of columns.
The vector or matrix on the top of the stack is @dfn{flattened} into a
plain vector.  If the number of columns is nonzero, this vector is
then formed into a matrix by taking successive groups of @var{n} elements.
If the number of columns does not evenly divide the number of elements
in the vector, the last row will be short and the result will not be
suitable for use as a matrix.  For example, with the matrix
@samp{[[1, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
@samp{[[1, 2, 3, 4]]} (a 
@texline @math{1\times4}
@infoline 1x4
matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a 
@texline @math{4\times1}
@infoline 4x1
matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original 
@texline @math{2\times2}
@infoline 2x2
matrix), @w{@kbd{v a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
matrix), and @kbd{v a 0} produces the flattened list 
@samp{[1, 2, @w{3, 4}]}.

@cindex Sorting data
@kindex V S
@kindex I V S
@pindex calc-sort
@tindex sort
@tindex rsort
The @kbd{V S} (@code{calc-sort}) [@code{sort}] command sorts the elements of
a vector into increasing order.  Real numbers, real infinities, and
constant interval forms come first in this ordering; next come other
kinds of numbers, then variables (in alphabetical order), then finally
come formulas and other kinds of objects; these are sorted according
to a kind of lexicographic ordering with the useful property that
one vector is less or greater than another if the first corresponding
unequal elements are less or greater, respectively.  Since quoted strings
are stored by Calc internally as vectors of ASCII character codes
(@pxref{Strings}), this means vectors of strings are also sorted into
alphabetical order by this command.

The @kbd{I V S} [@code{rsort}] command sorts a vector into decreasing order.

@cindex Permutation, inverse of
@cindex Inverse of permutation
@cindex Index tables
@cindex Rank tables
@kindex V G
@kindex I V G
@pindex calc-grade
@tindex grade
@tindex rgrade
The @kbd{V G} (@code{calc-grade}) [@code{grade}, @code{rgrade}] command
produces an index table or permutation vector which, if applied to the
input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
A permutation vector is just a vector of integers from 1 to @var{n}, where
each integer occurs exactly once.  One application of this is to sort a
matrix of data rows using one column as the sort key; extract that column,
grade it with @kbd{V G}, then use the result to reorder the original matrix
with @kbd{C-u v r}.  Another interesting property of the @code{V G} command
is that, if the input is itself a permutation vector, the result will
be the inverse of the permutation.  The inverse of an index table is
a rank table, whose @var{k}th element says where the @var{k}th original
vector element will rest when the vector is sorted.  To get a rank
table, just use @kbd{V G V G}.

With the Inverse flag, @kbd{I V G} produces an index table that would
sort the input into decreasing order.  Note that @kbd{V S} and @kbd{V G}
use a ``stable'' sorting algorithm, i.e., any two elements which are equal
will not be moved out of their original order.  Generally there is no way
to tell with @kbd{V S}, since two elements which are equal look the same,
but with @kbd{V G} this can be an important issue.  In the matrix-of-rows
example, suppose you have names and telephone numbers as two columns and
you wish to sort by phone number primarily, and by name when the numbers
are equal.  You can sort the data matrix by names first, and then again
by phone numbers.  Because the sort is stable, any two rows with equal
phone numbers will remain sorted by name even after the second sort.

@cindex Histograms
@kindex V H
@pindex calc-histogram
@ignore
@mindex histo@idots
@end ignore
@tindex histogram
The @kbd{V H} (@code{calc-histogram}) [@code{histogram}] command builds a
histogram of a vector of numbers.  Vector elements are assumed to be
integers or real numbers in the range [0..@var{n}) for some ``number of
bins'' @var{n}, which is the numeric prefix argument given to the
command.  The result is a vector of @var{n} counts of how many times
each value appeared in the original vector.  Non-integers in the input
are rounded down to integers.  Any vector elements outside the specified
range are ignored.  (You can tell if elements have been ignored by noting
that the counts in the result vector don't add up to the length of the
input vector.)

@kindex H V H
With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
The second-to-top vector is the list of numbers as before.  The top
vector is an equal-sized list of ``weights'' to attach to the elements
of the data vector.  For example, if the first data element is 4.2 and
the first weight is 10, then 10 will be added to bin 4 of the result
vector.  Without the hyperbolic flag, every element has a weight of one.

@kindex v t
@pindex calc-transpose
@tindex trn
The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
the transpose of the matrix at the top of the stack.  If the argument
is a plain vector, it is treated as a row vector and transposed into
a one-column matrix.

@kindex v v
@pindex calc-reverse-vector
@tindex rev
The @kbd{v v} (@code{calc-reverse-vector}) [@code{rev}] command reverses
a vector end-for-end.  Given a matrix, it reverses the order of the rows.
(To reverse the columns instead, just use @kbd{v t v v v t}.  The same
principle can be used to apply other vector commands to the columns of
a matrix.)

@kindex v m
@pindex calc-mask-vector
@tindex vmask
The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses
one vector as a mask to extract elements of another vector.  The mask
is in the second-to-top position; the target vector is on the top of
the stack.  These vectors must have the same length.  The result is
the same as the target vector, but with all elements which correspond
to zeros in the mask vector deleted.  Thus, for example,
@samp{vmask([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
@xref{Logical Operations}.

@kindex v e
@pindex calc-expand-vector
@tindex vexp
The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command
expands a vector according to another mask vector.  The result is a
vector the same length as the mask, but with nonzero elements replaced
by successive elements from the target vector.  The length of the target
vector is normally the number of nonzero elements in the mask.  If the
target vector is longer, its last few elements are lost.  If the target
vector is shorter, the last few nonzero mask elements are left
unreplaced in the result.  Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
produces @samp{[a, 0, b, 0, 7]}.

@kindex H v e
With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
top of the stack; the mask and target vectors come from the third and
second elements of the stack.  This filler is used where the mask is
zero:  @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
@samp{[a, z, c, z, 7]}.  If the filler value is itself a vector,
then successive values are taken from it, so that the effect is to
interleave two vectors according to the mask:
@samp{vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
@samp{[a, x, b, 7, y, 0]}.

Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
operation across the two vectors.  @xref{Logical Operations}.  Note that
the @code{? :} operation also discussed there allows other types of
masking using vectors.

@node Vector and Matrix Arithmetic, Set Operations, Manipulating Vectors, Matrix Functions
@section Vector and Matrix Arithmetic

@noindent
Basic arithmetic operations like addition and multiplication are defined
for vectors and matrices as well as for numbers.  Division of matrices, in
the sense of multiplying by the inverse, is supported.  (Division by a
matrix actually uses LU-decomposition for greater accuracy and speed.)
@xref{Basic Arithmetic}.

The following functions are applied element-wise if their arguments are
vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
@code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
@code{float}, @code{frac}.  @xref{Function Index}.

@kindex V J
@pindex calc-conj-transpose
@tindex ctrn
The @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes
the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.

@ignore
@mindex A
@end ignore
@kindex A (vectors)
@pindex calc-abs (vectors)
@ignore
@mindex abs
@end ignore
@tindex abs (vectors)
The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the
Frobenius norm of a vector or matrix argument.  This is the square
root of the sum of the squares of the absolute values of the
elements of the vector or matrix.  If the vector is interpreted as
a point in two- or three-dimensional space, this is the distance
from that point to the origin.

@kindex v n
@pindex calc-rnorm
@tindex rnorm
The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes
the row norm, or infinity-norm, of a vector or matrix.  For a plain
vector, this is the maximum of the absolute values of the elements.
For a matrix, this is the maximum of the row-absolute-value-sums,
i.e., of the sums of the absolute values of the elements along the
various rows.

@kindex V N
@pindex calc-cnorm
@tindex cnorm
The @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes
the column norm, or one-norm, of a vector or matrix.  For a plain
vector, this is the sum of the absolute values of the elements.
For a matrix, this is the maximum of the column-absolute-value-sums.
General @expr{k}-norms for @expr{k} other than one or infinity are
not provided.

@kindex V C
@pindex calc-cross
@tindex cross
The @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the
right-handed cross product of two vectors, each of which must have
exactly three elements.

@ignore
@mindex &
@end ignore
@kindex & (matrices)
@pindex calc-inv (matrices)
@ignore
@mindex inv
@end ignore
@tindex inv (matrices)
The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
inverse of a square matrix.  If the matrix is singular, the inverse
operation is left in symbolic form.  Matrix inverses are recorded so
that once an inverse (or determinant) of a particular matrix has been
computed, the inverse and determinant of the matrix can be recomputed
quickly in the future.

If the argument to @kbd{&} is a plain number @expr{x}, this
command simply computes @expr{1/x}.  This is okay, because the
@samp{/} operator also does a matrix inversion when dividing one
by a matrix.

@kindex V D
@pindex calc-mdet
@tindex det
The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
determinant of a square matrix.

@kindex V L
@pindex calc-mlud
@tindex lud
The @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the
LU decomposition of a matrix.  The result is a list of three matrices
which, when multiplied together left-to-right, form the original matrix.
The first is a permutation matrix that arises from pivoting in the
algorithm, the second is lower-triangular with ones on the diagonal,
and the third is upper-triangular.

@kindex V T
@pindex calc-mtrace
@tindex tr
The @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the
trace of a square matrix.  This is defined as the sum of the diagonal
elements of the matrix.

@node Set Operations, Statistical Operations, Vector and Matrix Arithmetic, Matrix Functions
@section Set Operations using Vectors

@noindent
@cindex Sets, as vectors
Calc includes several commands which interpret vectors as @dfn{sets} of
objects.  A set is a collection of objects; any given object can appear
only once in the set.  Calc stores sets as vectors of objects in
sorted order.  Objects in a Calc set can be any of the usual things,
such as numbers, variables, or formulas.  Two set elements are considered
equal if they are identical, except that numerically equal numbers like
the integer 4 and the float 4.0 are considered equal even though they
are not ``identical.''  Variables are treated like plain symbols without
attached values by the set operations; subtracting the set @samp{[b]}
from @samp{[a, b]} always yields the set @samp{[a]} even though if
the variables @samp{a} and @samp{b} both equaled 17, you might
expect the answer @samp{[]}.

If a set contains interval forms, then it is assumed to be a set of
real numbers.  In this case, all set operations require the elements
of the set to be only things that are allowed in intervals:  Real
numbers, plus and minus infinity, HMS forms, and date forms.  If
there are variables or other non-real objects present in a real set,
all set operations on it will be left in unevaluated form.

If the input to a set operation is a plain number or interval form
@var{a}, it is treated like the one-element vector @samp{[@var{a}]}.
The result is always a vector, except that if the set consists of a
single interval, the interval itself is returned instead.

@xref{Logical Operations}, for the @code{in} function which tests if
a certain value is a member of a given set.  To test if the set @expr{A}
is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.

@kindex V +
@pindex calc-remove-duplicates
@tindex rdup
The @kbd{V +} (@code{calc-remove-duplicates}) [@code{rdup}] command
converts an arbitrary vector into set notation.  It works by sorting
the vector as if by @kbd{V S}, then removing duplicates.  (For example,
@kbd{[a, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
reduced to @samp{[4, 5, a]}).  Overlapping intervals are merged as
necessary.  You rarely need to use @kbd{V +} explicitly, since all the
other set-based commands apply @kbd{V +} to their inputs before using
them.

@kindex V V
@pindex calc-set-union
@tindex vunion
The @kbd{V V} (@code{calc-set-union}) [@code{vunion}] command computes
the union of two sets.  An object is in the union of two sets if and
only if it is in either (or both) of the input sets.  (You could
accomplish the same thing by concatenating the sets with @kbd{|},
then using @kbd{V +}.)

@kindex V ^
@pindex calc-set-intersect
@tindex vint
The @kbd{V ^} (@code{calc-set-intersect}) [@code{vint}] command computes
the intersection of two sets.  An object is in the intersection if
and only if it is in both of the input sets.  Thus if the input
sets are disjoint, i.e., if they share no common elements, the result
will be the empty vector @samp{[]}.  Note that the characters @kbd{V}
and @kbd{^} were chosen to be close to the conventional mathematical
notation for set 
@texline union@tie{}(@math{A \cup B})
@infoline union
and 
@texline intersection@tie{}(@math{A \cap B}).
@infoline intersection.

@kindex V -
@pindex calc-set-difference
@tindex vdiff
The @kbd{V -} (@code{calc-set-difference}) [@code{vdiff}] command computes
the difference between two sets.  An object is in the difference
@expr{A - B} if and only if it is in @expr{A} but not in @expr{B}.
Thus subtracting @samp{[y,z]} from a set will remove the elements
@samp{y} and @samp{z} if they are present.  You can also think of this
as a general @dfn{set complement} operator; if @expr{A} is the set of
all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
Obviously this is only practical if the set of all possible values in
your problem is small enough to list in a Calc vector (or simple
enough to express in a few intervals).

@kindex V X
@pindex calc-set-xor
@tindex vxor
The @kbd{V X} (@code{calc-set-xor}) [@code{vxor}] command computes
the ``exclusive-or,'' or ``symmetric difference'' of two sets.
An object is in the symmetric difference of two sets if and only
if it is in one, but @emph{not} both, of the sets.  Objects that
occur in both sets ``cancel out.''

@kindex V ~
@pindex calc-set-complement
@tindex vcompl
The @kbd{V ~} (@code{calc-set-complement}) [@code{vcompl}] command
computes the complement of a set with respect to the real numbers.
Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
@samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}.

@kindex V F
@pindex calc-set-floor
@tindex vfloor
The @kbd{V F} (@code{calc-set-floor}) [@code{vfloor}] command
reinterprets a set as a set of integers.  Any non-integer values,
and intervals that do not enclose any integers, are removed.  Open
intervals are converted to equivalent closed intervals.  Successive
integers are converted into intervals of integers.  For example, the
complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
the complement with respect to the set of integers you could type
@kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.

@kindex V E
@pindex calc-set-enumerate
@tindex venum
The @kbd{V E} (@code{calc-set-enumerate}) [@code{venum}] command
converts a set of integers into an explicit vector.  Intervals in
the set are expanded out to lists of all integers encompassed by
the intervals.  This only works for finite sets (i.e., sets which
do not involve @samp{-inf} or @samp{inf}).

@kindex V :
@pindex calc-set-span
@tindex vspan
The @kbd{V :} (@code{calc-set-span}) [@code{vspan}] command converts any
set of reals into an interval form that encompasses all its elements.
The lower limit will be the smallest element in the set; the upper
limit will be the largest element.  For an empty set, @samp{vspan([])}
returns the empty interval @w{@samp{[0 .. 0)}}.

@kindex V #
@pindex calc-set-cardinality
@tindex vcard
The @kbd{V #} (@code{calc-set-cardinality}) [@code{vcard}] command counts
the number of integers in a set.  The result is the length of the vector
that would be produced by @kbd{V E}, although the computation is much
more efficient than actually producing that vector.

@cindex Sets, as binary numbers
Another representation for sets that may be more appropriate in some
cases is binary numbers.  If you are dealing with sets of integers
in the range 0 to 49, you can use a 50-bit binary number where a
particular bit is 1 if the corresponding element is in the set.
@xref{Binary Functions}, for a list of commands that operate on
binary numbers.  Note that many of the above set operations have
direct equivalents in binary arithmetic:  @kbd{b o} (@code{calc-or}),
@kbd{b a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
@kbd{b x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
respectively.  You can use whatever representation for sets is most
convenient to you.

@kindex b p
@kindex b u
@pindex calc-pack-bits
@pindex calc-unpack-bits
@tindex vpack
@tindex vunpack
The @kbd{b u} (@code{calc-unpack-bits}) [@code{vunpack}] command
converts an integer that represents a set in binary into a set
in vector/interval notation.  For example, @samp{vunpack(67)}
returns @samp{[[0 .. 1], 6]}.  If the input is negative, the set
it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
Use @kbd{V E} afterwards to expand intervals to individual
values if you wish.  Note that this command uses the @kbd{b}
(binary) prefix key.

The @kbd{b p} (@code{calc-pack-bits}) [@code{vpack}] command
converts the other way, from a vector or interval representing
a set of nonnegative integers into a binary integer describing
the same set.  The set may include positive infinity, but must
not include any negative numbers.  The input is interpreted as a
set of integers in the sense of @kbd{V F} (@code{vfloor}).  Beware
that a simple input like @samp{[100]} can result in a huge integer
representation 
@texline (@math{2^{100}}, a 31-digit integer, in this case).
@infoline (@expr{2^100}, a 31-digit integer, in this case).

@node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions
@section Statistical Operations on Vectors

@noindent
@cindex Statistical functions
The commands in this section take vectors as arguments and compute
various statistical measures on the data stored in the vectors.  The
references used in the definitions of these functions are Bevington's
@emph{Data Reduction and Error Analysis for the Physical Sciences},
and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
Vetterling.

The statistical commands use the @kbd{u} prefix key followed by
a shifted letter or other character.

@xref{Manipulating Vectors}, for a description of @kbd{V H}
(@code{calc-histogram}).

@xref{Curve Fitting}, for the @kbd{a F} command for doing
least-squares fits to statistical data.

@xref{Probability Distribution Functions}, for several common
probability distribution functions.

@menu
* Single-Variable Statistics::
* Paired-Sample Statistics::
@end menu

@node Single-Variable Statistics, Paired-Sample Statistics, Statistical Operations, Statistical Operations
@subsection Single-Variable Statistics

@noindent
These functions do various statistical computations on single
vectors.  Given a numeric prefix argument, they actually pop
@var{n} objects from the stack and combine them into a data
vector.  Each object may be either a number or a vector; if a
vector, any sub-vectors inside it are ``flattened'' as if by
@kbd{v a 0}; @pxref{Manipulating Vectors}.  By default one object
is popped, which (in order to be useful) is usually a vector.

If an argument is a variable name, and the value stored in that
variable is a vector, then the stored vector is used.  This method
has the advantage that if your data vector is large, you can avoid
the slow process of manipulating it directly on the stack.

These functions are left in symbolic form if any of their arguments
are not numbers or vectors, e.g., if an argument is a formula, or
a non-vector variable.  However, formulas embedded within vector
arguments are accepted; the result is a symbolic representation
of the computation, based on the assumption that the formula does
not itself represent a vector.  All varieties of numbers such as
error forms and interval forms are acceptable.

Some of the functions in this section also accept a single error form
or interval as an argument.  They then describe a property of the
normal or uniform (respectively) statistical distribution described
by the argument.  The arguments are interpreted in the same way as
the @var{M} argument of the random number function @kbd{k r}.  In
particular, an interval with integer limits is considered an integer
distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
An interval with at least one floating-point limit is a continuous
distribution:  @samp{[2.0 .. 6.0)} is @emph{not} the same as
@samp{[2.0 .. 5.0]}!

@kindex u #
@pindex calc-vector-count
@tindex vcount
The @kbd{u #} (@code{calc-vector-count}) [@code{vcount}] command
computes the number of data values represented by the inputs.
For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
If the argument is a single vector with no sub-vectors, this
simply computes the length of the vector.

@kindex u +
@kindex u *
@pindex calc-vector-sum
@pindex calc-vector-prod
@tindex vsum
@tindex vprod
@cindex Summations (statistical)
The @kbd{u +} (@code{calc-vector-sum}) [@code{vsum}] command
computes the sum of the data values.  The @kbd{u *}
(@code{calc-vector-prod}) [@code{vprod}] command computes the
product of the data values.  If the input is a single flat vector,
these are the same as @kbd{V R +} and @kbd{V R *}
(@pxref{Reducing and Mapping}).

@kindex u X
@kindex u N
@pindex calc-vector-max
@pindex calc-vector-min
@tindex vmax
@tindex vmin
The @kbd{u X} (@code{calc-vector-max}) [@code{vmax}] command
computes the maximum of the data values, and the @kbd{u N}
(@code{calc-vector-min}) [@code{vmin}] command computes the minimum.
If the argument is an interval, this finds the minimum or maximum
value in the interval.  (Note that @samp{vmax([2..6)) = 5} as
described above.)  If the argument is an error form, this returns
plus or minus infinity.

@kindex u M
@pindex calc-vector-mean
@tindex vmean
@cindex Mean of data values
The @kbd{u M} (@code{calc-vector-mean}) [@code{vmean}] command
computes the average (arithmetic mean) of the data values.
If the inputs are error forms 
@texline @math{x \pm \sigma},
@infoline @samp{x +/- s}, 
this is the weighted mean of the @expr{x} values with weights 
@texline @math{1 /\sigma^2}.
@infoline @expr{1 / s^2}.
@tex
\turnoffactive
$$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
           \displaystyle \sum { 1 \over \sigma_i^2 } } $$
@end tex
If the inputs are not error forms, this is simply the sum of the
values divided by the count of the values.

Note that a plain number can be considered an error form with
error 
@texline @math{\sigma = 0}.
@infoline @expr{s = 0}.  
If the input to @kbd{u M} is a mixture of
plain numbers and error forms, the result is the mean of the
plain numbers, ignoring all values with non-zero errors.  (By the
above definitions it's clear that a plain number effectively
has an infinite weight, next to which an error form with a finite
weight is completely negligible.)

This function also works for distributions (error forms or
intervals).  The mean of an error form `@var{a} @tfn{+/-} @var{b}' is simply
@expr{a}.  The mean of an interval is the mean of the minimum
and maximum values of the interval.

@kindex I u M
@pindex calc-vector-mean-error
@tindex vmeane
The @kbd{I u M} (@code{calc-vector-mean-error}) [@code{vmeane}]
command computes the mean of the data points expressed as an
error form.  This includes the estimated error associated with
the mean.  If the inputs are error forms, the error is the square
root of the reciprocal of the sum of the reciprocals of the squares
of the input errors.  (I.e., the variance is the reciprocal of the
sum of the reciprocals of the variances.)
@tex
\turnoffactive
$$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
@end tex
If the inputs are plain
numbers, the error is equal to the standard deviation of the values
divided by the square root of the number of values.  (This works
out to be equivalent to calculating the standard deviation and
then assuming each value's error is equal to this standard
deviation.)
@tex
\turnoffactive
$$ \sigma_\mu^2 = {\sigma^2 \over N} $$
@end tex

@kindex H u M
@pindex calc-vector-median
@tindex vmedian
@cindex Median of data values
The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}]
command computes the median of the data values.  The values are
first sorted into numerical order; the median is the middle
value after sorting.  (If the number of data values is even,
the median is taken to be the average of the two middle values.)
The median function is different from the other functions in
this section in that the arguments must all be real numbers;
variables are not accepted even when nested inside vectors.
(Otherwise it is not possible to sort the data values.)  If
any of the input values are error forms, their error parts are
ignored.

The median function also accepts distributions.  For both normal
(error form) and uniform (interval) distributions, the median is
the same as the mean.

@kindex H I u M
@pindex calc-vector-harmonic-mean
@tindex vhmean
@cindex Harmonic mean
The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}]
command computes the harmonic mean of the data values.  This is
defined as the reciprocal of the arithmetic mean of the reciprocals
of the values.
@tex
\turnoffactive
$$ { N \over \displaystyle \sum {1 \over x_i} } $$
@end tex

@kindex u G
@pindex calc-vector-geometric-mean
@tindex vgmean
@cindex Geometric mean
The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}]
command computes the geometric mean of the data values.  This
is the @var{n}th root of the product of the values.  This is also
equal to the @code{exp} of the arithmetic mean of the logarithms
of the data values.
@tex
\turnoffactive
$$ \exp \left ( \sum { \ln x_i } \right ) =
   \left ( \prod { x_i } \right)^{1 / N} $$
@end tex

@kindex H u G
@tindex agmean
The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric
mean'' of two numbers taken from the stack.  This is computed by
replacing the two numbers with their arithmetic mean and geometric
mean, then repeating until the two values converge.
@tex
\turnoffactive
$$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
@end tex

@cindex Root-mean-square
Another commonly used mean, the RMS (root-mean-square), can be computed
for a vector of numbers simply by using the @kbd{A} command.

@kindex u S
@pindex calc-vector-sdev
@tindex vsdev
@cindex Standard deviation
@cindex Sample statistics
The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command
computes the standard 
@texline deviation@tie{}@math{\sigma}
@infoline deviation
of the data values.  If the values are error forms, the errors are used
as weights just as for @kbd{u M}.  This is the @emph{sample} standard
deviation, whose value is the square root of the sum of the squares of
the differences between the values and the mean of the @expr{N} values,
divided by @expr{N-1}.
@tex
\turnoffactive
$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
@end tex

This function also applies to distributions.  The standard deviation
of a single error form is simply the error part.  The standard deviation
of a continuous interval happens to equal the difference between the
limits, divided by 
@texline @math{\sqrt{12}}.
@infoline @expr{sqrt(12)}.  
The standard deviation of an integer interval is the same as the
standard deviation of a vector of those integers.

@kindex I u S
@pindex calc-vector-pop-sdev
@tindex vpsdev
@cindex Population statistics
The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}]
command computes the @emph{population} standard deviation.
It is defined by the same formula as above but dividing
by @expr{N} instead of by @expr{N-1}.  The population standard
deviation is used when the input represents the entire set of
data values in the distribution; the sample standard deviation
is used when the input represents a sample of the set of all
data values, so that the mean computed from the input is itself
only an estimate of the true mean.
@tex
\turnoffactive
$$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
@end tex

For error forms and continuous intervals, @code{vpsdev} works
exactly like @code{vsdev}.  For integer intervals, it computes the
population standard deviation of the equivalent vector of integers.

@kindex H u S
@kindex H I u S
@pindex calc-vector-variance
@pindex calc-vector-pop-variance
@tindex vvar
@tindex vpvar
@cindex Variance of data values
The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and
@kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}]
commands compute the variance of the data values.  The variance
is the 
@texline square@tie{}@math{\sigma^2}
@infoline square
of the standard deviation, i.e., the sum of the
squares of the deviations of the data values from the mean.
(This definition also applies when the argument is a distribution.)

@ignore
@starindex
@end ignore
@tindex vflat
The @code{vflat} algebraic function returns a vector of its
arguments, interpreted in the same way as the other functions
in this section.  For example, @samp{vflat(1, [2, [3, 4]], 5)}
returns @samp{[1, 2, 3, 4, 5]}.

@node Paired-Sample Statistics,  , Single-Variable Statistics, Statistical Operations
@subsection Paired-Sample Statistics

@noindent
The functions in this section take two arguments, which must be
vectors of equal size.  The vectors are each flattened in the same
way as by the single-variable statistical functions.  Given a numeric
prefix argument of 1, these functions instead take one object from
the stack, which must be an 
@texline @math{N\times2}
@infoline Nx2
matrix of data values.  Once again, variable names can be used in place
of actual vectors and matrices.

@kindex u C
@pindex calc-vector-covariance
@tindex vcov
@cindex Covariance
The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command
computes the sample covariance of two vectors.  The covariance
of vectors @var{x} and @var{y} is the sum of the products of the
differences between the elements of @var{x} and the mean of @var{x}
times the differences between the corresponding elements of @var{y}
and the mean of @var{y}, all divided by @expr{N-1}.  Note that
the variance of a vector is just the covariance of the vector
with itself.  Once again, if the inputs are error forms the
errors are used as weight factors.  If both @var{x} and @var{y}
are composed of error forms, the error for a given data point
is taken as the square root of the sum of the squares of the two
input errors.
@tex
\turnoffactive
$$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
$$ \sigma_{x\!y}^2 =
    {\displaystyle {1 \over N-1}
                   \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
     \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
$$
@end tex

@kindex I u C
@pindex calc-vector-pop-covariance
@tindex vpcov
The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}]
command computes the population covariance, which is the same as the
sample covariance computed by @kbd{u C} except dividing by @expr{N}
instead of @expr{N-1}.

@kindex H u C
@pindex calc-vector-correlation
@tindex vcorr
@cindex Correlation coefficient
@cindex Linear correlation
The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}]
command computes the linear correlation coefficient of two vectors.
This is defined by the covariance of the vectors divided by the
product of their standard deviations.  (There is no difference
between sample or population statistics here.)
@tex
\turnoffactive
$$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
@end tex

@node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions
@section Reducing and Mapping Vectors

@noindent
The commands in this section allow for more general operations on the
elements of vectors.

@kindex V A
@pindex calc-apply
@tindex apply
The simplest of these operations is @kbd{V A} (@code{calc-apply})
[@code{apply}], which applies a given operator to the elements of a vector.
For example, applying the hypothetical function @code{f} to the vector
@w{@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2, 3)}.
Applying the @code{+} function to the vector @samp{[a, b]} gives
@samp{a + b}.  Applying @code{+} to the vector @samp{[a, b, c]} is an
error, since the @code{+} function expects exactly two arguments.

While @kbd{V A} is useful in some cases, you will usually find that either
@kbd{V R} or @kbd{V M}, described below, is closer to what you want.

@menu
* Specifying Operators::
* Mapping::
* Reducing::
* Nesting and Fixed Points::
* Generalized Products::
@end menu

@node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping
@subsection Specifying Operators

@noindent
Commands in this section (like @kbd{V A}) prompt you to press the key
corresponding to the desired operator.  Press @kbd{?} for a partial
list of the available operators.  Generally, an operator is any key or
sequence of keys that would normally take one or more arguments from
the stack and replace them with a result.  For example, @kbd{V A H C}
uses the hyperbolic cosine operator, @code{cosh}.  (Since @code{cosh}
expects one argument, @kbd{V A H C} requires a vector with a single
element as its argument.)

You can press @kbd{x} at the operator prompt to select any algebraic
function by name to use as the operator.  This includes functions you
have defined yourself using the @kbd{Z F} command.  (@xref{Algebraic
Definitions}.)  If you give a name for which no function has been
defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
Calc will prompt for the number of arguments the function takes if it
can't figure it out on its own (say, because you named a function that
is currently undefined).  It is also possible to type a digit key before
the function name to specify the number of arguments, e.g.,
@kbd{V M 3 x f @key{RET}} calls @code{f} with three arguments even if it
looks like it ought to have only two.  This technique may be necessary
if the function allows a variable number of arguments.  For example,
the @kbd{v e} [@code{vexp}] function accepts two or three arguments;
if you want to map with the three-argument version, you will have to
type @kbd{V M 3 v e}.

It is also possible to apply any formula to a vector by treating that
formula as a function.  When prompted for the operator to use, press
@kbd{'} (the apostrophe) and type your formula as an algebraic entry.
You will then be prompted for the argument list, which defaults to a
list of all variables that appear in the formula, sorted into alphabetic
order.  For example, suppose you enter the formula @w{@samp{x + 2y^x}}.
The default argument list would be @samp{(x y)}, which means that if
this function is applied to the arguments @samp{[3, 10]} the result will
be @samp{3 + 2*10^3}.  (If you plan to use a certain formula in this
way often, you might consider defining it as a function with @kbd{Z F}.)

Another way to specify the arguments to the formula you enter is with
@kbd{$}, @kbd{$$}, and so on.  For example, @kbd{V A ' $$ + 2$^$$}
has the same effect as the previous example.  The argument list is
automatically taken to be @samp{($$ $)}.  (The order of the arguments
may seem backwards, but it is analogous to the way normal algebraic
entry interacts with the stack.)

If you press @kbd{$} at the operator prompt, the effect is similar to
the apostrophe except that the relevant formula is taken from top-of-stack
instead.  The actual vector arguments of the @kbd{V A $} or related command
then start at the second-to-top stack position.  You will still be
prompted for an argument list.

@cindex Nameless functions
@cindex Generic functions
A function can be written without a name using the notation @samp{<#1 - #2>},
which means ``a function of two arguments that computes the first
argument minus the second argument.''  The symbols @samp{#1} and @samp{#2}
are placeholders for the arguments.  You can use any names for these
placeholders if you wish, by including an argument list followed by a
colon:  @samp{<x, y : x - y>}.  When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
to map across the vectors.  When you type @kbd{V A ' x + 2y^x @key{RET} @key{RET}},
Calc builds the nameless function @w{@samp{<x, y : x + 2 y^x>}}.  In both
cases, Calc also writes the nameless function to the Trail so that you
can get it back later if you wish.

If there is only one argument, you can write @samp{#} in place of @samp{#1}.
(Note that @samp{< >} notation is also used for date forms.  Calc tells
that @samp{<@var{stuff}>} is a nameless function by the presence of
@samp{#} signs inside @var{stuff}, or by the fact that @var{stuff}
begins with a list of variables followed by a colon.)

You can type a nameless function directly to @kbd{V A '}, or put one on
the stack and use it with @w{@kbd{V A $}}.  Calc will not prompt for an
argument list in this case, since the nameless function specifies the
argument list as well as the function itself.  In @kbd{V A '}, you can
omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2> @key{RET}},
which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.

@cindex Lambda expressions
@ignore
@starindex
@end ignore
@tindex lambda
The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
(The word @code{lambda} derives from Lisp notation and the theory of
functions.)  The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
ArgB, ArgA + ArgB)}.  Note that there is no actual Calc function called
@code{lambda}; the whole point is that the @code{lambda} expression is
used in its symbolic form, not evaluated for an answer until it is applied
to specific arguments by a command like @kbd{V A} or @kbd{V M}.

(Actually, @code{lambda} does have one special property:  Its arguments
are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
will not simplify the @samp{2/3} until the nameless function is actually
called.)

@tindex add
@tindex sub
@ignore
@mindex @idots
@end ignore
@tindex mul
@ignore
@mindex @null
@end ignore
@tindex div
@ignore
@mindex @null
@end ignore
@tindex pow
@ignore
@mindex @null
@end ignore
@tindex neg
@ignore
@mindex @null
@end ignore
@tindex mod
@ignore
@mindex @null
@end ignore
@tindex vconcat
As usual, commands like @kbd{V A} have algebraic function name equivalents.
For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
@samp{apply(gcd, v)}.  The first argument specifies the operator name,
and is either a variable whose name is the same as the function name,
or a nameless function like @samp{<#^3+1>}.  Operators that are normally
written as algebraic symbols have the names @code{add}, @code{sub},
@code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
@code{vconcat}.

@ignore
@starindex
@end ignore
@tindex call
The @code{call} function builds a function call out of several arguments:
@samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
in turn is the same as @samp{gcd(x, y)}.  The first argument of @code{call},
like the other functions described here, may be either a variable naming a
function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
as @samp{x + 2y}).

(Experts will notice that it's not quite proper to use a variable to name
a function, since the name @code{gcd} corresponds to the Lisp variable
@code{var-gcd} but to the Lisp function @code{calcFunc-gcd}.  Calc
automatically makes this translation, so you don't have to worry
about it.)

@node Mapping, Reducing, Specifying Operators, Reducing and Mapping
@subsection Mapping

@noindent
@kindex V M
@pindex calc-map
@tindex map
The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given
operator elementwise to one or more vectors.  For example, mapping
@code{A} [@code{abs}] produces a vector of the absolute values of the
elements in the input vector.  Mapping @code{+} pops two vectors from
the stack, which must be of equal length, and produces a vector of the
pairwise sums of the elements.  If either argument is a non-vector, it
is duplicated for each element of the other vector.  For example,
@kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
With the 2 listed first, it would have computed a vector of powers of
two.  Mapping a user-defined function pops as many arguments from the
stack as the function requires.  If you give an undefined name, you will
be prompted for the number of arguments to use.

If any argument to @kbd{V M} is a matrix, the operator is normally mapped
across all elements of the matrix.  For example, given the matrix
@expr{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
produce another 
@texline @math{3\times2}
@infoline 3x2
matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.

@tindex mapr
The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the
operator prompt) maps by rows instead.  For example, @kbd{V M _ A} views
the above matrix as a vector of two 3-element row vectors.  It produces
a new vector which contains the absolute values of those row vectors,
namely @expr{[3.74, 8.77]}.  (Recall, the absolute value of a vector is
defined as the square root of the sum of the squares of the elements.)
Some operators accept vectors and return new vectors; for example,
@kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row
of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.

Sometimes a vector of vectors (representing, say, strings, sets, or lists)
happens to look like a matrix.  If so, remember to use @kbd{V M _} if you
want to map a function across the whole strings or sets rather than across
their individual elements.

@tindex mapc
The command @kbd{V M :} [@code{mapc}] maps by columns.  Basically, it
transposes the input matrix, maps by rows, and then, if the result is a
matrix, transposes again.  For example, @kbd{V M : A} takes the absolute
values of the three columns of the matrix, treating each as a 2-vector,
and @kbd{V M : v v} reverses the columns to get the matrix
@expr{[[-4, 5, -6], [1, -2, 3]]}.

(The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
and column-like appearances, and were not already taken by useful
operators.  Also, they appear shifted on most keyboards so they are easy
to type after @kbd{V M}.)

The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
not matrices (so if none of the arguments are matrices, they have no
effect at all).  If some of the arguments are matrices and others are
plain numbers, the plain numbers are held constant for all rows of the
matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
a vector takes a dot product of the vector with itself).

If some of the arguments are vectors with the same lengths as the
rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
arguments, those vectors are also held constant for every row or
column.

Sometimes it is useful to specify another mapping command as the operator
to use with @kbd{V M}.  For example, @kbd{V M _ V A +} applies @kbd{V A +}
to each row of the input matrix, which in turn adds the two values on that
row.  If you give another vector-operator command as the operator for
@kbd{V M}, it automatically uses map-by-rows mode if you don't specify
otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}.  (If
you really want to map-by-elements another mapping command, you can use
a triple-nested mapping command:  @kbd{V M V M V A +} means to map
@kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
mapped over the elements of each row.)

@tindex mapa
@tindex mapd
Previous versions of Calc had ``map across'' and ``map down'' modes
that are now considered obsolete; the old ``map across'' is now simply
@kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}.  The algebraic
functions @code{mapa} and @code{mapd} are still supported, though.
Note also that, while the old mapping modes were persistent (once you
set the mode, it would apply to later mapping commands until you reset
it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
mapping command.  The default @kbd{V M} always means map-by-elements.

@xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like
@kbd{V M} but for equations and inequalities instead of vectors.
@xref{Storing Variables}, for the @kbd{s m} command which modifies a
variable's stored value using a @kbd{V M}-like operator.

@node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
@subsection Reducing

@noindent
@kindex V R
@pindex calc-reduce
@tindex reduce
The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given
binary operator across all the elements of a vector.  A binary operator is
a function such as @code{+} or @code{max} which takes two arguments.  For
example, reducing @code{+} over a vector computes the sum of the elements
of the vector.  Reducing @code{-} computes the first element minus each of
the remaining elements.  Reducing @code{max} computes the maximum element
and so on.  In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
produces @samp{f(f(f(a, b), c), d)}.

@kindex I V R
@tindex rreduce
The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
that works from right to left through the vector.  For example, plain
@kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d}
but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
or @samp{a - b + c - d}.  This ``alternating sum'' occurs frequently
in power series expansions.

@kindex V U
@tindex accum
The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
accumulation operation.  Here Calc does the corresponding reduction
operation, but instead of producing only the final result, it produces
a vector of all the intermediate results.  Accumulating @code{+} over
the vector @samp{[a, b, c, d]} produces the vector
@samp{[a, a + b, a + b + c, a + b + c + d]}.

@kindex I V U
@tindex raccum
The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
vector @samp{[a - b + c - d, b - c + d, c - d, d]}.

@tindex reducea
@tindex rreducea
@tindex reduced
@tindex rreduced
As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise.  For
example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
compute @expr{a + b + c + d + e + f}.  You can type @kbd{V R _} or
@kbd{V R :} to modify this behavior.  The @kbd{V R _} [@code{reducea}]
command reduces ``across'' the matrix; it reduces each row of the matrix
as a vector, then collects the results.  Thus @kbd{V R _ +} of this
matrix would produce @expr{[a + b + c, d + e + f]}.  Similarly, @kbd{V R :}
[@code{reduced}] reduces down; @kbd{V R : +} would produce @expr{[a + d,
b + e, c + f]}.

@tindex reducer
@tindex rreducer
There is a third ``by rows'' mode for reduction that is occasionally
useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over
the rows of the matrix themselves.  Thus @kbd{V R = +} on the above
matrix would get the same result as @kbd{V R : +}, since adding two
row vectors is equivalent to adding their elements.  But @kbd{V R = *}
would multiply the two rows (to get a single number, their dot product),
while @kbd{V R : *} would produce a vector of the products of the columns.

These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
but they are not currently supported with @kbd{V U} or @kbd{I V U}.

@tindex reducec
@tindex rreducec
The obsolete reduce-by-columns function, @code{reducec}, is still
supported but there is no way to get it through the @kbd{V R} command.

The commands @kbd{C-x * :} and @kbd{C-x * _} are equivalent to typing
@kbd{C-x * r} to grab a rectangle of data into Calc, and then typing
@kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or
rows of the matrix.  @xref{Grabbing From Buffers}.

@node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping
@subsection Nesting and Fixed Points

@noindent
@kindex H V R
@tindex nest
The @kbd{H V R} [@code{nest}] command applies a function to a given
argument repeatedly.  It takes two values, @samp{a} and @samp{n}, from
the stack, where @samp{n} must be an integer.  It then applies the
function nested @samp{n} times; if the function is @samp{f} and @samp{n}
is 3, the result is @samp{f(f(f(a)))}.  The number @samp{n} may be
negative if Calc knows an inverse for the function @samp{f}; for
example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.

@kindex H V U
@tindex anest
The @kbd{H V U} [@code{anest}] command is an accumulating version of
@code{nest}:  It returns a vector of @samp{n+1} values, e.g.,
@samp{[a, f(a), f(f(a)), f(f(f(a)))]}.  If @samp{n} is negative and
@samp{F} is the inverse of @samp{f}, then the result is of the
form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.

@kindex H I V R
@tindex fixp
@cindex Fixed points
The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except
that it takes only an @samp{a} value from the stack; the function is
applied until it reaches a ``fixed point,'' i.e., until the result
no longer changes.

@kindex H I V U
@tindex afixp
The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
The first element of the return vector will be the initial value @samp{a};
the last element will be the final result that would have been returned
by @code{fixp}.

For example, 0.739085 is a fixed point of the cosine function (in radians):
@samp{cos(0.739085) = 0.739085}.  You can find this value by putting, say,
1.0 on the stack and typing @kbd{H I V U C}.  (We use the accumulating
version so we can see the intermediate results:  @samp{[1, 0.540302, 0.857553,
0.65329, ...]}.  With a precision of six, this command will take 36 steps
to converge to 0.739085.)

Newton's method for finding roots is a classic example of iteration
to a fixed point.  To find the square root of five starting with an
initial guess, Newton's method would look for a fixed point of the
function @samp{(x + 5/x) / 2}.  Putting a guess of 1 on the stack
and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
2.23607.  This is equivalent to using the @kbd{a R} (@code{calc-find-root})
command to find a root of the equation @samp{x^2 = 5}.

These examples used numbers for @samp{a} values.  Calc keeps applying
the function until two successive results are equal to within the
current precision.  For complex numbers, both the real parts and the
imaginary parts must be equal to within the current precision.  If
@samp{a} is a formula (say, a variable name), then the function is
applied until two successive results are exactly the same formula.
It is up to you to ensure that the function will eventually converge;
if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.

The algebraic @code{fixp} function takes two optional arguments, @samp{n}
and @samp{tol}.  The first is the maximum number of steps to be allowed,
and must be either an integer or the symbol @samp{inf} (infinity, the
default).  The second is a convergence tolerance.  If a tolerance is
specified, all results during the calculation must be numbers, not
formulas, and the iteration stops when the magnitude of the difference
between two successive results is less than or equal to the tolerance.
(This implies that a tolerance of zero iterates until the results are
exactly equal.)

Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
computes the square root of @samp{A} given the initial guess @samp{B},
stopping when the result is correct within the specified tolerance, or
when 20 steps have been taken, whichever is sooner.

@node Generalized Products,  , Nesting and Fixed Points, Reducing and Mapping
@subsection Generalized Products

@kindex V O
@pindex calc-outer-product
@tindex outer
The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies
a given binary operator to all possible pairs of elements from two
vectors, to produce a matrix.  For example, @kbd{V O *} with @samp{[a, b]}
and @samp{[x, y, z]} on the stack produces a multiplication table:
@samp{[[a x, a y, a z], [b x, b y, b z]]}.  Element @var{r},@var{c} of
the result matrix is obtained by applying the operator to element @var{r}
of the lefthand vector and element @var{c} of the righthand vector.

@kindex V I
@pindex calc-inner-product
@tindex inner
The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes
the generalized inner product of two vectors or matrices, given a
``multiplicative'' operator and an ``additive'' operator.  These can each
actually be any binary operators; if they are @samp{*} and @samp{+},
respectively, the result is a standard matrix multiplication.  Element
@var{r},@var{c} of the result matrix is obtained by mapping the
multiplicative operator across row @var{r} of the lefthand matrix and
column @var{c} of the righthand matrix, and then reducing with the additive
operator.  Just as for the standard @kbd{*} command, this can also do a
vector-matrix or matrix-vector inner product, or a vector-vector
generalized dot product.

Since @kbd{V I} requires two operators, it prompts twice.  In each case,
you can use any of the usual methods for entering the operator.  If you
use @kbd{$} twice to take both operator formulas from the stack, the
first (multiplicative) operator is taken from the top of the stack
and the second (additive) operator is taken from second-to-top.

@node Vector and Matrix Formats,  , Reducing and Mapping, Matrix Functions
@section Vector and Matrix Display Formats

@noindent
Commands for controlling vector and matrix display use the @kbd{v} prefix
instead of the usual @kbd{d} prefix.  But they are display modes; in
particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
in the same way (@pxref{Display Modes}).  Matrix display is also
influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
@pxref{Normal Language Modes}.

@kindex V <
@pindex calc-matrix-left-justify
@kindex V =
@pindex calc-matrix-center-justify
@kindex V >
@pindex calc-matrix-right-justify
The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
(@code{calc-matrix-right-justify}), and @w{@kbd{v =}}
(@code{calc-matrix-center-justify}) control whether matrix elements
are justified to the left, right, or center of their columns.

@kindex V [
@pindex calc-vector-brackets
@kindex V @{
@pindex calc-vector-braces
@kindex V (
@pindex calc-vector-parens
The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
brackets that surround vectors and matrices displayed in the stack on
and off.  The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
(@code{calc-vector-parens}) commands use curly braces or parentheses,
respectively, instead of square brackets.  For example, @kbd{v @{} might
be used in preparation for yanking a matrix into a buffer running
Mathematica.  (In fact, the Mathematica language mode uses this mode;
@pxref{Mathematica Language Mode}.)  Note that, regardless of the
display mode, either brackets or braces may be used to enter vectors,
and parentheses may never be used for this purpose.

@kindex V ]
@pindex calc-matrix-brackets
The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
``big'' style display of matrices.  It prompts for a string of code
letters; currently implemented letters are @code{R}, which enables
brackets on each row of the matrix; @code{O}, which enables outer
brackets in opposite corners of the matrix; and @code{C}, which
enables commas or semicolons at the ends of all rows but the last.
The default format is @samp{RO}.  (Before Calc 2.00, the format
was fixed at @samp{ROC}.)  Here are some example matrices:

@example
@group
[ [ 123,  0,   0  ]       [ [ 123,  0,   0  ],
  [  0,  123,  0  ]         [  0,  123,  0  ],
  [  0,   0,  123 ] ]       [  0,   0,  123 ] ]

         RO                        ROC

@end group
@end example
@noindent
@example
@group
  [ 123,  0,   0            [ 123,  0,   0 ;
     0,  123,  0               0,  123,  0 ;
     0,   0,  123 ]            0,   0,  123 ]

          O                        OC

@end group
@end example
@noindent
@example
@group
  [ 123,  0,   0  ]           123,  0,   0
  [  0,  123,  0  ]            0,  123,  0
  [  0,   0,  123 ]            0,   0,  123

          R                       @r{blank}
@end group
@end example

@noindent
Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
@samp{OC} are all recognized as matrices during reading, while
the others are useful for display only.

@kindex V ,
@pindex calc-vector-commas
The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
off in vector and matrix display.

In vectors of length one, and in all vectors when commas have been
turned off, Calc adds extra parentheses around formulas that might
otherwise be ambiguous.  For example, @samp{[a b]} could be a vector
of the one formula @samp{a b}, or it could be a vector of two
variables with commas turned off.  Calc will display the former
case as @samp{[(a b)]}.  You can disable these extra parentheses
(to make the output less cluttered at the expense of allowing some
ambiguity) by adding the letter @code{P} to the control string you
give to @kbd{v ]} (as described above).

@kindex V .
@pindex calc-full-vectors
The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
display of long vectors on and off.  In this mode, vectors of six
or more elements, or matrices of six or more rows or columns, will
be displayed in an abbreviated form that displays only the first
three elements and the last element:  @samp{[a, b, c, ..., z]}.
When very large vectors are involved this will substantially
improve Calc's display speed.

@kindex t .
@pindex calc-full-trail-vectors
The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
similar mode for recording vectors in the Trail.  If you turn on
this mode, vectors of six or more elements and matrices of six or
more rows or columns will be abbreviated when they are put in the
Trail.  The @kbd{t y} (@code{calc-trail-yank}) command will be
unable to recover those vectors.  If you are working with very
large vectors, this mode will improve the speed of all operations
that involve the trail.

@kindex V /
@pindex calc-break-vectors
The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
vector display on and off.  Normally, matrices are displayed with one
row per line but all other types of vectors are displayed in a single
line.  This mode causes all vectors, whether matrices or not, to be
displayed with a single element per line.  Sub-vectors within the
vectors will still use the normal linear form.

@node Algebra, Units, Matrix Functions, Top
@chapter Algebra

@noindent
This section covers the Calc features that help you work with
algebraic formulas.  First, the general sub-formula selection
mechanism is described; this works in conjunction with any Calc
commands.  Then, commands for specific algebraic operations are
described.  Finally, the flexible @dfn{rewrite rule} mechanism
is discussed.

The algebraic commands use the @kbd{a} key prefix; selection
commands use the @kbd{j} (for ``just a letter that wasn't used
for anything else'') prefix.

@xref{Editing Stack Entries}, to see how to manipulate formulas
using regular Emacs editing commands.

When doing algebraic work, you may find several of the Calculator's
modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
or No-Simplification mode (@kbd{m O}),
Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
Symbolic mode (@kbd{m s}).  @xref{Mode Settings}, for discussions
of these modes.  You may also wish to select Big display mode (@kbd{d B}).
@xref{Normal Language Modes}.

@menu
* Selecting Subformulas::
* Algebraic Manipulation::
* Simplifying Formulas::
* Polynomials::
* Calculus::
* Solving Equations::
* Numerical Solutions::
* Curve Fitting::
* Summations::
* Logical Operations::
* Rewrite Rules::
@end menu

@node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
@section Selecting Sub-Formulas

@noindent
@cindex Selections
@cindex Sub-formulas
@cindex Parts of formulas
When working with an algebraic formula it is often necessary to
manipulate a portion of the formula rather than the formula as a
whole.  Calc allows you to ``select'' a portion of any formula on
the stack.  Commands which would normally operate on that stack
entry will now operate only on the sub-formula, leaving the
surrounding part of the stack entry alone.

One common non-algebraic use for selection involves vectors.  To work
on one element of a vector in-place, simply select that element as a
``sub-formula'' of the vector.

@menu
* Making Selections::
* Changing Selections::
* Displaying Selections::
* Operating on Selections::
* Rearranging with Selections::
@end menu

@node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas
@subsection Making Selections

@noindent
@kindex j s
@pindex calc-select-here
To select a sub-formula, move the Emacs cursor to any character in that
sub-formula, and press @w{@kbd{j s}} (@code{calc-select-here}).  Calc will
highlight the smallest portion of the formula that contains that
character.  By default the sub-formula is highlighted by blanking out
all of the rest of the formula with dots.  Selection works in any
display mode but is perhaps easiest in Big mode (@kbd{d B}).
Suppose you enter the following formula:

@smallexample
@group
           3    ___
    (a + b)  + V c
1:  ---------------
        2 x + 1
@end group
@end smallexample

@noindent
(by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}).  If you move the
cursor to the letter @samp{b} and press @w{@kbd{j s}}, the display changes
to

@smallexample
@group
           .    ...
    .. . b.  . . .
1*  ...............
        . . . .
@end group
@end smallexample

@noindent
Every character not part of the sub-formula @samp{b} has been changed
to a dot.  The @samp{*} next to the line number is to remind you that
the formula has a portion of it selected.  (In this case, it's very
obvious, but it might not always be.  If Embedded mode is enabled,
the word @samp{Sel} also appears in the mode line because the stack
may not be visible.  @pxref{Embedded Mode}.)

If you had instead placed the cursor on the parenthesis immediately to
the right of the @samp{b}, the selection would have been:

@smallexample
@group
           .    ...
    (a + b)  . . .
1*  ...............
        . . . .
@end group
@end smallexample

@noindent
The portion selected is always large enough to be considered a complete
formula all by itself, so selecting the parenthesis selects the whole
formula that it encloses.  Putting the cursor on the @samp{+} sign
would have had the same effect.

(Strictly speaking, the Emacs cursor is really the manifestation of
the Emacs ``point,'' which is a position @emph{between} two characters
in the buffer.  So purists would say that Calc selects the smallest
sub-formula which contains the character to the right of ``point.'')

If you supply a numeric prefix argument @var{n}, the selection is
expanded to the @var{n}th enclosing sub-formula.  Thus, positioning
the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
@samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
and so on.

If the cursor is not on any part of the formula, or if you give a
numeric prefix that is too large, the entire formula is selected.

If the cursor is on the @samp{.} line that marks the top of the stack
(i.e., its normal ``rest position''), this command selects the entire
formula at stack level 1.  Most selection commands similarly operate
on the formula at the top of the stack if you haven't positioned the
cursor on any stack entry.

@kindex j a
@pindex calc-select-additional
The @kbd{j a} (@code{calc-select-additional}) command enlarges the
current selection to encompass the cursor.  To select the smallest
sub-formula defined by two different points, move to the first and
press @kbd{j s}, then move to the other and press @kbd{j a}.  This
is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
select the two ends of a region of text during normal Emacs editing.

@kindex j o
@pindex calc-select-once
The @kbd{j o} (@code{calc-select-once}) command selects a formula in
exactly the same way as @kbd{j s}, except that the selection will
last only as long as the next command that uses it.  For example,
@kbd{j o 1 +} is a handy way to add one to the sub-formula indicated
by the cursor.

(A somewhat more precise definition: The @kbd{j o} command sets a flag
such that the next command involving selected stack entries will clear
the selections on those stack entries afterwards.  All other selection
commands except @kbd{j a} and @kbd{j O} clear this flag.)

@kindex j S
@kindex j O
@pindex calc-select-here-maybe
@pindex calc-select-once-maybe
The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
(@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
and @kbd{j o}, respectively, except that if the formula already
has a selection they have no effect.  This is analogous to the
behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
@pxref{Selections with Rewrite Rules}) and is mainly intended to be
used in keyboard macros that implement your own selection-oriented
commands.

Selection of sub-formulas normally treats associative terms like
@samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula.
If you place the cursor anywhere inside @samp{a + b - c + d} except
on one of the variable names and use @kbd{j s}, you will select the
entire four-term sum.

@kindex j b
@pindex calc-break-selections
The @kbd{j b} (@code{calc-break-selections}) command controls a mode
in which the ``deep structure'' of these associative formulas shows
through.  Calc actually stores the above formulas as @samp{((a + b) - c) + d}
and @samp{x * (y * z)}.  (Note that for certain obscure reasons, Calc
treats multiplication as right-associative.)  Once you have enabled
@kbd{j b} mode, selecting with the cursor on the @samp{-} sign would
only select the @samp{a + b - c} portion, which makes sense when the
deep structure of the sum is considered.  There is no way to select
the @samp{b - c + d} portion; although this might initially look
like just as legitimate a sub-formula as @samp{a + b - c}, the deep
structure shows that it isn't.  The @kbd{d U} command can be used
to view the deep structure of any formula (@pxref{Normal Language Modes}).

When @kbd{j b} mode has not been enabled, the deep structure is
generally hidden by the selection commands---what you see is what
you get.

@kindex j u
@pindex calc-unselect
The @kbd{j u} (@code{calc-unselect}) command unselects the formula
that the cursor is on.  If there was no selection in the formula,
this command has no effect.  With a numeric prefix argument, it
unselects the @var{n}th stack element rather than using the cursor
position.

@kindex j c
@pindex calc-clear-selections
The @kbd{j c} (@code{calc-clear-selections}) command unselects all
stack elements.

@node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas
@subsection Changing Selections

@noindent
@kindex j m
@pindex calc-select-more
Once you have selected a sub-formula, you can expand it using the
@w{@kbd{j m}} (@code{calc-select-more}) command.  If @samp{a + b} is
selected, pressing @w{@kbd{j m}} repeatedly works as follows:

@smallexample
@group
           3    ...                3    ___                3    ___
    (a + b)  . . .          (a + b)  + V c          (a + b)  + V c
1*  ...............     1*  ...............     1*  ---------------
        . . . .                 . . . .                 2 x + 1
@end group
@end smallexample

@noindent
In the last example, the entire formula is selected.  This is roughly
the same as having no selection at all, but because there are subtle
differences the @samp{*} character is still there on the line number.

With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
times (or until the entire formula is selected).  Note that @kbd{j s}
with argument @var{n} is equivalent to plain @kbd{j s} followed by
@kbd{j m} with argument @var{n}.  If @w{@kbd{j m}} is used when there
is no current selection, it is equivalent to @w{@kbd{j s}}.

Even though @kbd{j m} does not explicitly use the location of the
cursor within the formula, it nevertheless uses the cursor to determine
which stack element to operate on.  As usual, @kbd{j m} when the cursor
is not on any stack element operates on the top stack element.

@kindex j l
@pindex calc-select-less
The @kbd{j l} (@code{calc-select-less}) command reduces the current
selection around the cursor position.  That is, it selects the
immediate sub-formula of the current selection which contains the
cursor, the opposite of @kbd{j m}.  If the cursor is not inside the
current selection, the command de-selects the formula.

@kindex j 1-9
@pindex calc-select-part
The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
select the @var{n}th sub-formula of the current selection.  They are
like @kbd{j l} (@code{calc-select-less}) except they use counting
rather than the cursor position to decide which sub-formula to select.
For example, if the current selection is @kbd{a + b + c} or
@kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
@kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
these cases, @kbd{j 4} through @kbd{j 9} would be errors.

If there is no current selection, @kbd{j 1} through @kbd{j 9} select
the @var{n}th top-level sub-formula.  (In other words, they act as if
the entire stack entry were selected first.)  To select the @var{n}th
sub-formula where @var{n} is greater than nine, you must instead invoke
@w{@kbd{j 1}} with @var{n} as a numeric prefix argument.

@kindex j n
@kindex j p
@pindex calc-select-next
@pindex calc-select-previous
The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
(@code{calc-select-previous}) commands change the current selection
to the next or previous sub-formula at the same level.  For example,
if @samp{b} is selected in @w{@samp{2 + a*b*c + x}}, then @kbd{j n}
selects @samp{c}.  Further @kbd{j n} commands would be in error because,
even though there is something to the right of @samp{c} (namely, @samp{x}),
it is not at the same level; in this case, it is not a term of the
same product as @samp{b} and @samp{c}.  However, @kbd{j m} (to select
the whole product @samp{a*b*c} as a term of the sum) followed by
@w{@kbd{j n}} would successfully select the @samp{x}.

Similarly, @kbd{j p} moves the selection from the @samp{b} in this
sample formula to the @samp{a}.  Both commands accept numeric prefix
arguments to move several steps at a time.

It is interesting to compare Calc's selection commands with the
Emacs Info system's commands for navigating through hierarchically
organized documentation.  Calc's @kbd{j n} command is completely
analogous to Info's @kbd{n} command.  Likewise, @kbd{j p} maps to
@kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
(Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
@kbd{j l}; in each case, you can jump directly to a sub-component
of the hierarchy simply by pointing to it with the cursor.

@node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas
@subsection Displaying Selections

@noindent
@kindex j d
@pindex calc-show-selections
The @kbd{j d} (@code{calc-show-selections}) command controls how
selected sub-formulas are displayed.  One of the alternatives is
illustrated in the above examples; if we press @kbd{j d} we switch
to the other style in which the selected portion itself is obscured
by @samp{#} signs:

@smallexample
@group
           3    ...                  #    ___
    (a + b)  . . .            ## # ##  + V c
1*  ...............       1*  ---------------
        . . . .                   2 x + 1
@end group
@end smallexample

@node Operating on Selections, Rearranging with Selections, Displaying Selections, Selecting Subformulas
@subsection Operating on Selections

@noindent
Once a selection is made, all Calc commands that manipulate items
on the stack will operate on the selected portions of the items
instead.  (Note that several stack elements may have selections
at once, though there can be only one selection at a time in any
given stack element.)

@kindex j e
@pindex calc-enable-selections
The @kbd{j e} (@code{calc-enable-selections}) command disables the
effect that selections have on Calc commands.  The current selections
still exist, but Calc commands operate on whole stack elements anyway.
This mode can be identified by the fact that the @samp{*} markers on
the line numbers are gone, even though selections are visible.  To
reactivate the selections, press @kbd{j e} again.

To extract a sub-formula as a new formula, simply select the
sub-formula and press @key{RET}.  This normally duplicates the top
stack element; here it duplicates only the selected portion of that
element.

To replace a sub-formula with something different, you can enter the
new value onto the stack and press @key{TAB}.  This normally exchanges
the top two stack elements; here it swaps the value you entered into
the selected portion of the formula, returning the old selected
portion to the top of the stack.

@smallexample
@group
           3    ...                    ...                    ___
    (a + b)  . . .           17 x y . . .           17 x y + V c
2*  ...............      2*  .............      2:  -------------
        . . . .                 . . . .                2 x + 1

                                    3                      3
1:  17 x y               1:  (a + b)            1:  (a + b)
@end group
@end smallexample

In this example we select a sub-formula of our original example,
enter a new formula, @key{TAB} it into place, then deselect to see
the complete, edited formula.

If you want to swap whole formulas around even though they contain
selections, just use @kbd{j e} before and after.

@kindex j '
@pindex calc-enter-selection
The @kbd{j '} (@code{calc-enter-selection}) command is another way
to replace a selected sub-formula.  This command does an algebraic
entry just like the regular @kbd{'} key.  When you press @key{RET},
the formula you type replaces the original selection.  You can use
the @samp{$} symbol in the formula to refer to the original
selection.  If there is no selection in the formula under the cursor,
the cursor is used to make a temporary selection for the purposes of
the command.  Thus, to change a term of a formula, all you have to
do is move the Emacs cursor to that term and press @kbd{j '}.

@kindex j `
@pindex calc-edit-selection
The @kbd{j `} (@code{calc-edit-selection}) command is a similar
analogue of the @kbd{`} (@code{calc-edit}) command.  It edits the
selected sub-formula in a separate buffer.  If there is no
selection, it edits the sub-formula indicated by the cursor.

To delete a sub-formula, press @key{DEL}.  This generally replaces
the sub-formula with the constant zero, but in a few suitable contexts
it uses the constant one instead.  The @key{DEL} key automatically
deselects and re-simplifies the entire formula afterwards.  Thus:

@smallexample
@group
              ###
    17 x y + # #          17 x y         17 # y          17 y
1*  -------------     1:  -------    1*  -------    1:  -------
       2 x + 1            2 x + 1        2 x + 1        2 x + 1
@end group
@end smallexample

In this example, we first delete the @samp{sqrt(c)} term; Calc
accomplishes this by replacing @samp{sqrt(c)} with zero and
resimplifying.  We then delete the @kbd{x} in the numerator;
since this is part of a product, Calc replaces it with @samp{1}
and resimplifies.

If you select an element of a vector and press @key{DEL}, that
element is deleted from the vector.  If you delete one side of
an equation or inequality, only the opposite side remains.

@kindex j @key{DEL}
@pindex calc-del-selection
The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
@key{DEL} but with the auto-selecting behavior of @kbd{j '} and
@kbd{j `}.  It deletes the selected portion of the formula
indicated by the cursor, or, in the absence of a selection, it
deletes the sub-formula indicated by the cursor position.

@kindex j @key{RET}
@pindex calc-grab-selection
(There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
command.)

Normal arithmetic operations also apply to sub-formulas.  Here we
select the denominator, press @kbd{5 -} to subtract five from the
denominator, press @kbd{n} to negate the denominator, then
press @kbd{Q} to take the square root.

@smallexample
@group
     .. .           .. .           .. .             .. .
1*  .......    1*  .......    1*  .......    1*  ..........
    2 x + 1        2 x - 4        4 - 2 x         _________
                                                 V 4 - 2 x
@end group
@end smallexample

Certain types of operations on selections are not allowed.  For
example, for an arithmetic function like @kbd{-} no more than one of
the arguments may be a selected sub-formula.  (As the above example
shows, the result of the subtraction is spliced back into the argument
which had the selection; if there were more than one selection involved,
this would not be well-defined.)  If you try to subtract two selections,
the command will abort with an error message.

Operations on sub-formulas sometimes leave the formula as a whole
in an ``un-natural'' state.  Consider negating the @samp{2 x} term
of our sample formula by selecting it and pressing @kbd{n}
(@code{calc-change-sign}).

@smallexample
@group
       .. .                .. .
1*  ..........      1*  ...........
     .........           ..........
    . . . 2 x           . . . -2 x
@end group
@end smallexample

Unselecting the sub-formula reveals that the minus sign, which would
normally have cancelled out with the subtraction automatically, has
not been able to do so because the subtraction was not part of the
selected portion.  Pressing @kbd{=} (@code{calc-evaluate}) or doing
any other mathematical operation on the whole formula will cause it
to be simplified.

@smallexample
@group
       17 y                17 y
1:  -----------     1:  ----------
     __________          _________
    V 4 - -2 x          V 4 + 2 x
@end group
@end smallexample

@node Rearranging with Selections,  , Operating on Selections, Selecting Subformulas
@subsection Rearranging Formulas using Selections

@noindent
@kindex j R
@pindex calc-commute-right
The @kbd{j R} (@code{calc-commute-right}) command moves the selected
sub-formula to the right in its surrounding formula.  Generally the
selection is one term of a sum or product; the sum or product is
rearranged according to the commutative laws of algebra.

As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used
if there is no selection in the current formula.  All commands described
in this section share this property.  In this example, we place the
cursor on the @samp{a} and type @kbd{j R}, then repeat.

@smallexample
1:  a + b - c          1:  b + a - c          1:  b - c + a
@end smallexample

@noindent
Note that in the final step above, the @samp{a} is switched with
the @samp{c} but the signs are adjusted accordingly.  When moving
terms of sums and products, @kbd{j R} will never change the
mathematical meaning of the formula.

The selected term may also be an element of a vector or an argument
of a function.  The term is exchanged with the one to its right.
In this case, the ``meaning'' of the vector or function may of
course be drastically changed.

@smallexample
1:  [a, b, c]          1:  [b, a, c]          1:  [b, c, a]

1:  f(a, b, c)         1:  f(b, a, c)         1:  f(b, c, a)
@end smallexample

@kindex j L
@pindex calc-commute-left
The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R}
except that it swaps the selected term with the one to its left.

With numeric prefix arguments, these commands move the selected
term several steps at a time.  It is an error to try to move a
term left or right past the end of its enclosing formula.
With numeric prefix arguments of zero, these commands move the
selected term as far as possible in the given direction.

@kindex j D
@pindex calc-sel-distribute
The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected
sum or product into the surrounding formula using the distributive
law.  For example, in @samp{a * (b - c)} with the @samp{b - c}
selected, the result is @samp{a b - a c}.  This also distributes
products or quotients into surrounding powers, and can also do
transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)},
where @samp{a + b} is the selected term, and @samp{ln(a ^ b)}
to @samp{ln(a) b}, where @samp{a ^ b} is the selected term.

For multiple-term sums or products, @kbd{j D} takes off one term
at a time:  @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b}
with the @samp{c - d} selected so that you can type @kbd{j D}
repeatedly to expand completely.  The @kbd{j D} command allows a
numeric prefix argument which specifies the maximum number of
times to expand at once; the default is one time only.

@vindex DistribRules
The @kbd{j D} command is implemented using rewrite rules.
@xref{Selections with Rewrite Rules}.  The rules are stored in
the Calc variable @code{DistribRules}.  A convenient way to view
these rules is to use @kbd{s e} (@code{calc-edit-variable}) which
displays and edits the stored value of a variable.  Press @kbd{C-c C-c}
to return from editing mode; be careful not to make any actual changes
or else you will affect the behavior of future @kbd{j D} commands!

To extend @kbd{j D} to handle new cases, just edit @code{DistribRules}
as described above.  You can then use the @kbd{s p} command to save
this variable's value permanently for future Calc sessions.
@xref{Operations on Variables}.

@kindex j M
@pindex calc-sel-merge
@vindex MergeRules
The @kbd{j M} (@code{calc-sel-merge}) command is the complement
of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or
@samp{a c} selected, the result is @samp{a * (b - c)}.  Once
again, @kbd{j M} can also merge calls to functions like @code{exp}
and @code{ln}; examine the variable @code{MergeRules} to see all
the relevant rules.

@kindex j C
@pindex calc-sel-commute
@vindex CommuteRules
The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments
of the selected sum, product, or equation.  It always behaves as
if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is
treated as the nested sums @samp{(a + b) + c} by this command.
If you put the cursor on the first @samp{+}, the result is
@samp{(b + a) + c}; if you put the cursor on the second @samp{+}, the
result is @samp{c + (a + b)} (which the default simplifications
will rearrange to @samp{(c + a) + b}).  The relevant rules are stored
in the variable @code{CommuteRules}.

You may need to turn default simplifications off (with the @kbd{m O}
command) in order to get the full benefit of @kbd{j C}.  For example,
commuting @samp{a - b} produces @samp{-b + a}, but the default
simplifications will ``simplify'' this right back to @samp{a - b} if
you don't turn them off.  The same is true of some of the other
manipulations described in this section.

@kindex j N
@pindex calc-sel-negate
@vindex NegateRules
The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected
term with the negative of that term, then adjusts the surrounding
formula in order to preserve the meaning.  For example, given
@samp{exp(a - b)} where @samp{a - b} is selected, the result is
@samp{1 / exp(b - a)}.  By contrast, selecting a term and using the
regular @kbd{n} (@code{calc-change-sign}) command negates the
term without adjusting the surroundings, thus changing the meaning
of the formula as a whole.  The rules variable is @code{NegateRules}.

@kindex j &
@pindex calc-sel-invert
@vindex InvertRules
The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N}
except it takes the reciprocal of the selected term.  For example,
given @samp{a - ln(b)} with @samp{b} selected, the result is
@samp{a + ln(1/b)}.  The rules variable is @code{InvertRules}.

@kindex j E
@pindex calc-sel-jump-equals
@vindex JumpRules
The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the
selected term from one side of an equation to the other.  Given
@samp{a + b = c + d} with @samp{c} selected, the result is
@samp{a + b - c = d}.  This command also works if the selected
term is part of a @samp{*}, @samp{/}, or @samp{^} formula.  The
relevant rules variable is @code{JumpRules}.

@kindex j I
@kindex H j I
@pindex calc-sel-isolate
The @kbd{j I} (@code{calc-sel-isolate}) command isolates the
selected term on its side of an equation.  It uses the @kbd{a S}
(@code{calc-solve-for}) command to solve the equation, and the
Hyperbolic flag affects it in the same way.  @xref{Solving Equations}.
When it applies, @kbd{j I} is often easier to use than @kbd{j E}.
It understands more rules of algebra, and works for inequalities
as well as equations.

@kindex j *
@kindex j /
@pindex calc-sel-mult-both-sides
@pindex calc-sel-div-both-sides
The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a
formula using algebraic entry, then multiplies both sides of the
selected quotient or equation by that formula.  It simplifies each
side with @kbd{a s} (@code{calc-simplify}) before re-forming the
quotient or equation.  You can suppress this simplification by
providing any numeric prefix argument.  There is also a @kbd{j /}
(@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but
dividing instead of multiplying by the factor you enter.

As a special feature, if the numerator of the quotient is 1, then
the denominator is expanded at the top level using the distributive
law (i.e., using the @kbd{C-u -1 a x} command).  Suppose the
formula on the stack is @samp{1 / (sqrt(a) + 1)}, and you wish
to eliminate the square root in the denominator by multiplying both
sides by @samp{sqrt(a) - 1}.  Calc's default simplifications would
change the result @samp{(sqrt(a) - 1) / (sqrt(a) - 1) (sqrt(a) + 1)}
right back to the original form by cancellation; Calc expands the
denominator to @samp{sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1} to prevent
this.  (You would now want to use an @kbd{a x} command to expand
the rest of the way, whereupon the denominator would cancel out to
the desired form, @samp{a - 1}.)  When the numerator is not 1, this
initial expansion is not necessary because Calc's default
simplifications will not notice the potential cancellation.

If the selection is an inequality, @kbd{j *} and @kbd{j /} will
accept any factor, but will warn unless they can prove the factor
is either positive or negative.  (In the latter case the direction
of the inequality will be switched appropriately.)  @xref{Declarations},
for ways to inform Calc that a given variable is positive or
negative.  If Calc can't tell for sure what the sign of the factor
will be, it will assume it is positive and display a warning
message.

For selections that are not quotients, equations, or inequalities,
these commands pull out a multiplicative factor:  They divide (or
multiply) by the entered formula, simplify, then multiply (or divide)
back by the formula.

@kindex j +
@kindex j -
@pindex calc-sel-add-both-sides
@pindex calc-sel-sub-both-sides
The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
(@code{calc-sel-sub-both-sides}) commands analogously add to or
subtract from both sides of an equation or inequality.  For other
types of selections, they extract an additive factor.  A numeric
prefix argument suppresses simplification of the intermediate
results.

@kindex j U
@pindex calc-sel-unpack
The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
selected function call with its argument.  For example, given
@samp{a + sin(x^2)} with @samp{sin(x^2)} selected, the result
is @samp{a + x^2}.  (The @samp{x^2} will remain selected; if you
wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
now to take the cosine of the selected part.)

@kindex j v
@pindex calc-sel-evaluate
The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
normal default simplifications on the selected sub-formula.
These are the simplifications that are normally done automatically
on all results, but which may have been partially inhibited by
previous selection-related operations, or turned off altogether
by the @kbd{m O} command.  This command is just an auto-selecting
version of the @w{@kbd{a v}} command (@pxref{Algebraic Manipulation}).

With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
the @kbd{a s} (@code{calc-simplify}) command to the selected
sub-formula.  With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
applies the @kbd{a e} (@code{calc-simplify-extended}) command.
@xref{Simplifying Formulas}.  With a negative prefix argument
it simplifies at the top level only, just as with @kbd{a v}.
Here the ``top'' level refers to the top level of the selected
sub-formula.

@kindex j "
@pindex calc-sel-expand-formula
The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
(@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.

You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
to define other algebraic operations on sub-formulas.  @xref{Rewrite Rules}.

@node Algebraic Manipulation, Simplifying Formulas, Selecting Subformulas, Algebra
@section Algebraic Manipulation

@noindent
The commands in this section perform general-purpose algebraic
manipulations.  They work on the whole formula at the top of the
stack (unless, of course, you have made a selection in that
formula).

Many algebra commands prompt for a variable name or formula.  If you
answer the prompt with a blank line, the variable or formula is taken
from top-of-stack, and the normal argument for the command is taken
from the second-to-top stack level.

@kindex a v
@pindex calc-alg-evaluate
The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
default simplifications on a formula; for example, @samp{a - -b} is
changed to @samp{a + b}.  These simplifications are normally done
automatically on all Calc results, so this command is useful only if
you have turned default simplifications off with an @kbd{m O}
command.  @xref{Simplification Modes}.

It is often more convenient to type @kbd{=}, which is like @kbd{a v}
but which also substitutes stored values for variables in the formula.
Use @kbd{a v} if you want the variables to ignore their stored values.

If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
as if in Algebraic Simplification mode.  This is equivalent to typing
@kbd{a s}; @pxref{Simplifying Formulas}.  If you give a numeric prefix
of 3 or more, it uses Extended Simplification mode (@kbd{a e}).

If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or @mathit{-3},
it simplifies in the corresponding mode but only works on the top-level
function call of the formula.  For example, @samp{(2 + 3) * (2 + 3)} will
simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
@samp{2 + 3}.  As another example, typing @kbd{V R +} to sum the vector
@samp{[1, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
in No-Simplify mode.  Using @kbd{a v} will evaluate this all the way to
10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
(@xref{Reducing and Mapping}.)

@tindex evalv
@tindex evalvn
The @kbd{=} command corresponds to the @code{evalv} function, and
the related @kbd{N} command, which is like @kbd{=} but temporarily
disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
to the @code{evalvn} function.  (These commands interpret their prefix
arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
the number of stack elements to evaluate at once, and @kbd{N} treats
it as a temporary different working precision.)

The @code{evalvn} function can take an alternate working precision
as an optional second argument.  This argument can be either an
integer, to set the precision absolutely, or a vector containing
a single integer, to adjust the precision relative to the current
precision.  Note that @code{evalvn} with a larger than current
precision will do the calculation at this higher precision, but the
result will as usual be rounded back down to the current precision
afterward.  For example, @samp{evalvn(pi - 3.1415)} at a precision
of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
will return @samp{9.26535897932e-5} (computing a 25-digit result which
is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
will return @samp{9.2654e-5}.

@kindex a "
@pindex calc-expand-formula
The @kbd{a "} (@code{calc-expand-formula}) command expands functions
into their defining formulas wherever possible.  For example,
@samp{deg(x^2)} is changed to @samp{180 x^2 / pi}.  Most functions,
like @code{sin} and @code{gcd}, are not defined by simple formulas
and so are unaffected by this command.  One important class of
functions which @emph{can} be expanded is the user-defined functions
created by the @kbd{Z F} command.  @xref{Algebraic Definitions}.
Other functions which @kbd{a "} can expand include the probability
distribution functions, most of the financial functions, and the
hyperbolic and inverse hyperbolic functions.  A numeric prefix argument
affects @kbd{a "} in the same way as it does @kbd{a v}:  A positive
argument expands all functions in the formula and then simplifies in
various ways; a negative argument expands and simplifies only the
top-level function call.

@kindex a M
@pindex calc-map-equation
@tindex mapeq
The @kbd{a M} (@code{calc-map-equation}) [@code{mapeq}] command applies
a given function or operator to one or more equations.  It is analogous
to @kbd{V M}, which operates on vectors instead of equations.
@pxref{Reducing and Mapping}.  For example, @kbd{a M S} changes
@samp{x = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
@samp{x = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
With two equations on the stack, @kbd{a M +} would add the lefthand
sides together and the righthand sides together to get the two
respective sides of a new equation.

Mapping also works on inequalities.  Mapping two similar inequalities
produces another inequality of the same type.  Mapping an inequality
with an equation produces an inequality of the same type.  Mapping a
@samp{<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
are mapped, the direction of the second inequality is reversed to
match the first:  Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
reverses the latter to get @samp{2 < a}, which then allows the
combination @samp{a + 2 < b + a}, which the @kbd{a s} command can
then simplify to get @samp{2 < b}.

Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
or invert an inequality will reverse the direction of the inequality.
Other adjustments to inequalities are @emph{not} done automatically;
@kbd{a M S} will change @w{@samp{x < y}} to @samp{sin(x) < sin(y)} even
though this is not true for all values of the variables.

@kindex H a M
@tindex mapeqp
With the Hyperbolic flag, @kbd{H a M} [@code{mapeqp}] does a plain
mapping operation without reversing the direction of any inequalities.
Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
(This change is mathematically incorrect, but perhaps you were
fixing an inequality which was already incorrect.)

@kindex I a M
@tindex mapeqr
With the Inverse flag, @kbd{I a M} [@code{mapeqr}] always reverses
the direction of the inequality.  You might use @kbd{I a M C} to
change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
working with small positive angles.

@kindex a b
@pindex calc-substitute
@tindex subst
The @kbd{a b} (@code{calc-substitute}) [@code{subst}] command substitutes
all occurrences
of some variable or sub-expression of an expression with a new
sub-expression.  For example, substituting @samp{sin(x)} with @samp{cos(y)}
in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
@samp{2 cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
Note that this is a purely structural substitution; the lone @samp{x} and
the @samp{sin(2 x)} stayed the same because they did not look like
@samp{sin(x)}.  @xref{Rewrite Rules}, for a more general method for
doing substitutions.

The @kbd{a b} command normally prompts for two formulas, the old
one and the new one.  If you enter a blank line for the first
prompt, all three arguments are taken from the stack (new, then old,
then target expression).  If you type an old formula but then enter a
blank line for the new one, the new formula is taken from top-of-stack
and the target from second-to-top.  If you answer both prompts, the
target is taken from top-of-stack as usual.

Note that @kbd{a b} has no understanding of commutativity or
associativity.  The pattern @samp{x+y} will not match the formula
@samp{y+x}.  Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
because the @samp{+} operator is left-associative, so the ``deep
structure'' of that formula is @samp{(x+y) + z}.  Use @kbd{d U}
(@code{calc-unformatted-language}) mode to see the true structure of
a formula.  The rewrite rule mechanism, discussed later, does not have
these limitations.

As an algebraic function, @code{subst} takes three arguments:
Target expression, old, new.  Note that @code{subst} is always
evaluated immediately, even if its arguments are variables, so if
you wish to put a call to @code{subst} onto the stack you must
turn the default simplifications off first (with @kbd{m O}).

@node Simplifying Formulas, Polynomials, Algebraic Manipulation, Algebra
@section Simplifying Formulas

@noindent
@kindex a s
@pindex calc-simplify
@tindex simplify
The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies
various algebraic rules to simplify a formula.  This includes rules which
are not part of the default simplifications because they may be too slow
to apply all the time, or may not be desirable all of the time.  For
example, non-adjacent terms of sums are combined, as in @samp{a + b + 2 a}
to @samp{b + 3 a}, and some formulas like @samp{sin(arcsin(x))} are
simplified to @samp{x}.

The sections below describe all the various kinds of algebraic
simplifications Calc provides in full detail.  None of Calc's
simplification commands are designed to pull rabbits out of hats;
they simply apply certain specific rules to put formulas into
less redundant or more pleasing forms.  Serious algebra in Calc
must be done manually, usually with a combination of selections
and rewrite rules.  @xref{Rearranging with Selections}.
@xref{Rewrite Rules}.

@xref{Simplification Modes}, for commands to control what level of
simplification occurs automatically.  Normally only the ``default
simplifications'' occur.

@menu
* Default Simplifications::
* Algebraic Simplifications::
* Unsafe Simplifications::
* Simplification of Units::
@end menu

@node Default Simplifications, Algebraic Simplifications, Simplifying Formulas, Simplifying Formulas
@subsection Default Simplifications

@noindent
@cindex Default simplifications
This section describes the ``default simplifications,'' those which are
normally applied to all results.  For example, if you enter the variable
@expr{x} on the stack twice and push @kbd{+}, Calc's default
simplifications automatically change @expr{x + x} to @expr{2 x}.

The @kbd{m O} command turns off the default simplifications, so that
@expr{x + x} will remain in this form unless you give an explicit
``simplify'' command like @kbd{=} or @kbd{a v}.  @xref{Algebraic
Manipulation}.  The @kbd{m D} command turns the default simplifications
back on.

The most basic default simplification is the evaluation of functions.
For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@tfn{sqrt}(9)}
is evaluated to @expr{3}.  Evaluation does not occur if the arguments
to a function are somehow of the wrong type @expr{@tfn{tan}([2,3,4])}),
range (@expr{@tfn{tan}(90)}), or number (@expr{@tfn{tan}(3,5)}), 
or if the function name is not recognized (@expr{@tfn{f}(5)}), or if
Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
(@expr{@tfn{sqrt}(2)}).

Calc simplifies (evaluates) the arguments to a function before it
simplifies the function itself.  Thus @expr{@tfn{sqrt}(5+4)} is
simplified to @expr{@tfn{sqrt}(9)} before the @code{sqrt} function
itself is applied.  There are very few exceptions to this rule:
@code{quote}, @code{lambda}, and @code{condition} (the @code{::}
operator) do not evaluate their arguments, @code{if} (the @code{? :}
operator) does not evaluate all of its arguments, and @code{evalto}
does not evaluate its lefthand argument.

Most commands apply the default simplifications to all arguments they
take from the stack, perform a particular operation, then simplify
the result before pushing it back on the stack.  In the common special
case of regular arithmetic commands like @kbd{+} and @kbd{Q} [@code{sqrt}],
the arguments are simply popped from the stack and collected into a
suitable function call, which is then simplified (the arguments being
simplified first as part of the process, as described above).

The default simplifications are too numerous to describe completely
here, but this section will describe the ones that apply to the
major arithmetic operators.  This list will be rather technical in
nature, and will probably be interesting to you only if you are
a serious user of Calc's algebra facilities.

@tex
\bigskip
@end tex

As well as the simplifications described here, if you have stored
any rewrite rules in the variable @code{EvalRules} then these rules
will also be applied before any built-in default simplifications.
@xref{Automatic Rewrites}, for details.

@tex
\bigskip
@end tex

And now, on with the default simplifications:

Arithmetic operators like @kbd{+} and @kbd{*} always take two
arguments in Calc's internal form.  Sums and products of three or
more terms are arranged by the associative law of algebra into
a left-associative form for sums, @expr{((a + b) + c) + d}, and
a right-associative form for products, @expr{a * (b * (c * d))}.
Formulas like @expr{(a + b) + (c + d)} are rearranged to
left-associative form, though this rarely matters since Calc's
algebra commands are designed to hide the inner structure of
sums and products as much as possible.  Sums and products in
their proper associative form will be written without parentheses
in the examples below.

Sums and products are @emph{not} rearranged according to the
commutative law (@expr{a + b} to @expr{b + a}) except in a few
special cases described below.  Some algebra programs always
rearrange terms into a canonical order, which enables them to
see that @expr{a b + b a} can be simplified to @expr{2 a b}.
Calc assumes you have put the terms into the order you want
and generally leaves that order alone, with the consequence
that formulas like the above will only be simplified if you
explicitly give the @kbd{a s} command.  @xref{Algebraic
Simplifications}.

Differences @expr{a - b} are treated like sums @expr{a + (-b)}
for purposes of simplification; one of the default simplifications
is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
represents a ``negative-looking'' term, into @expr{a - b} form.
``Negative-looking'' means negative numbers, negated formulas like
@expr{-x}, and products or quotients in which either term is
negative-looking.

Other simplifications involving negation are @expr{-(-x)} to @expr{x};
@expr{-(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
negative-looking, simplified by negating that term, or else where
@expr{a} or @expr{b} is any number, by negating that number;
@expr{-(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
(This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
cases where the order of terms in a sum is changed by the default
simplifications.)

The distributive law is used to simplify sums in some cases:
@expr{a x + b x} to @expr{(a + b) x}, where @expr{a} represents
a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
and similarly for @expr{b}.  Use the @kbd{a c}, @w{@kbd{a f}}, or
@kbd{j M} commands to merge sums with non-numeric coefficients
using the distributive law.

The distributive law is only used for sums of two terms, or
for adjacent terms in a larger sum.  Thus @expr{a + b + b + c}
is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
is not simplified.  The reason is that comparing all terms of a
sum with one another would require time proportional to the
square of the number of terms; Calc relegates potentially slow
operations like this to commands that have to be invoked
explicitly, like @kbd{a s}.

Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
A consequence of the above rules is that @expr{0 - a} is simplified
to @expr{-a}.

@tex
\bigskip
@end tex

The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
@expr{(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
@expr{0 a} and @expr{a 0} are simplified to @expr{0}, except that
in Matrix mode where @expr{a} is not provably scalar the result
is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
infinite the result is @samp{nan}.

Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
where this occurs for negated formulas but not for regular negative
numbers.

Products are commuted only to move numbers to the front:
@expr{a b 2} is commuted to @expr{2 a b}.

The product @expr{a (b + c)} is distributed over the sum only if
@expr{a} and at least one of @expr{b} and @expr{c} are numbers:
@expr{2 (x + 3)} goes to @expr{2 x + 6}.  The formula
@expr{(-a) (b - c)}, where @expr{-a} is a negative number, is
rewritten to @expr{a (c - b)}.

The distributive law of products and powers is used for adjacent
terms of the product: @expr{x^a x^b} goes to 
@texline @math{x^{a+b}}
@infoline @expr{x^(a+b)}
where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
or the implicit one-half of @expr{@tfn{sqrt}(x)}, and similarly for
@expr{b}.  The result is written using @samp{sqrt} or @samp{1/sqrt}
if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
If the sum of the powers is zero, the product is simplified to
@expr{1} or to @samp{idn(1)} if Matrix mode is enabled.

The product of a negative power times anything but another negative
power is changed to use division:  
@texline @math{x^{-2} y}
@infoline @expr{x^(-2) y} 
goes to @expr{y / x^2} unless Matrix mode is
in effect and neither @expr{x} nor @expr{y} are scalar (in which
case it is considered unsafe to rearrange the order of the terms).

Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
@expr{(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.

@tex
\bigskip
@end tex

Simplifications for quotients are analogous to those for products.
The quotient @expr{0 / x} is simplified to @expr{0}, with the same
exceptions that were noted for @expr{0 x}.  Likewise, @expr{x / 1}
and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
respectively.

The quotient @expr{x / 0} is left unsimplified or changed to an
infinite quantity, as directed by the current infinite mode.
@xref{Infinite Mode}.

The expression 
@texline @math{a / b^{-c}}
@infoline @expr{a / b^(-c)} 
is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
power.  Also, @expr{1 / b^c} is changed to 
@texline @math{b^{-c}}
@infoline @expr{b^(-c)} 
for any power @expr{c}.

Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
@expr{(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
goes to @expr{(a c) / b} unless Matrix mode prevents this
rearrangement.  Similarly, @expr{a / (b:c)} is simplified to
@expr{(c:b) a} for any fraction @expr{b:c}.

The distributive law is applied to @expr{(a + b) / c} only if
@expr{c} and at least one of @expr{a} and @expr{b} are numbers.
Quotients of powers and square roots are distributed just as
described for multiplication.

Quotients of products cancel only in the leading terms of the
numerator and denominator.  In other words, @expr{a x b / a y b}
is cancelled to @expr{x b / y b} but not to @expr{x / y}.  Once
again this is because full cancellation can be slow; use @kbd{a s}
to cancel all terms of the quotient.

Quotients of negative-looking values are simplified according
to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.

@tex
\bigskip
@end tex

The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
in Matrix mode.  The formula @expr{0^x} is simplified to @expr{0}
unless @expr{x} is a negative number, complex number or zero.
If @expr{x} is negative, complex or @expr{0.0}, @expr{0^x} is an
infinity or an unsimplified formula according to the current infinite
mode.  The expression @expr{0^0} is simplified to @expr{1}.

Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
is an integer, or if either @expr{a} or @expr{b} are nonnegative
real numbers.  Powers of powers @expr{(a^b)^c} are simplified to
@texline @math{a^{b c}}
@infoline @expr{a^(b c)} 
only when @expr{c} is an integer and @expr{b c} also
evaluates to an integer.  Without these restrictions these simplifications
would not be safe because of problems with principal values.
(In other words, 
@texline @math{((-3)^{1/2})^2}
@infoline @expr{((-3)^1:2)^2} 
is safe to simplify, but
@texline @math{((-3)^2)^{1/2}}
@infoline @expr{((-3)^2)^1:2} 
is not.)  @xref{Declarations}, for ways to inform Calc that your
variables satisfy these requirements.

As a special case of this rule, @expr{@tfn{sqrt}(x)^n} is simplified to
@texline @math{x^{n/2}}
@infoline @expr{x^(n/2)} 
only for even integers @expr{n}.

If @expr{a} is known to be real, @expr{b} is an even integer, and
@expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is
simplified to @expr{@tfn{abs}(a^(b c))}.

Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
for any negative-looking expression @expr{-a}.

Square roots @expr{@tfn{sqrt}(x)} generally act like one-half powers
@texline @math{x^{1:2}}
@infoline @expr{x^1:2} 
for the purposes of the above-listed simplifications.

Also, note that 
@texline @math{1 / x^{1:2}}
@infoline @expr{1 / x^1:2} 
is changed to 
@texline @math{x^{-1:2}},
@infoline @expr{x^(-1:2)},
but @expr{1 / @tfn{sqrt}(x)} is left alone.

@tex
\bigskip
@end tex

Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
following rules:  @expr{@tfn{idn}(a) + b} to @expr{a + b} if @expr{b}
is provably scalar, or expanded out if @expr{b} is a matrix;
@expr{@tfn{idn}(a) + @tfn{idn}(b)} to @expr{@tfn{idn}(a + b)}; 
@expr{-@tfn{idn}(a)} to @expr{@tfn{idn}(-a)}; @expr{a @tfn{idn}(b)} to 
@expr{@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b} 
if @expr{a} is provably non-scalar;  @expr{@tfn{idn}(a) @tfn{idn}(b)} to
@expr{@tfn{idn}(a b)}; analogous simplifications for quotients involving
@code{idn}; and @expr{@tfn{idn}(a)^n} to @expr{@tfn{idn}(a^n)} where
@expr{n} is an integer.

@tex
\bigskip
@end tex

The @code{floor} function and other integer truncation functions
vanish if the argument is provably integer-valued, so that
@expr{@tfn{floor}(@tfn{round}(x))} simplifies to @expr{@tfn{round}(x)}.
Also, combinations of @code{float}, @code{floor} and its friends,
and @code{ffloor} and its friends, are simplified in appropriate
ways.  @xref{Integer Truncation}.

The expression @expr{@tfn{abs}(-x)} changes to @expr{@tfn{abs}(x)}.
The expression @expr{@tfn{abs}(@tfn{abs}(x))} changes to
@expr{@tfn{abs}(x)};  in fact, @expr{@tfn{abs}(x)} changes to @expr{x} or
@expr{-x} if @expr{x} is provably nonnegative or nonpositive
(@pxref{Declarations}). 

While most functions do not recognize the variable @code{i} as an
imaginary number, the @code{arg} function does handle the two cases
@expr{@tfn{arg}(@tfn{i})} and @expr{@tfn{arg}(-@tfn{i})} just for convenience.

The expression @expr{@tfn{conj}(@tfn{conj}(x))} simplifies to @expr{x}.
Various other expressions involving @code{conj}, @code{re}, and
@code{im} are simplified, especially if some of the arguments are
provably real or involve the constant @code{i}.  For example,
@expr{@tfn{conj}(a + b i)} is changed to 
@expr{@tfn{conj}(a) - @tfn{conj}(b) i},  or to @expr{a - b i} if @expr{a}
and @expr{b} are known to be real.

Functions like @code{sin} and @code{arctan} generally don't have
any default simplifications beyond simply evaluating the functions
for suitable numeric arguments and infinity.  The @kbd{a s} command
described in the next section does provide some simplifications for
these functions, though.

One important simplification that does occur is that
@expr{@tfn{ln}(@tfn{e})} is simplified to 1, and @expr{@tfn{ln}(@tfn{e}^x)} is
simplified to @expr{x} for any @expr{x}.  This occurs even if you have
stored a different value in the Calc variable @samp{e}; but this would
be a bad idea in any case if you were also using natural logarithms!

Among the logical functions, @tfn{!(@var{a} <= @var{b})} changes to
@tfn{@var{a} > @var{b}} and so on.  Equations and inequalities where both sides
are either negative-looking or zero are simplified by negating both sides
and reversing the inequality.  While it might seem reasonable to simplify
@expr{!!x} to @expr{x}, this would not be valid in general because
@expr{!!2} is 1, not 2.

Most other Calc functions have few if any default simplifications
defined, aside of course from evaluation when the arguments are
suitable numbers.

@node Algebraic Simplifications, Unsafe Simplifications, Default Simplifications, Simplifying Formulas
@subsection Algebraic Simplifications

@noindent
@cindex Algebraic simplifications
The @kbd{a s} command makes simplifications that may be too slow to
do all the time, or that may not be desirable all of the time.
If you find these simplifications are worthwhile, you can type
@kbd{m A} to have Calc apply them automatically.

This section describes all simplifications that are performed by
the @kbd{a s} command.  Note that these occur in addition to the
default simplifications; even if the default simplifications have
been turned off by an @kbd{m O} command, @kbd{a s} will turn them
back on temporarily while it simplifies the formula.

There is a variable, @code{AlgSimpRules}, in which you can put rewrites
to be applied by @kbd{a s}.  Its use is analogous to @code{EvalRules},
but without the special restrictions.  Basically, the simplifier does
@samp{@w{a r} AlgSimpRules} with an infinite repeat count on the whole
expression being simplified, then it traverses the expression applying
the built-in rules described below.  If the result is different from
the original expression, the process repeats with the default
simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
then the built-in simplifications, and so on.

@tex
\bigskip
@end tex

Sums are simplified in two ways.  Constant terms are commuted to the
end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
The only exception is that a constant will not be commuted away
from the first position of a difference, i.e., @expr{2 - x} is not
commuted to @expr{-x + 2}.

Also, terms of sums are combined by the distributive law, as in
@expr{x + y + 2 x} to @expr{y + 3 x}.  This always occurs for
adjacent terms, but @kbd{a s} compares all pairs of terms including
non-adjacent ones.

@tex
\bigskip
@end tex

Products are sorted into a canonical order using the commutative
law.  For example, @expr{b c a} is commuted to @expr{a b c}.
This allows easier comparison of products; for example, the default
simplifications will not change @expr{x y + y x} to @expr{2 x y},
but @kbd{a s} will; it first rewrites the sum to @expr{x y + x y},
and then the default simplifications are able to recognize a sum
of identical terms.

The canonical ordering used to sort terms of products has the
property that real-valued numbers, interval forms and infinities
come first, and are sorted into increasing order.  The @kbd{V S}
command uses the same ordering when sorting a vector.

Sorting of terms of products is inhibited when Matrix mode is
turned on; in this case, Calc will never exchange the order of
two terms unless it knows at least one of the terms is a scalar.

Products of powers are distributed by comparing all pairs of
terms, using the same method that the default simplifications
use for adjacent terms of products.

Even though sums are not sorted, the commutative law is still
taken into account when terms of a product are being compared.
Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
be simplified to @expr{-(x - y)^2}; Calc does not notice that
one term can be written as a constant times the other, even if
that constant is @mathit{-1}.

A fraction times any expression, @expr{(a:b) x}, is changed to
a quotient involving integers:  @expr{a x / b}.  This is not
done for floating-point numbers like @expr{0.5}, however.  This
is one reason why you may find it convenient to turn Fraction mode
on while doing algebra; @pxref{Fraction Mode}.

@tex
\bigskip
@end tex

Quotients are simplified by comparing all terms in the numerator
with all terms in the denominator for possible cancellation using
the distributive law.  For example, @expr{a x^2 b / c x^3 d} will
cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
(The terms in the denominator will then be rearranged to @expr{c d x}
as described above.)  If there is any common integer or fractional
factor in the numerator and denominator, it is cancelled out;
for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.

Non-constant common factors are not found even by @kbd{a s}.  To
cancel the factor @expr{a} in @expr{(a x + a) / a^2} you could first
use @kbd{j M} on the product @expr{a x} to Merge the numerator to
@expr{a (1+x)}, which can then be simplified successfully.

@tex
\bigskip
@end tex

Integer powers of the variable @code{i} are simplified according
to the identity @expr{i^2 = -1}.  If you store a new value other
than the complex number @expr{(0,1)} in @code{i}, this simplification
will no longer occur.  This is done by @kbd{a s} instead of by default
in case someone (unwisely) uses the name @code{i} for a variable
unrelated to complex numbers; it would be unfortunate if Calc
quietly and automatically changed this formula for reasons the
user might not have been thinking of.

Square roots of integer or rational arguments are simplified in
several ways.  (Note that these will be left unevaluated only in
Symbolic mode.)  First, square integer or rational factors are
pulled out so that @expr{@tfn{sqrt}(8)} is rewritten as
@texline @math{2\,@tfn{sqrt}(2)}.
@infoline @expr{2 sqrt(2)}.  
Conceptually speaking this implies factoring the argument into primes
and moving pairs of primes out of the square root, but for reasons of
efficiency Calc only looks for primes up to 29.

Square roots in the denominator of a quotient are moved to the
numerator:  @expr{1 / @tfn{sqrt}(3)} changes to @expr{@tfn{sqrt}(3) / 3}.
The same effect occurs for the square root of a fraction:
@expr{@tfn{sqrt}(2:3)} changes to @expr{@tfn{sqrt}(6) / 3}.

@tex
\bigskip
@end tex

The @code{%} (modulo) operator is simplified in several ways
when the modulus @expr{M} is a positive real number.  First, if
the argument is of the form @expr{x + n} for some real number
@expr{n}, then @expr{n} is itself reduced modulo @expr{M}.  For
example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.

If the argument is multiplied by a constant, and this constant
has a common integer divisor with the modulus, then this factor is
cancelled out.  For example, @samp{12 x % 15} is changed to
@samp{3 (4 x % 5)} by factoring out 3.  Also, @samp{(12 x + 1) % 15}
is changed to @samp{3 ((4 x + 1:3) % 5)}.  While these forms may
not seem ``simpler,'' they allow Calc to discover useful information
about modulo forms in the presence of declarations.

If the modulus is 1, then Calc can use @code{int} declarations to
evaluate the expression.  For example, the idiom @samp{x % 2} is
often used to check whether a number is odd or even.  As described
above, @w{@samp{2 n % 2}} and @samp{(2 n + 1) % 2} are simplified to
@samp{2 (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
can simplify these to 0 and 1 (respectively) if @code{n} has been
declared to be an integer.

@tex
\bigskip
@end tex

Trigonometric functions are simplified in several ways.  Whenever a
products of two trigonometric functions can be replaced by a single
function, the replacement is made; for example,
@expr{@tfn{tan}(x) @tfn{cos}(x)} is simplified to @expr{@tfn{sin}(x)}. 
Reciprocals of trigonometric functions are replaced by their reciprocal
function; for example, @expr{1/@tfn{sec}(x)} is simplified to
@expr{@tfn{cos}(x)}.  The corresponding simplifications for the
hyperbolic functions are also handled.

Trigonometric functions of their inverse functions are
simplified. The expression @expr{@tfn{sin}(@tfn{arcsin}(x))} is
simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.  
Trigonometric functions of inverses of different trigonometric
functions can also be simplified, as in @expr{@tfn{sin}(@tfn{arccos}(x))}
to @expr{@tfn{sqrt}(1 - x^2)}.

If the argument to @code{sin} is negative-looking, it is simplified to
@expr{-@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
Finally, certain special values of the argument are recognized;
@pxref{Trigonometric and Hyperbolic Functions}.

Hyperbolic functions of their inverses and of negative-looking
arguments are also handled, as are exponentials of inverse
hyperbolic functions.

No simplifications for inverse trigonometric and hyperbolic
functions are known, except for negative arguments of @code{arcsin},
@code{arctan}, @code{arcsinh}, and @code{arctanh}.  Note that
@expr{@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
@expr{x}, since this only correct within an integer multiple of 
@texline @math{2 \pi}
@infoline @expr{2 pi} 
radians or 360 degrees.  However, @expr{@tfn{arcsinh}(@tfn{sinh}(x))} is
simplified to @expr{x} if @expr{x} is known to be real.

Several simplifications that apply to logarithms and exponentials
are that @expr{@tfn{exp}(@tfn{ln}(x))}, 
@texline @tfn{e}@math{^{\ln(x)}},
@infoline @expr{e^@tfn{ln}(x)}, 
and
@texline @math{10^{{\rm log10}(x)}}
@infoline @expr{10^@tfn{log10}(x)} 
all reduce to @expr{x}.  Also, @expr{@tfn{ln}(@tfn{exp}(x))}, etc., can
reduce to @expr{x} if @expr{x} is provably real.  The form
@expr{@tfn{exp}(x)^y} is simplified to @expr{@tfn{exp}(x y)}.  If @expr{x}
is a suitable multiple of 
@texline @math{\pi i} 
@infoline @expr{pi i}
(as described above for the trigonometric functions), then
@expr{@tfn{exp}(x)} or @expr{e^x} will be expanded.  Finally,
@expr{@tfn{ln}(x)} is simplified to a form involving @code{pi} and
@code{i} where @expr{x} is provably negative, positive imaginary, or
negative imaginary. 

The error functions @code{erf} and @code{erfc} are simplified when
their arguments are negative-looking or are calls to the @code{conj}
function.

@tex
\bigskip
@end tex

Equations and inequalities are simplified by cancelling factors
of products, quotients, or sums on both sides.  Inequalities
change sign if a negative multiplicative factor is cancelled.
Non-constant multiplicative factors as in @expr{a b = a c} are
cancelled from equations only if they are provably nonzero (generally
because they were declared so; @pxref{Declarations}).  Factors
are cancelled from inequalities only if they are nonzero and their
sign is known.

Simplification also replaces an equation or inequality with
1 or 0 (``true'' or ``false'') if it can through the use of
declarations.  If @expr{x} is declared to be an integer greater
than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
all simplified to 0, but @expr{x > 3} is simplified to 1.
By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
as is @expr{x^2 >= 0} if @expr{x} is known to be real.

@node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas
@subsection ``Unsafe'' Simplifications

@noindent
@cindex Unsafe simplifications
@cindex Extended simplification
@kindex a e
@pindex calc-simplify-extended
@ignore
@mindex esimpl@idots
@end ignore
@tindex esimplify
The @kbd{a e} (@code{calc-simplify-extended}) [@code{esimplify}] command
is like @kbd{a s}
except that it applies some additional simplifications which are not
``safe'' in all cases.  Use this only if you know the values in your
formula lie in the restricted ranges for which these simplifications
are valid.  The symbolic integrator uses @kbd{a e};
one effect of this is that the integrator's results must be used with
caution.  Where an integral table will often attach conditions like
``for positive @expr{a} only,'' Calc (like most other symbolic
integration programs) will simply produce an unqualified result.

Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
to type @kbd{C-u -3 a v}, which does extended simplification only
on the top level of the formula without affecting the sub-formulas.
In fact, @kbd{C-u -3 j v} allows you to target extended simplification
to any specific part of a formula.

The variable @code{ExtSimpRules} contains rewrites to be applied by
the @kbd{a e} command.  These are applied in addition to
@code{EvalRules} and @code{AlgSimpRules}.  (The @kbd{a r AlgSimpRules}
step described above is simply followed by an @kbd{a r ExtSimpRules} step.)

Following is a complete list of ``unsafe'' simplifications performed
by @kbd{a e}.

@tex
\bigskip
@end tex

Inverse trigonometric or hyperbolic functions, called with their
corresponding non-inverse functions as arguments, are simplified
by @kbd{a e}.  For example, @expr{@tfn{arcsin}(@tfn{sin}(x))} changes
to @expr{x}.  Also, @expr{@tfn{arcsin}(@tfn{cos}(x))} and
@expr{@tfn{arccos}(@tfn{sin}(x))} both change to @expr{@tfn{pi}/2 - x}.
These simplifications are unsafe because they are valid only for
values of @expr{x} in a certain range; outside that range, values
are folded down to the 360-degree range that the inverse trigonometric
functions always produce.

Powers of powers @expr{(x^a)^b} are simplified to 
@texline @math{x^{a b}}
@infoline @expr{x^(a b)}
for all @expr{a} and @expr{b}.  These results will be valid only
in a restricted range of @expr{x}; for example, in 
@texline @math{(x^2)^{1:2}}
@infoline @expr{(x^2)^1:2}
the powers cancel to get @expr{x}, which is valid for positive values
of @expr{x} but not for negative or complex values.

Similarly, @expr{@tfn{sqrt}(x^a)} and @expr{@tfn{sqrt}(x)^a} are both
simplified (possibly unsafely) to 
@texline @math{x^{a/2}}.
@infoline @expr{x^(a/2)}.

Forms like @expr{@tfn{sqrt}(1 - sin(x)^2)} are simplified to, e.g.,
@expr{@tfn{cos}(x)}.  Calc has identities of this sort for @code{sin},
@code{cos}, @code{tan}, @code{sinh}, and @code{cosh}.

Arguments of square roots are partially factored to look for
squared terms that can be extracted.  For example,
@expr{@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to 
@expr{a b @tfn{sqrt}(a+b)}.

The simplifications of @expr{@tfn{ln}(@tfn{exp}(x))},
@expr{@tfn{ln}(@tfn{e}^x)}, and @expr{@tfn{log10}(10^x)} to @expr{x} are also
unsafe because of problems with principal values (although these
simplifications are safe if @expr{x} is known to be real).

Common factors are cancelled from products on both sides of an
equation, even if those factors may be zero:  @expr{a x / b x}
to @expr{a / b}.  Such factors are never cancelled from
inequalities:  Even @kbd{a e} is not bold enough to reduce
@expr{a x < b x} to @expr{a < b} (or @expr{a > b}, depending
on whether you believe @expr{x} is positive or negative).
The @kbd{a M /} command can be used to divide a factor out of
both sides of an inequality.

@node Simplification of Units,  , Unsafe Simplifications, Simplifying Formulas
@subsection Simplification of Units

@noindent
The simplifications described in this section are applied by the
@kbd{u s} (@code{calc-simplify-units}) command.  These are in addition
to the regular @kbd{a s} (but not @kbd{a e}) simplifications described
earlier.  @xref{Basic Operations on Units}.

The variable @code{UnitSimpRules} contains rewrites to be applied by
the @kbd{u s} command.  These are applied in addition to @code{EvalRules}
and @code{AlgSimpRules}.

Scalar mode is automatically put into effect when simplifying units.
@xref{Matrix Mode}.

Sums @expr{a + b} involving units are simplified by extracting the
units of @expr{a} as if by the @kbd{u x} command (call the result
@expr{u_a}), then simplifying the expression @expr{b / u_a}
using @kbd{u b} and @kbd{u s}.  If the result has units then the sum
is inconsistent and is left alone.  Otherwise, it is rewritten
in terms of the units @expr{u_a}.

If units auto-ranging mode is enabled, products or quotients in
which the first argument is a number which is out of range for the
leading unit are modified accordingly.

When cancelling and combining units in products and quotients,
Calc accounts for unit names that differ only in the prefix letter.
For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
However, compatible but different units like @code{ft} and @code{in}
are not combined in this way.

Quotients @expr{a / b} are simplified in three additional ways.  First,
if @expr{b} is a number or a product beginning with a number, Calc
computes the reciprocal of this number and moves it to the numerator.

Second, for each pair of unit names from the numerator and denominator
of a quotient, if the units are compatible (e.g., they are both
units of area) then they are replaced by the ratio between those
units.  For example, in @samp{3 s in N / kg cm} the units
@samp{in / cm} will be replaced by @expr{2.54}.

Third, if the units in the quotient exactly cancel out, so that
a @kbd{u b} command on the quotient would produce a dimensionless
number for an answer, then the quotient simplifies to that number.

For powers and square roots, the ``unsafe'' simplifications
@expr{(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
and @expr{(a^b)^c} to 
@texline @math{a^{b c}}
@infoline @expr{a^(b c)} 
are done if the powers are real numbers.  (These are safe in the context
of units because all numbers involved can reasonably be assumed to be
real.)

Also, if a unit name is raised to a fractional power, and the
base units in that unit name all occur to powers which are a
multiple of the denominator of the power, then the unit name
is expanded out into its base units, which can then be simplified
according to the previous paragraph.  For example, @samp{acre^1.5}
is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
is defined in terms of @samp{m^2}, and that the 2 in the power of
@code{m} is a multiple of 2 in @expr{3:2}.  Thus, @code{acre^1.5} is
replaced by approximately 
@texline @math{(4046 m^2)^{1.5}}
@infoline @expr{(4046 m^2)^1.5}, 
which is then changed to 
@texline @math{4046^{1.5} \, (m^2)^{1.5}},
@infoline @expr{4046^1.5 (m^2)^1.5}, 
then to @expr{257440 m^3}.

The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
as well as @code{floor} and the other integer truncation functions,
applied to unit names or products or quotients involving units, are
simplified.  For example, @samp{round(1.6 in)} is changed to
@samp{round(1.6) round(in)}; the lefthand term evaluates to 2,
and the righthand term simplifies to @code{in}.

The functions @code{sin}, @code{cos}, and @code{tan} with arguments
that have angular units like @code{rad} or @code{arcmin} are
simplified by converting to base units (radians), then evaluating
with the angular mode temporarily set to radians.

@node Polynomials, Calculus, Simplifying Formulas, Algebra
@section Polynomials

A @dfn{polynomial} is a sum of terms which are coefficients times
various powers of a ``base'' variable.  For example, @expr{2 x^2 + 3 x - 4}
is a polynomial in @expr{x}.  Some formulas can be considered
polynomials in several different variables:  @expr{1 + 2 x + 3 y + 4 x y^2}
is a polynomial in both @expr{x} and @expr{y}.  Polynomial coefficients
are often numbers, but they may in general be any formulas not
involving the base variable.

@kindex a f
@pindex calc-factor
@tindex factor
The @kbd{a f} (@code{calc-factor}) [@code{factor}] command factors a
polynomial into a product of terms.  For example, the polynomial
@expr{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}.  As another
example, @expr{a c + b d + b c + a d} is factored into the product
@expr{(a + b) (c + d)}.

Calc currently has three algorithms for factoring.  Formulas which are
linear in several variables, such as the second example above, are
merged according to the distributive law.  Formulas which are
polynomials in a single variable, with constant integer or fractional
coefficients, are factored into irreducible linear and/or quadratic
terms.  The first example above factors into three linear terms
(@expr{x}, @expr{x+1}, and @expr{x+1} again).  Finally, formulas
which do not fit the above criteria are handled by the algebraic
rewrite mechanism.

Calc's polynomial factorization algorithm works by using the general
root-finding command (@w{@kbd{a P}}) to solve for the roots of the
polynomial.  It then looks for roots which are rational numbers
or complex-conjugate pairs, and converts these into linear and
quadratic terms, respectively.  Because it uses floating-point
arithmetic, it may be unable to find terms that involve large
integers (whose number of digits approaches the current precision).
Also, irreducible factors of degree higher than quadratic are not
found, and polynomials in more than one variable are not treated.
(A more robust factorization algorithm may be included in a future
version of Calc.)

@vindex FactorRules
@ignore
@starindex
@end ignore
@tindex thecoefs
@ignore
@starindex
@end ignore
@ignore
@mindex @idots
@end ignore
@tindex thefactors
The rewrite-based factorization method uses rules stored in the variable
@code{FactorRules}.  @xref{Rewrite Rules}, for a discussion of the
operation of rewrite rules.  The default @code{FactorRules} are able
to factor quadratic forms symbolically into two linear terms,
@expr{(a x + b) (c x + d)}.  You can edit these rules to include other
cases if you wish.  To use the rules, Calc builds the formula
@samp{thecoefs(x, [a, b, c, ...])} where @code{x} is the polynomial
base variable and @code{a}, @code{b}, etc., are polynomial coefficients
(which may be numbers or formulas).  The constant term is written first,
i.e., in the @code{a} position.  When the rules complete, they should have
changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
Calc then multiplies these terms together to get the complete
factored form of the polynomial.  If the rules do not change the
@code{thecoefs} call to a @code{thefactors} call, @kbd{a f} leaves the
polynomial alone on the assumption that it is unfactorable.  (Note that
the function names @code{thecoefs} and @code{thefactors} are used only
as placeholders; there are no actual Calc functions by those names.)

@kindex H a f
@tindex factors
The @kbd{H a f} [@code{factors}] command also factors a polynomial,
but it returns a list of factors instead of an expression which is the
product of the factors.  Each factor is represented by a sub-vector
of the factor, and the power with which it appears.  For example,
@expr{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
If there is an overall numeric factor, it always comes first in the list.
The functions @code{factor} and @code{factors} allow a second argument
when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
respect to the specific variable @expr{v}.  The default is to factor with
respect to all the variables that appear in @expr{x}.

@kindex a c
@pindex calc-collect
@tindex collect
The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a
formula as a
polynomial in a given variable, ordered in decreasing powers of that
variable.  For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
The polynomial will be expanded out using the distributive law as
necessary:  Collecting @expr{x} in @expr{(x - 1)^3} produces
@expr{x^3 - 3 x^2 + 3 x - 1}.  Terms not involving @expr{x} will
not be expanded.

The ``variable'' you specify at the prompt can actually be any
expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
by @samp{ln(x+1)} or integer powers thereof.  If @samp{x} also appears
in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.

@kindex a x
@pindex calc-expand
@tindex expand
The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an
expression by applying the distributive law everywhere.  It applies to
products, quotients, and powers involving sums.  By default, it fully
distributes all parts of the expression.  With a numeric prefix argument,
the distributive law is applied only the specified number of times, then
the partially expanded expression is left on the stack.

The @kbd{a x} and @kbd{j D} commands are somewhat redundant.  Use
@kbd{a x} if you want to expand all products of sums in your formula.
Use @kbd{j D} if you want to expand a particular specified term of
the formula.  There is an exactly analogous correspondence between
@kbd{a f} and @kbd{j M}.  (The @kbd{j D} and @kbd{j M} commands
also know many other kinds of expansions, such as
@samp{exp(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
do not do.)

Calc's automatic simplifications will sometimes reverse a partial
expansion.  For example, the first step in expanding @expr{(x+1)^3} is
to write @expr{(x+1) (x+1)^2}.  If @kbd{a x} stops there and tries
to put this formula onto the stack, though, Calc will automatically
simplify it back to @expr{(x+1)^3} form.  The solution is to turn
simplification off first (@pxref{Simplification Modes}), or to run
@kbd{a x} without a numeric prefix argument so that it expands all
the way in one step.

@kindex a a
@pindex calc-apart
@tindex apart
The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a
rational function by partial fractions.  A rational function is the
quotient of two polynomials; @code{apart} pulls this apart into a
sum of rational functions with simple denominators.  In algebraic
notation, the @code{apart} function allows a second argument that
specifies which variable to use as the ``base''; by default, Calc
chooses the base variable automatically.

@kindex a n
@pindex calc-normalize-rat
@tindex nrat
The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command
attempts to arrange a formula into a quotient of two polynomials.
For example, given @expr{1 + (a + b/c) / d}, the result would be
@expr{(b + a c + c d) / c d}.  The quotient is reduced, so that
@kbd{a n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.

@kindex a \
@pindex calc-poly-div
@tindex pdiv
The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides
two polynomials @expr{u} and @expr{v}, yielding a new polynomial
@expr{q}.  If several variables occur in the inputs, the inputs are
considered multivariate polynomials.  (Calc divides by the variable
with the largest power in @expr{u} first, or, in the case of equal
powers, chooses the variables in alphabetical order.)  For example,
dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
The remainder from the division, if any, is reported at the bottom
of the screen and is also placed in the Trail along with the quotient.

Using @code{pdiv} in algebraic notation, you can specify the particular
variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
If @code{pdiv} is given only two arguments (as is always the case with
the @kbd{a \} command), then it does a multivariate division as outlined
above.

@kindex a %
@pindex calc-poly-rem
@tindex prem
The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides
two polynomials and keeps the remainder @expr{r}.  The quotient
@expr{q} is discarded.  For any formulas @expr{a} and @expr{b}, the
results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
(This is analogous to plain @kbd{\} and @kbd{%}, which compute the
integer quotient and remainder from dividing two numbers.)

@kindex a /
@kindex H a /
@pindex calc-poly-div-rem
@tindex pdivrem
@tindex pdivide
The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command
divides two polynomials and reports both the quotient and the
remainder as a vector @expr{[q, r]}.  The @kbd{H a /} [@code{pdivide}]
command divides two polynomials and constructs the formula
@expr{q + r/b} on the stack.  (Naturally if the remainder is zero,
this will immediately simplify to @expr{q}.)

@kindex a g
@pindex calc-poly-gcd
@tindex pgcd
The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes
the greatest common divisor of two polynomials.  (The GCD actually
is unique only to within a constant multiplier; Calc attempts to
choose a GCD which will be unsurprising.)  For example, the @kbd{a n}
command uses @kbd{a g} to take the GCD of the numerator and denominator
of a quotient, then divides each by the result using @kbd{a \}.  (The
definition of GCD ensures that this division can take place without
leaving a remainder.)

While the polynomials used in operations like @kbd{a /} and @kbd{a g}
often have integer coefficients, this is not required.  Calc can also
deal with polynomials over the rationals or floating-point reals.
Polynomials with modulo-form coefficients are also useful in many
applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
automatically transforms this into a polynomial over the field of
integers mod 5:  @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.

Congratulations and thanks go to Ove Ewerlid
(@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
polynomial routines used in the above commands.

@xref{Decomposing Polynomials}, for several useful functions for
extracting the individual coefficients of a polynomial.

@node Calculus, Solving Equations, Polynomials, Algebra
@section Calculus

@noindent
The following calculus commands do not automatically simplify their
inputs or outputs using @code{calc-simplify}.  You may find it helps
to do this by hand by typing @kbd{a s} or @kbd{a e}.  It may also help
to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
readable way.

@menu
* Differentiation::
* Integration::
* Customizing the Integrator::
* Numerical Integration::
* Taylor Series::
@end menu

@node Differentiation, Integration, Calculus, Calculus
@subsection Differentiation

@noindent
@kindex a d
@kindex H a d
@pindex calc-derivative
@tindex deriv
@tindex tderiv
The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes
the derivative of the expression on the top of the stack with respect to
some variable, which it will prompt you to enter.  Normally, variables
in the formula other than the specified differentiation variable are
considered constant, i.e., @samp{deriv(y,x)} is reduced to zero.  With
the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
instead, in which derivatives of variables are not reduced to zero
unless those variables are known to be ``constant,'' i.e., independent
of any other variables.  (The built-in special variables like @code{pi}
are considered constant, as are variables that have been declared
@code{const}; @pxref{Declarations}.)

With a numeric prefix argument @var{n}, this command computes the
@var{n}th derivative.

When working with trigonometric functions, it is best to switch to
Radians mode first (with @w{@kbd{m r}}).  The derivative of @samp{sin(x)}
in degrees is @samp{(pi/180) cos(x)}, probably not the expected
answer!

If you use the @code{deriv} function directly in an algebraic formula,
you can write @samp{deriv(f,x,x0)} which represents the derivative
of @expr{f} with respect to @expr{x}, evaluated at the point 
@texline @math{x=x_0}.
@infoline @expr{x=x0}.

If the formula being differentiated contains functions which Calc does
not know, the derivatives of those functions are produced by adding
primes (apostrophe characters).  For example, @samp{deriv(f(2x), x)}
produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
derivative of @code{f}.

For functions you have defined with the @kbd{Z F} command, Calc expands
the functions according to their defining formulas unless you have
also defined @code{f'} suitably.  For example, suppose we define
@samp{sinc(x) = sin(x)/x} using @kbd{Z F}.  If we then differentiate
the formula @samp{sinc(2 x)}, the formula will be expanded to
@samp{sin(2 x) / (2 x)} and differentiated.  However, if we also
define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
result as @samp{2 dsinc(2 x)}.  @xref{Algebraic Definitions}.

For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
to the first argument is written @samp{f'(x,y,z)}; derivatives with
respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
Various higher-order derivatives can be formed in the obvious way, e.g.,
@samp{f'@var{}'(x)} (the second derivative of @code{f}) or
@samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
argument once).

@node Integration, Customizing the Integrator, Differentiation, Calculus
@subsection Integration

@noindent
@kindex a i
@pindex calc-integral
@tindex integ
The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the
indefinite integral of the expression on the top of the stack with
respect to a prompted-for variable.  The integrator is not guaranteed to
work for all integrable functions, but it is able to integrate several
large classes of formulas.  In particular, any polynomial or rational
function (a polynomial divided by a polynomial) is acceptable.
(Rational functions don't have to be in explicit quotient form, however; 
@texline @math{x/(1+x^{-2})}
@infoline @expr{x/(1+x^-2)}
is not strictly a quotient of polynomials, but it is equivalent to
@expr{x^3/(x^2+1)}, which is.)  Also, square roots of terms involving
@expr{x} and @expr{x^2} may appear in rational functions being
integrated.  Finally, rational functions involving trigonometric or
hyperbolic functions can be integrated.

With an argument (@kbd{C-u a i}), this command will compute the definite
integral of the expression on top of the stack.  In this case, the
command will again prompt for an integration variable, then prompt for a
lower limit and an upper limit.

@ifinfo
If you use the @code{integ} function directly in an algebraic formula,
you can also write @samp{integ(f,x,v)} which expresses the resulting
indefinite integral in terms of variable @code{v} instead of @code{x}.
With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
integral from @code{a} to @code{b}.
@end ifinfo
@tex
If you use the @code{integ} function directly in an algebraic formula,
you can also write @samp{integ(f,x,v)} which expresses the resulting
indefinite integral in terms of variable @code{v} instead of @code{x}.
With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
integral $\int_a^b f(x) \, dx$.
@end tex

Please note that the current implementation of Calc's integrator sometimes
produces results that are significantly more complex than they need to
be.  For example, the integral Calc finds for 
@texline @math{1/(x+\sqrt{x^2+1})}
@infoline @expr{1/(x+sqrt(x^2+1))}
is several times more complicated than the answer Mathematica
returns for the same input, although the two forms are numerically
equivalent.  Also, any indefinite integral should be considered to have
an arbitrary constant of integration added to it, although Calc does not
write an explicit constant of integration in its result.  For example,
Calc's solution for 
@texline @math{1/(1+\tan x)}
@infoline @expr{1/(1+tan(x))} 
differs from the solution given in the @emph{CRC Math Tables} by a
constant factor of  
@texline @math{\pi i / 2}
@infoline @expr{pi i / 2},
due to a different choice of constant of integration.

The Calculator remembers all the integrals it has done.  If conditions
change in a way that would invalidate the old integrals, say, a switch
from Degrees to Radians mode, then they will be thrown out.  If you
suspect this is not happening when it should, use the
@code{calc-flush-caches} command; @pxref{Caches}.

@vindex IntegLimit
Calc normally will pursue integration by substitution or integration by
parts up to 3 nested times before abandoning an approach as fruitless.
If the integrator is taking too long, you can lower this limit by storing
a number (like 2) in the variable @code{IntegLimit}.  (The @kbd{s I}
command is a convenient way to edit @code{IntegLimit}.)  If this variable
has no stored value or does not contain a nonnegative integer, a limit
of 3 is used.  The lower this limit is, the greater the chance that Calc
will be unable to integrate a function it could otherwise handle.  Raising
this limit allows the Calculator to solve more integrals, though the time
it takes may grow exponentially.  You can monitor the integrator's actions
by creating an Emacs buffer called @code{*Trace*}.  If such a buffer
exists, the @kbd{a i} command will write a log of its actions there.

If you want to manipulate integrals in a purely symbolic way, you can
set the integration nesting limit to 0 to prevent all but fast
table-lookup solutions of integrals.  You might then wish to define
rewrite rules for integration by parts, various kinds of substitutions,
and so on.  @xref{Rewrite Rules}.

@node Customizing the Integrator, Numerical Integration, Integration, Calculus
@subsection Customizing the Integrator

@noindent
@vindex IntegRules
Calc has two built-in rewrite rules called @code{IntegRules} and
@code{IntegAfterRules} which you can edit to define new integration
methods.  @xref{Rewrite Rules}.  At each step of the integration process,
Calc wraps the current integrand in a call to the fictitious function
@samp{integtry(@var{expr},@var{var})}, where @var{expr} is the
integrand and @var{var} is the integration variable.  If your rules
rewrite this to be a plain formula (not a call to @code{integtry}), then
Calc will use this formula as the integral of @var{expr}.  For example,
the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
integrate a function @code{mysin} that acts like the sine function.
Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
will produce the integral @samp{-2 mycos(2y+1)}.  Note that Calc has
automatically made various transformations on the integral to allow it
to use your rule; integral tables generally give rules for
@samp{mysin(a x + b)}, but you don't need to use this much generality
in your @code{IntegRules}.

@cindex Exponential integral Ei(x)
@ignore
@starindex
@end ignore
@tindex Ei
As a more serious example, the expression @samp{exp(x)/x} cannot be
integrated in terms of the standard functions, so the ``exponential
integral'' function 
@texline @math{{\rm Ei}(x)}
@infoline @expr{Ei(x)} 
was invented to describe it.
We can get Calc to do this integral in terms of a made-up @code{Ei}
function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
to @code{IntegRules}.  Now entering @samp{exp(2x)/x} on the stack
and typing @kbd{a i x} yields @samp{Ei(2 x)}.  This new rule will
work with Calc's various built-in integration methods (such as
integration by substitution) to solve a variety of other problems
involving @code{Ei}:  For example, now Calc will also be able to
integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).

Your rule may do further integration by calling @code{integ}.  For
example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
Note that @code{integ} was called with only one argument.  This notation
is allowed only within @code{IntegRules}; it means ``integrate this
with respect to the same integration variable.''  If Calc is unable
to integrate @code{u}, the integration that invoked @code{IntegRules}
also fails.  Thus integrating @samp{twice(f(x))} fails, returning the
unevaluated integral @samp{integ(twice(f(x)), x)}.  It is still valid
to call @code{integ} with two or more arguments, however; in this case,
if @code{u} is not integrable, @code{twice} itself will still be
integrated:  If the above rule is changed to @samp{... := twice(integ(u,x))},
then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.

If a rule instead produces the formula @samp{integsubst(@var{sexpr},
@var{svar})}, either replacing the top-level @code{integtry} call or
nested anywhere inside the expression, then Calc will apply the
substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to
integrate the original @var{expr}.  For example, the rule
@samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
a square root in the integrand, it should attempt the substitution
@samp{u = sqrt(x)}.  (This particular rule is unnecessary because
Calc always tries ``obvious'' substitutions where @var{sexpr} actually
appears in the integrand.)  The variable @var{svar} may be the same
as the @var{var} that appeared in the call to @code{integtry}, but
it need not be.

When integrating according to an @code{integsubst}, Calc uses the
equation solver to find the inverse of @var{sexpr} (if the integrand
refers to @var{var} anywhere except in subexpressions that exactly
match @var{sexpr}).  It uses the differentiator to find the derivative
of @var{sexpr} and/or its inverse (it has two methods that use one
derivative or the other).  You can also specify these items by adding
extra arguments to the @code{integsubst} your rules construct; the
general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
@var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still
written as a function of @var{svar}), and @var{sprime} is the
derivative of @var{sexpr} with respect to @var{svar}.  If you don't
specify these things, and Calc is not able to work them out on its
own with the information it knows, then your substitution rule will
work only in very specific, simple cases.

Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
in other words, Calc stops rewriting as soon as any rule in your rule
set succeeds.  (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
example above would keep on adding layers of @code{integsubst} calls
forever!)

@vindex IntegSimpRules
Another set of rules, stored in @code{IntegSimpRules}, are applied
every time the integrator uses @kbd{a s} to simplify an intermediate
result.  For example, putting the rule @samp{twice(x) := 2 x} into
@code{IntegSimpRules} would tell Calc to convert the @code{twice}
function into a form it knows whenever integration is attempted.

One more way to influence the integrator is to define a function with
the @kbd{Z F} command (@pxref{Algebraic Definitions}).  Calc's
integrator automatically expands such functions according to their
defining formulas, even if you originally asked for the function to
be left unevaluated for symbolic arguments.  (Certain other Calc
systems, such as the differentiator and the equation solver, also
do this.)

@vindex IntegAfterRules
Sometimes Calc is able to find a solution to your integral, but it
expresses the result in a way that is unnecessarily complicated.  If
this happens, you can either use @code{integsubst} as described
above to try to hint at a more direct path to the desired result, or
you can use @code{IntegAfterRules}.  This is an extra rule set that
runs after the main integrator returns its result; basically, Calc does
an @kbd{a r IntegAfterRules} on the result before showing it to you.
(It also does an @kbd{a s}, without @code{IntegSimpRules}, after that
to further simplify the result.)  For example, Calc's integrator
sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
the default @code{IntegAfterRules} rewrite this into the more readable
form @samp{2 arctanh(x)}.  Note that, unlike @code{IntegRules},
@code{IntegSimpRules} and @code{IntegAfterRules} are applied any number
of times until no further changes are possible.  Rewriting by
@code{IntegAfterRules} occurs only after the main integrator has
finished, not at every step as for @code{IntegRules} and
@code{IntegSimpRules}.

@node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus
@subsection Numerical Integration

@noindent
@kindex a I
@pindex calc-num-integral
@tindex ninteg
If you want a purely numerical answer to an integration problem, you can
use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command.  This
command prompts for an integration variable, a lower limit, and an
upper limit.  Except for the integration variable, all other variables
that appear in the integrand formula must have stored values.  (A stored
value, if any, for the integration variable itself is ignored.)

Numerical integration works by evaluating your formula at many points in
the specified interval.  Calc uses an ``open Romberg'' method; this means
that it does not evaluate the formula actually at the endpoints (so that
it is safe to integrate @samp{sin(x)/x} from zero, for example).  Also,
the Romberg method works especially well when the function being
integrated is fairly smooth.  If the function is not smooth, Calc will
have to evaluate it at quite a few points before it can accurately
determine the value of the integral.

Integration is much faster when the current precision is small.  It is
best to set the precision to the smallest acceptable number of digits
before you use @kbd{a I}.  If Calc appears to be taking too long, press
@kbd{C-g} to halt it and try a lower precision.  If Calc still appears
to need hundreds of evaluations, check to make sure your function is
well-behaved in the specified interval.

It is possible for the lower integration limit to be @samp{-inf} (minus
infinity).  Likewise, the upper limit may be plus infinity.  Calc
internally transforms the integral into an equivalent one with finite
limits.  However, integration to or across singularities is not supported:
The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
by Calc's symbolic integrator, for example), but @kbd{a I} will fail
because the integrand goes to infinity at one of the endpoints.

@node Taylor Series,  , Numerical Integration, Calculus
@subsection Taylor Series

@noindent
@kindex a t
@pindex calc-taylor
@tindex taylor
The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a
power series expansion or Taylor series of a function.  You specify the
variable and the desired number of terms.  You may give an expression of
the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead
of just a variable to produce a Taylor expansion about the point @var{a}.
You may specify the number of terms with a numeric prefix argument;
otherwise the command will prompt you for the number of terms.  Note that
many series expansions have coefficients of zero for some terms, so you
may appear to get fewer terms than you asked for.

If the @kbd{a i} command is unable to find a symbolic integral for a
function, you can get an approximation by integrating the function's
Taylor series.

@node Solving Equations, Numerical Solutions, Calculus, Algebra
@section Solving Equations

@noindent
@kindex a S
@pindex calc-solve-for
@tindex solve
@cindex Equations, solving
@cindex Solving equations
The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges
an equation to solve for a specific variable.  An equation is an
expression of the form @expr{L = R}.  For example, the command @kbd{a S x}
will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}.  If the
input is not an equation, it is treated like an equation of the
form @expr{X = 0}.

This command also works for inequalities, as in @expr{y < 3x + 6}.
Some inequalities cannot be solved where the analogous equation could
be; for example, solving 
@texline @math{a < b \, c}
@infoline @expr{a < b c} 
for @expr{b} is impossible
without knowing the sign of @expr{c}.  In this case, @kbd{a S} will
produce the result 
@texline @math{b \mathbin{\hbox{\code{!=}}} a/c}
@infoline @expr{b != a/c} 
(using the not-equal-to operator) to signify that the direction of the
inequality is now unknown.  The inequality 
@texline @math{a \le b \, c}
@infoline @expr{a <= b c} 
is not even partially solved.  @xref{Declarations}, for a way to tell
Calc that the signs of the variables in a formula are in fact known.

Two useful commands for working with the result of @kbd{a S} are
@kbd{a .} (@pxref{Logical Operations}), which converts @expr{x = y/3 - 2}
to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
another formula with @expr{x} set equal to @expr{y/3 - 2}.

@menu
* Multiple Solutions::
* Solving Systems of Equations::
* Decomposing Polynomials::
@end menu

@node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations
@subsection Multiple Solutions

@noindent
@kindex H a S
@tindex fsolve
Some equations have more than one solution.  The Hyperbolic flag
(@code{H a S}) [@code{fsolve}] tells the solver to report the fully
general family of solutions.  It will invent variables @code{n1},
@code{n2}, @dots{}, which represent independent arbitrary integers, and
@code{s1}, @code{s2}, @dots{}, which represent independent arbitrary
signs (either @mathit{+1} or @mathit{-1}).  If you don't use the Hyperbolic
flag, Calc will use zero in place of all arbitrary integers, and plus
one in place of all arbitrary signs.  Note that variables like @code{n1}
and @code{s1} are not given any special interpretation in Calc except by
the equation solver itself.  As usual, you can use the @w{@kbd{s l}}
(@code{calc-let}) command to obtain solutions for various actual values
of these variables.

For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
equation are @samp{sqrt(y)} and @samp{-sqrt(y)}.  Another way to
think about it is that the square-root operation is really a
two-valued function; since every Calc function must return a
single result, @code{sqrt} chooses to return the positive result.
Then @kbd{H a S} doctors this result using @code{s1} to indicate
the full set of possible values of the mathematical square-root.

There is a similar phenomenon going the other direction:  Suppose
we solve @samp{sqrt(y) = x} for @code{y}.  Calc squares both sides
to get @samp{y = x^2}.  This is correct, except that it introduces
some dubious solutions.  Consider solving @samp{sqrt(y) = -3}:
Calc will report @expr{y = 9} as a valid solution, which is true
in the mathematical sense of square-root, but false (there is no
solution) for the actual Calc positive-valued @code{sqrt}.  This
happens for both @kbd{a S} and @kbd{H a S}.

@cindex @code{GenCount} variable
@vindex GenCount
@ignore
@starindex
@end ignore
@tindex an
@ignore
@starindex
@end ignore
@tindex as
If you store a positive integer in the Calc variable @code{GenCount},
then Calc will generate formulas of the form @samp{as(@var{n})} for
arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
where @var{n} represents successive values taken by incrementing
@code{GenCount} by one.  While the normal arbitrary sign and
integer symbols start over at @code{s1} and @code{n1} with each
new Calc command, the @code{GenCount} approach will give each
arbitrary value a name that is unique throughout the entire Calc
session.  Also, the arbitrary values are function calls instead
of variables, which is advantageous in some cases.  For example,
you can make a rewrite rule that recognizes all arbitrary signs
using a pattern like @samp{as(n)}.  The @kbd{s l} command only works
on variables, but you can use the @kbd{a b} (@code{calc-substitute})
command to substitute actual values for function calls like @samp{as(3)}.

The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
way to create or edit this variable.  Press @kbd{C-c C-c} to finish.

If you have not stored a value in @code{GenCount}, or if the value
in that variable is not a positive integer, the regular
@code{s1}/@code{n1} notation is used.

@kindex I a S
@kindex H I a S
@tindex finv
@tindex ffinv
With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression
on top of the stack as a function of the specified variable and solves
to find the inverse function, written in terms of the same variable.
For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a
fully general inverse, as described above.

@kindex a P
@pindex calc-poly-roots
@tindex roots
Some equations, specifically polynomials, have a known, finite number
of solutions.  The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}]
command uses @kbd{H a S} to solve an equation in general form, then, for
all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
variables like @code{n1} for which @code{n1} only usefully varies over
a finite range, it expands these variables out to all their possible
values.  The results are collected into a vector, which is returned.
For example, @samp{roots(x^4 = 1, x)} returns the four solutions
@samp{[1, -1, (0, 1), (0, -1)]}.  Generally an @var{n}th degree
polynomial will always have @var{n} roots on the complex plane.
(If you have given a @code{real} declaration for the solution
variable, then only the real-valued solutions, if any, will be
reported; @pxref{Declarations}.)

Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
symbolic solutions if the polynomial has symbolic coefficients.  Also
note that Calc's solver is not able to get exact symbolic solutions
to all polynomials.  Polynomials containing powers up to @expr{x^4}
can always be solved exactly; polynomials of higher degree sometimes
can be:  @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
which can be solved for @expr{x^3} using the quadratic equation, and then
for @expr{x} by taking cube roots.  But in many cases, like
@expr{x^6 + x + 1}, Calc does not know how to rewrite the polynomial
into a form it can solve.  The @kbd{a P} command can still deliver a
list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
is not turned on.  (If you work with Symbolic mode on, recall that the
@kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the
formula on the stack with Symbolic mode temporarily off.)  Naturally,
@kbd{a P} can only provide numerical roots if the polynomial coefficients
are all numbers (real or complex).

@node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations
@subsection Solving Systems of Equations

@noindent
@cindex Systems of equations, symbolic
You can also use the commands described above to solve systems of
simultaneous equations.  Just create a vector of equations, then
specify a vector of variables for which to solve.  (You can omit
the surrounding brackets when entering the vector of variables
at the prompt.)

For example, putting @samp{[x + y = a, x - y = b]} on the stack
and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
@samp{[x = a - (a-b)/2, y = (a-b)/2]}.  The result vector will
have the same length as the variables vector, and the variables
will be listed in the same order there.  Note that the solutions
are not always simplified as far as possible; the solution for
@expr{x} here could be improved by an application of the @kbd{a n}
command.

Calc's algorithm works by trying to eliminate one variable at a
time by solving one of the equations for that variable and then
substituting into the other equations.  Calc will try all the
possibilities, but you can speed things up by noting that Calc
first tries to eliminate the first variable with the first
equation, then the second variable with the second equation,
and so on.  It also helps to put the simpler (e.g., more linear)
equations toward the front of the list.  Calc's algorithm will
solve any system of linear equations, and also many kinds of
nonlinear systems.

@ignore
@starindex
@end ignore
@tindex elim
Normally there will be as many variables as equations.  If you
give fewer variables than equations (an ``over-determined'' system
of equations), Calc will find a partial solution.  For example,
typing @kbd{a S y @key{RET}} with the above system of equations
would produce @samp{[y = a - x]}.  There are now several ways to
express this solution in terms of the original variables; Calc uses
the first one that it finds.  You can control the choice by adding
variable specifiers of the form @samp{elim(@var{v})} to the
variables list.  This says that @var{v} should be eliminated from
the equations; the variable will not appear at all in the solution.
For example, typing @kbd{a S y,elim(x)} would yield
@samp{[y = a - (b+a)/2]}.

If the variables list contains only @code{elim} specifiers,
Calc simply eliminates those variables from the equations
and then returns the resulting set of equations.  For example,
@kbd{a S elim(x)} produces @samp{[a - 2 y = b]}.  Every variable
eliminated will reduce the number of equations in the system
by one.

Again, @kbd{a S} gives you one solution to the system of
equations.  If there are several solutions, you can use @kbd{H a S}
to get a general family of solutions, or, if there is a finite
number of solutions, you can use @kbd{a P} to get a list.  (In
the latter case, the result will take the form of a matrix where
the rows are different solutions and the columns correspond to the
variables you requested.)

Another way to deal with certain kinds of overdetermined systems of
equations is the @kbd{a F} command, which does least-squares fitting
to satisfy the equations.  @xref{Curve Fitting}.

@node Decomposing Polynomials,  , Solving Systems of Equations, Solving Equations
@subsection Decomposing Polynomials

@noindent
@ignore
@starindex
@end ignore
@tindex poly
The @code{poly} function takes a polynomial and a variable as
arguments, and returns a vector of polynomial coefficients (constant
coefficient first).  For example, @samp{poly(x^3 + 2 x, x)} returns
@expr{[0, 2, 0, 1]}.  If the input is not a polynomial in @expr{x},
the call to @code{poly} is left in symbolic form.  If the input does
not involve the variable @expr{x}, the input is returned in a list
of length one, representing a polynomial with only a constant
coefficient.  The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
The last element of the returned vector is guaranteed to be nonzero;
note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
Note also that @expr{x} may actually be any formula; for example,
@samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.

@cindex Coefficients of polynomial
@cindex Degree of polynomial
To get the @expr{x^k} coefficient of polynomial @expr{p}, use
@samp{poly(p, x)_(k+1)}.  To get the degree of polynomial @expr{p},
use @samp{vlen(poly(p, x)) - 1}.  For example, @samp{poly((x+1)^4, x)}
returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
gives the @expr{x^2} coefficient of this polynomial, 6.

@ignore
@starindex
@end ignore
@tindex gpoly
One important feature of the solver is its ability to recognize
formulas which are ``essentially'' polynomials.  This ability is
made available to the user through the @code{gpoly} function, which
is used just like @code{poly}:  @samp{gpoly(@var{expr}, @var{var})}.
If @var{expr} is a polynomial in some term which includes @var{var}, then
this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]}
where @var{x} is the term that depends on @var{var}, @var{c} is a
vector of polynomial coefficients (like the one returned by @code{poly}),
and @var{a} is a multiplier which is usually 1.  Basically,
@samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
@var{c}_3 @var{x}^2 + ...)}.  The last element of @var{c} is
guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
(i.e., the trivial decomposition @var{expr} = @var{x} is not
considered a polynomial).  One side effect is that @samp{gpoly(x, x)}
and @samp{gpoly(6, x)}, both of which might be expected to recognize
their arguments as polynomials, will not because the decomposition
is considered trivial.

For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.

The term @var{x} may itself be a polynomial in @var{var}.  This is
done to reduce the size of the @var{c} vector.  For example,
@samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
since a quadratic polynomial in @expr{x^2} is easier to solve than
a quartic polynomial in @expr{x}.

A few more examples of the kinds of polynomials @code{gpoly} can
discover:

@smallexample
sin(x) - 1               [sin(x), [-1, 1], 1]
x + 1/x - 1              [x, [1, -1, 1], 1/x]
x + 1/x                  [x^2, [1, 1], 1/x]
x^3 + 2 x                [x^2, [2, 1], x]
x + x^2:3 + sqrt(x)      [x^1:6, [1, 1, 0, 1], x^1:2]
x^(2a) + 2 x^a + 5       [x^a, [5, 2, 1], 1]
(exp(-x) + exp(x)) / 2   [e^(2 x), [0.5, 0.5], e^-x]
@end smallexample

The @code{poly} and @code{gpoly} functions accept a third integer argument
which specifies the largest degree of polynomial that is acceptable.
If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
or less will be returned.  Otherwise, the @code{poly} or @code{gpoly}
call will remain in symbolic form.  For example, the equation solver
can handle quartics and smaller polynomials, so it calls
@samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
can be treated by its linear, quadratic, cubic, or quartic formulas.

@ignore
@starindex
@end ignore
@tindex pdeg
The @code{pdeg} function computes the degree of a polynomial;
@samp{pdeg(p,x)} is the highest power of @code{x} that appears in
@code{p}.  This is the same as @samp{vlen(poly(p,x))-1}, but is
much more efficient.  If @code{p} is constant with respect to @code{x},
then @samp{pdeg(p,x) = 0}.  If @code{p} is not a polynomial in @code{x}
(e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
It is possible to omit the second argument @code{x}, in which case
@samp{pdeg(p)} returns the highest total degree of any term of the
polynomial, counting all variables that appear in @code{p}.  Note
that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
the degree of the constant zero is considered to be @code{-inf}
(minus infinity).

@ignore
@starindex
@end ignore
@tindex plead
The @code{plead} function finds the leading term of a polynomial.
Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
though again more efficient.  In particular, @samp{plead((2x+1)^10, x)}
returns 1024 without expanding out the list of coefficients.  The
value of @code{plead(p,x)} will be zero only if @expr{p = 0}.

@ignore
@starindex
@end ignore
@tindex pcont
The @code{pcont} function finds the @dfn{content} of a polynomial.  This
is the greatest common divisor of all the coefficients of the polynomial.
With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
to get a list of coefficients, then uses @code{pgcd} (the polynomial
GCD function) to combine these into an answer.  For example,
@samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}.  The content is
basically the ``biggest'' polynomial that can be divided into @code{p}
exactly.  The sign of the content is the same as the sign of the leading
coefficient.

With only one argument, @samp{pcont(p)} computes the numerical
content of the polynomial, i.e., the @code{gcd} of the numerical
coefficients of all the terms in the formula.  Note that @code{gcd}
is defined on rational numbers as well as integers; it computes
the @code{gcd} of the numerators and the @code{lcm} of the
denominators.  Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
Dividing the polynomial by this number will clear all the
denominators, as well as dividing by any common content in the
numerators.  The numerical content of a polynomial is negative only
if all the coefficients in the polynomial are negative.

@ignore
@starindex
@end ignore
@tindex pprim
The @code{pprim} function finds the @dfn{primitive part} of a
polynomial, which is simply the polynomial divided (using @code{pdiv}
if necessary) by its content.  If the input polynomial has rational
coefficients, the result will have integer coefficients in simplest
terms.

@node Numerical Solutions, Curve Fitting, Solving Equations, Algebra
@section Numerical Solutions

@noindent
Not all equations can be solved symbolically.  The commands in this
section use numerical algorithms that can find a solution to a specific
instance of an equation to any desired accuracy.  Note that the
numerical commands are slower than their algebraic cousins; it is a
good idea to try @kbd{a S} before resorting to these commands.

(@xref{Curve Fitting}, for some other, more specialized, operations
on numerical data.)

@menu
* Root Finding::
* Minimization::
* Numerical Systems of Equations::
@end menu

@node Root Finding, Minimization, Numerical Solutions, Numerical Solutions
@subsection Root Finding

@noindent
@kindex a R
@pindex calc-find-root
@tindex root
@cindex Newton's method
@cindex Roots of equations
@cindex Numerical root-finding
The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a
numerical solution (or @dfn{root}) of an equation.  (This command treats
inequalities the same as equations.  If the input is any other kind
of formula, it is interpreted as an equation of the form @expr{X = 0}.)

The @kbd{a R} command requires an initial guess on the top of the
stack, and a formula in the second-to-top position.  It prompts for a
solution variable, which must appear in the formula.  All other variables
that appear in the formula must have assigned values, i.e., when
a value is assigned to the solution variable and the formula is
evaluated with @kbd{=}, it should evaluate to a number.  Any assigned
value for the solution variable itself is ignored and unaffected by
this command.

When the command completes, the initial guess is replaced on the stack
by a vector of two numbers:  The value of the solution variable that
solves the equation, and the difference between the lefthand and
righthand sides of the equation at that value.  Ordinarily, the second
number will be zero or very nearly zero.  (Note that Calc uses a
slightly higher precision while finding the root, and thus the second
number may be slightly different from the value you would compute from
the equation yourself.)

The @kbd{v h} (@code{calc-head}) command is a handy way to extract
the first element of the result vector, discarding the error term.

The initial guess can be a real number, in which case Calc searches
for a real solution near that number, or a complex number, in which
case Calc searches the whole complex plane near that number for a
solution, or it can be an interval form which restricts the search
to real numbers inside that interval.

Calc tries to use @kbd{a d} to take the derivative of the equation.
If this succeeds, it uses Newton's method.  If the equation is not
differentiable Calc uses a bisection method.  (If Newton's method
appears to be going astray, Calc switches over to bisection if it
can, or otherwise gives up.  In this case it may help to try again
with a slightly different initial guess.)  If the initial guess is a
complex number, the function must be differentiable.

If the formula (or the difference between the sides of an equation)
is negative at one end of the interval you specify and positive at
the other end, the root finder is guaranteed to find a root.
Otherwise, Calc subdivides the interval into small parts looking for
positive and negative values to bracket the root.  When your guess is
an interval, Calc will not look outside that interval for a root.

@kindex H a R
@tindex wroot
The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except
that if the initial guess is an interval for which the function has
the same sign at both ends, then rather than subdividing the interval
Calc attempts to widen it to enclose a root.  Use this mode if
you are not sure if the function has a root in your interval.

If the function is not differentiable, and you give a simple number
instead of an interval as your initial guess, Calc uses this widening
process even if you did not type the Hyperbolic flag.  (If the function
@emph{is} differentiable, Calc uses Newton's method which does not
require a bounding interval in order to work.)

If Calc leaves the @code{root} or @code{wroot} function in symbolic
form on the stack, it will normally display an explanation for why
no root was found.  If you miss this explanation, press @kbd{w}
(@code{calc-why}) to get it back.

@node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions
@subsection Minimization

@noindent
@kindex a N
@kindex H a N
@kindex a X
@kindex H a X
@pindex calc-find-minimum
@pindex calc-find-maximum
@tindex minimize
@tindex maximize
@cindex Minimization, numerical
The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command
finds a minimum value for a formula.  It is very similar in operation
to @kbd{a R} (@code{calc-find-root}):  You give the formula and an initial
guess on the stack, and are prompted for the name of a variable.  The guess
may be either a number near the desired minimum, or an interval enclosing
the desired minimum.  The function returns a vector containing the
value of the variable which minimizes the formula's value, along
with the minimum value itself.

Note that this command looks for a @emph{local} minimum.  Many functions
have more than one minimum; some, like 
@texline @math{x \sin x},
@infoline @expr{x sin(x)}, 
have infinitely many.  In fact, there is no easy way to define the
``global'' minimum of 
@texline @math{x \sin x}
@infoline @expr{x sin(x)} 
but Calc can still locate any particular local minimum
for you.  Calc basically goes downhill from the initial guess until it
finds a point at which the function's value is greater both to the left
and to the right.  Calc does not use derivatives when minimizing a function.

If your initial guess is an interval and it looks like the minimum
occurs at one or the other endpoint of the interval, Calc will return
that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
@expr{(2..3]} would report no minimum found.  In general, you should
use closed intervals to find literally the minimum value in that
range of @expr{x}, or open intervals to find the local minimum, if
any, that happens to lie in that range.

Most functions are smooth and flat near their minimum values.  Because
of this flatness, if the current precision is, say, 12 digits, the
variable can only be determined meaningfully to about six digits.  Thus
you should set the precision to twice as many digits as you need in your
answer.

@ignore
@mindex wmin@idots
@end ignore
@tindex wminimize
@ignore
@mindex wmax@idots
@end ignore
@tindex wmaximize
The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R},
expands the guess interval to enclose a minimum rather than requiring
that the minimum lie inside the interval you supply.

The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and
@kbd{H a X} [@code{wmaximize}] commands effectively minimize the
negative of the formula you supply.

The formula must evaluate to a real number at all points inside the
interval (or near the initial guess if the guess is a number).  If
the initial guess is a complex number the variable will be minimized
over the complex numbers; if it is real or an interval it will
be minimized over the reals.

@node Numerical Systems of Equations,  , Minimization, Numerical Solutions
@subsection Systems of Equations

@noindent
@cindex Systems of equations, numerical
The @kbd{a R} command can also solve systems of equations.  In this
case, the equation should instead be a vector of equations, the
guess should instead be a vector of numbers (intervals are not
supported), and the variable should be a vector of variables.  You
can omit the brackets while entering the list of variables.  Each
equation must be differentiable by each variable for this mode to
work.  The result will be a vector of two vectors:  The variable
values that solved the system of equations, and the differences
between the sides of the equations with those variable values.
There must be the same number of equations as variables.  Since
only plain numbers are allowed as guesses, the Hyperbolic flag has
no effect when solving a system of equations.

It is also possible to minimize over many variables with @kbd{a N}
(or maximize with @kbd{a X}).  Once again the variable name should
be replaced by a vector of variables, and the initial guess should
be an equal-sized vector of initial guesses.  But, unlike the case of
multidimensional @kbd{a R}, the formula being minimized should
still be a single formula, @emph{not} a vector.  Beware that
multidimensional minimization is currently @emph{very} slow.

@node Curve Fitting, Summations, Numerical Solutions, Algebra
@section Curve Fitting

@noindent
The @kbd{a F} command fits a set of data to a @dfn{model formula},
such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
to be determined.  For a typical set of measured data there will be
no single @expr{m} and @expr{b} that exactly fit the data; in this
case, Calc chooses values of the parameters that provide the closest
possible fit.

@menu
* Linear Fits::
* Polynomial and Multilinear Fits::
* Error Estimates for Fits::
* Standard Nonlinear Models::
* Curve Fitting Details::
* Interpolation::
@end menu

@node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting
@subsection Linear Fits

@noindent
@kindex a F
@pindex calc-curve-fit
@tindex fit
@cindex Linear regression
@cindex Least-squares fits
The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts
to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
straight line, polynomial, or other function of @expr{x}.  For the
moment we will consider only the case of fitting to a line, and we
will ignore the issue of whether or not the model was in fact a good
fit for the data.

In a standard linear least-squares fit, we have a set of @expr{(x,y)}
data points that we wish to fit to the model @expr{y = m x + b}
by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
values calculated from the formula be as close as possible to the actual
@expr{y} values in the data set.  (In a polynomial fit, the model is
instead, say, @expr{y = a x^3 + b x^2 + c x + d}.  In a multilinear fit,
we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
@expr{y = a x_1 + b x_2 + c x_3 + d}.  These will be discussed later.)

In the model formula, variables like @expr{x} and @expr{x_2} are called
the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
variable}.  Variables like @expr{m}, @expr{a}, and @expr{b} are called
the @dfn{parameters} of the model.

The @kbd{a F} command takes the data set to be fitted from the stack.
By default, it expects the data in the form of a matrix.  For example,
for a linear or polynomial fit, this would be a 
@texline @math{2\times N}
@infoline 2xN
matrix where the first row is a list of @expr{x} values and the second
row has the corresponding @expr{y} values.  For the multilinear fit
shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
@expr{x_3}, and @expr{y}, respectively).

If you happen to have an 
@texline @math{N\times2}
@infoline Nx2
matrix instead of a 
@texline @math{2\times N}
@infoline 2xN
matrix, just press @kbd{v t} first to transpose the matrix.

After you type @kbd{a F}, Calc prompts you to select a model.  For a
linear fit, press the digit @kbd{1}.

Calc then prompts for you to name the variables.  By default it chooses
high letters like @expr{x} and @expr{y} for independent variables and
low letters like @expr{a} and @expr{b} for parameters.  (The dependent
variable doesn't need a name.)  The two kinds of variables are separated
by a semicolon.  Since you generally care more about the names of the
independent variables than of the parameters, Calc also allows you to
name only those and let the parameters use default names.

For example, suppose the data matrix

@ifinfo
@example
@group
[ [ 1, 2, 3, 4,  5  ]
  [ 5, 7, 9, 11, 13 ] ]
@end group
@end example
@end ifinfo
@tex
\turnoffactive
\turnoffactive
\beforedisplay
$$ \pmatrix{ 1 & 2 & 3 & 4  & 5  \cr
             5 & 7 & 9 & 11 & 13 }
$$
\afterdisplay
@end tex

@noindent
is on the stack and we wish to do a simple linear fit.  Type
@kbd{a F}, then @kbd{1} for the model, then @key{RET} to use
the default names.  The result will be the formula @expr{3 + 2 x}
on the stack.  Calc has created the model expression @kbd{a + b x},
then found the optimal values of @expr{a} and @expr{b} to fit the
data.  (In this case, it was able to find an exact fit.)  Calc then
substituted those values for @expr{a} and @expr{b} in the model
formula.

The @kbd{a F} command puts two entries in the trail.  One is, as
always, a copy of the result that went to the stack; the other is
a vector of the actual parameter values, written as equations:
@expr{[a = 3, b = 2]}, in case you'd rather read them in a list
than pick them out of the formula.  (You can type @kbd{t y}
to move this vector to the stack; see @ref{Trail Commands}.

Specifying a different independent variable name will affect the
resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
the equations that go into the trail.

@tex
\bigskip
@end tex

To see what happens when the fit is not exact, we could change
the number 13 in the data matrix to 14 and try the fit again.
The result is:

@example
2.6 + 2.2 x
@end example

Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $ @key{RET}}, shows
a reasonably close match to the y-values in the data.

@example
[4.8, 7., 9.2, 11.4, 13.6]
@end example

Since there is no line which passes through all the @var{n} data points,
Calc has chosen a line that best approximates the data points using
the method of least squares.  The idea is to define the @dfn{chi-square}
error measure

@ifinfo
@example
chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
\afterdisplay
@end tex

@noindent
which is clearly zero if @expr{a + b x} exactly fits all data points,
and increases as various @expr{a + b x_i} values fail to match the
corresponding @expr{y_i} values.  There are several reasons why the
summand is squared, one of them being to ensure that 
@texline @math{\chi^2 \ge 0}.
@infoline @expr{chi^2 >= 0}.
Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
for which the error 
@texline @math{\chi^2}
@infoline @expr{chi^2} 
is as small as possible.

Other kinds of models do the same thing but with a different model
formula in place of @expr{a + b x_i}.

@tex
\bigskip
@end tex

A numeric prefix argument causes the @kbd{a F} command to take the
data in some other form than one big matrix.  A positive argument @var{n}
will take @var{N} items from the stack, corresponding to the @var{n} rows
of a data matrix.  In the linear case, @var{n} must be 2 since there
is always one independent variable and one dependent variable.

A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
items from the stack, an @var{n}-row matrix of @expr{x} values, and a
vector of @expr{y} values.  If there is only one independent variable,
the @expr{x} values can be either a one-row matrix or a plain vector,
in which case the @kbd{C-u} prefix is the same as a @w{@kbd{C-u 2}} prefix.

@node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting
@subsection Polynomial and Multilinear Fits

@noindent
To fit the data to higher-order polynomials, just type one of the
digits @kbd{2} through @kbd{9} when prompted for a model.  For example,
we could fit the original data matrix from the previous section
(with 13, not 14) to a parabola instead of a line by typing
@kbd{a F 2 @key{RET}}.

@example
2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
@end example

Note that since the constant and linear terms are enough to fit the
data exactly, it's no surprise that Calc chose a tiny contribution
for @expr{x^2}.  (The fact that it's not exactly zero is due only
to roundoff error.  Since our data are exact integers, we could get
an exact answer by typing @kbd{m f} first to get Fraction mode.
Then the @expr{x^2} term would vanish altogether.  Usually, though,
the data being fitted will be approximate floats so Fraction mode
won't help.)

Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
gives a much larger @expr{x^2} contribution, as Calc bends the
line slightly to improve the fit.

@example
0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
@end example

An important result from the theory of polynomial fitting is that it
is always possible to fit @var{n} data points exactly using a polynomial
of degree @mathit{@var{n}-1}, sometimes called an @dfn{interpolating polynomial}.
Using the modified (14) data matrix, a model number of 4 gives
a polynomial that exactly matches all five data points:

@example
0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
@end example

The actual coefficients we get with a precision of 12, like
@expr{0.0416666663588}, clearly suffer from loss of precision.
It is a good idea to increase the working precision to several
digits beyond what you need when you do a fitting operation.
Or, if your data are exact, use Fraction mode to get exact
results.

You can type @kbd{i} instead of a digit at the model prompt to fit
the data exactly to a polynomial.  This just counts the number of
columns of the data matrix to choose the degree of the polynomial
automatically.

Fitting data ``exactly'' to high-degree polynomials is not always
a good idea, though.  High-degree polynomials have a tendency to
wiggle uncontrollably in between the fitting data points.  Also,
if the exact-fit polynomial is going to be used to interpolate or
extrapolate the data, it is numerically better to use the @kbd{a p}
command described below.  @xref{Interpolation}.

@tex
\bigskip
@end tex

Another generalization of the linear model is to assume the
@expr{y} values are a sum of linear contributions from several
@expr{x} values.  This is a @dfn{multilinear} fit, and it is also
selected by the @kbd{1} digit key.  (Calc decides whether the fit
is linear or multilinear by counting the rows in the data matrix.)

Given the data matrix,

@example
@group
[ [  1,   2,   3,    4,   5  ]
  [  7,   2,   3,    5,   2  ]
  [ 14.5, 15, 18.5, 22.5, 24 ] ]
@end group
@end example

@noindent
the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
second row @expr{y}, and will fit the values in the third row to the
model @expr{a + b x + c y}.

@example
8. + 3. x + 0.5 y
@end example

Calc can do multilinear fits with any number of independent variables
(i.e., with any number of data rows).

@tex
\bigskip
@end tex

Yet another variation is @dfn{homogeneous} linear models, in which
the constant term is known to be zero.  In the linear case, this
means the model formula is simply @expr{a x}; in the multilinear
case, the model might be @expr{a x + b y + c z}; and in the polynomial
case, the model could be @expr{a x + b x^2 + c x^3}.  You can get
a homogeneous linear or multilinear model by pressing the letter
@kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}.

It is certainly possible to have other constrained linear models,
like @expr{2.3 + a x} or @expr{a - 4 x}.  While there is no single
key to select models like these, a later section shows how to enter
any desired model by hand.  In the first case, for example, you
would enter @kbd{a F ' 2.3 + a x}.

Another class of models that will work but must be entered by hand
are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.

@node Error Estimates for Fits, Standard Nonlinear Models, Polynomial and Multilinear Fits, Curve Fitting
@subsection Error Estimates for Fits

@noindent
@kindex H a F
@tindex efit
With the Hyperbolic flag, @kbd{H a F} [@code{efit}] performs the same
fitting operation as @kbd{a F}, but reports the coefficients as error
forms instead of plain numbers.  Fitting our two data matrices (first
with 13, then with 14) to a line with @kbd{H a F} gives the results,

@example
3. + 2. x
2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
@end example

In the first case the estimated errors are zero because the linear
fit is perfect.  In the second case, the errors are nonzero but
moderately small, because the data are still very close to linear.

It is also possible for the @emph{input} to a fitting operation to
contain error forms.  The data values must either all include errors
or all be plain numbers.  Error forms can go anywhere but generally
go on the numbers in the last row of the data matrix.  If the last
row contains error forms
@texline `@var{y_i}@w{ @tfn{+/-} }@math{\sigma_i}', 
@infoline `@var{y_i}@w{ @tfn{+/-} }@var{sigma_i}', 
then the 
@texline @math{\chi^2}
@infoline @expr{chi^2}
statistic is now,

@ifinfo
@example
chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
@end example
@end ifinfo
@tex
\turnoffactive
\beforedisplay
$$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
\afterdisplay
@end tex

@noindent
so that data points with larger error estimates contribute less to
the fitting operation.

If there are error forms on other rows of the data matrix, all the
errors for a given data point are combined; the square root of the
sum of the squares of the errors forms the 
@texline @math{\sigma_i}
@infoline @expr{sigma_i} 
used for the data point.

Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
matrix, although if you are concerned about error analysis you will
probably use @kbd{H a F} so that the output also contains error
estimates.

If the input contains error forms but all the 
@texline @math{\sigma_i}
@infoline @expr{sigma_i} 
values are the same, it is easy to see that the resulting fitted model
will be the same as if the input did not have error forms at all 
@texline (@math{\chi^2}
@infoline (@expr{chi^2}
is simply scaled uniformly by 
@texline @math{1 / \sigma^2},
@infoline @expr{1 / sigma^2}, 
which doesn't affect where it has a minimum).  But there @emph{will} be
a difference in the estimated errors of the coefficients reported by
@kbd{H a F}. 

Consult any text on statistical modeling of data for a discussion
of where these error estimates come from and how they should be
interpreted.

@tex
\bigskip
@end tex

@kindex I a F
@tindex xfit
With the Inverse flag, @kbd{I a F} [@code{xfit}] produces even more
information.  The result is a vector of six items:

@enumerate
@item
The model formula with error forms for its coefficients or
parameters.  This is the result that @kbd{H a F} would have
produced.

@item
A vector of ``raw'' parameter values for the model.  These are the
polynomial coefficients or other parameters as plain numbers, in the
same order as the parameters appeared in the final prompt of the
@kbd{I a F} command.  For polynomials of degree @expr{d}, this vector
will have length @expr{M = d+1} with the constant term first.

@item
The covariance matrix @expr{C} computed from the fit.  This is
an @var{m}x@var{m} symmetric matrix; the diagonal elements
@texline @math{C_{jj}}
@infoline @expr{C_j_j} 
are the variances 
@texline @math{\sigma_j^2}
@infoline @expr{sigma_j^2} 
of the parameters.  The other elements are covariances
@texline @math{\sigma_{ij}^2} 
@infoline @expr{sigma_i_j^2} 
that describe the correlation between pairs of parameters.  (A related
set of numbers, the @dfn{linear correlation coefficients} 
@texline @math{r_{ij}},
@infoline @expr{r_i_j},
are defined as 
@texline @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
@infoline @expr{sigma_i_j^2 / sigma_i sigma_j}.)

@item
A vector of @expr{M} ``parameter filter'' functions whose
meanings are described below.  If no filters are necessary this
will instead be an empty vector; this is always the case for the
polynomial and multilinear fits described so far.

@item
The value of 
@texline @math{\chi^2}
@infoline @expr{chi^2} 
for the fit, calculated by the formulas shown above.  This gives a
measure of the quality of the fit; statisticians consider
@texline @math{\chi^2 \approx N - M}
@infoline @expr{chi^2 = N - M} 
to indicate a moderately good fit (where again @expr{N} is the number of
data points and @expr{M} is the number of parameters).

@item
A measure of goodness of fit expressed as a probability @expr{Q}.
This is computed from the @code{utpc} probability distribution
function using 
@texline @math{\chi^2}
@infoline @expr{chi^2} 
with @expr{N - M} degrees of freedom.  A
value of 0.5 implies a good fit; some texts recommend that often
@expr{Q = 0.1} or even 0.001 can signify an acceptable fit.  In
particular, 
@texline @math{\chi^2}
@infoline @expr{chi^2} 
statistics assume the errors in your inputs
follow a normal (Gaussian) distribution; if they don't, you may
have to accept smaller values of @expr{Q}.

The @expr{Q} value is computed only if the input included error
estimates.  Otherwise, Calc will report the symbol @code{nan}
for @expr{Q}.  The reason is that in this case the 
@texline @math{\chi^2}
@infoline @expr{chi^2}
value has effectively been used to estimate the original errors
in the input, and thus there is no redundant information left
over to use for a confidence test.
@end enumerate

@node Standard Nonlinear Models, Curve Fitting Details, Error Estimates for Fits, Curve Fitting
@subsection Standard Nonlinear Models

@noindent
The @kbd{a F} command also accepts other kinds of models besides
lines and polynomials.  Some common models have quick single-key
abbreviations; others must be entered by hand as algebraic formulas.

Here is a complete list of the standard models recognized by @kbd{a F}:

@table @kbd
@item 1
Linear or multilinear.  @mathit{a + b x + c y + d z}.
@item 2-9
Polynomials.  @mathit{a + b x + c x^2 + d x^3}.
@item e
Exponential.  @mathit{a} @tfn{exp}@mathit{(b x)} @tfn{exp}@mathit{(c y)}.
@item E
Base-10 exponential.  @mathit{a} @tfn{10^}@mathit{(b x)} @tfn{10^}@mathit{(c y)}.
@item x
Exponential (alternate notation).  @tfn{exp}@mathit{(a + b x + c y)}.
@item X
Base-10 exponential (alternate).  @tfn{10^}@mathit{(a + b x + c y)}.
@item l
Logarithmic.  @mathit{a + b} @tfn{ln}@mathit{(x) + c} @tfn{ln}@mathit{(y)}.
@item L
Base-10 logarithmic.  @mathit{a + b} @tfn{log10}@mathit{(x) + c} @tfn{log10}@mathit{(y)}.
@item ^
General exponential.  @mathit{a b^x c^y}.
@item p
Power law.  @mathit{a x^b y^c}.
@item q
Quadratic.  @mathit{a + b (x-c)^2 + d (x-e)^2}.
@item g
Gaussian.  
@texline @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}.
@infoline @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
@end table

All of these models are used in the usual way; just press the appropriate
letter at the model prompt, and choose variable names if you wish.  The
result will be a formula as shown in the above table, with the best-fit
values of the parameters substituted.  (You may find it easier to read
the parameter values from the vector that is placed in the trail.)

All models except Gaussian and polynomials can generalize as shown to any
number of independent variables.  Also, all the built-in models have an
additive or multiplicative parameter shown as @expr{a} in the above table
which can be replaced by zero or one, as appropriate, by typing @kbd{h}
before the model key.

Note that many of these models are essentially equivalent, but express
the parameters slightly differently.  For example, @expr{a b^x} and
the other two exponential models are all algebraic rearrangements of
each other.  Also, the ``quadratic'' model is just a degree-2 polynomial
with the parameters expressed differently.  Use whichever form best
matches the problem.

The HP-28/48 calculators support four different models for curve
fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
@samp{a exp(b x)}, and @samp{a x^b}, respectively.  In each case,
@expr{a} is what the HP-48 identifies as the ``intercept,'' and
@expr{b} is what it calls the ``slope.''

@tex
\bigskip
@end tex

If the model you want doesn't appear on this list, press @kbd{'}
(the apostrophe key) at the model prompt to enter any algebraic
formula, such as @kbd{m x - b}, as the model.  (Not all models
will work, though---see the next section for details.)

The model can also be an equation like @expr{y = m x + b}.
In this case, Calc thinks of all the rows of the data matrix on
equal terms; this model effectively has two parameters
(@expr{m} and @expr{b}) and two independent variables (@expr{x}
and @expr{y}), with no ``dependent'' variables.  Model equations
do not need to take this @expr{y =} form.  For example, the
implicit line equation @expr{a x + b y = 1} works fine as a
model.

When you enter a model, Calc makes an alphabetical list of all
the variables that appear in the model.  These are used for the
default parameters, independent variables, and dependent variable
(in that order).  If you enter a plain formula (not an equation),
Calc assumes the dependent variable does not appear in the formula
and thus does not need a name.

For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
and the data matrix has three rows (meaning two independent variables),
Calc will use @expr{a,mu,sigma} as the default parameters, and the
data rows will be named @expr{t} and @expr{x}, respectively.  If you
enter an equation instead of a plain formula, Calc will use @expr{a,mu}
as the parameters, and @expr{sigma,t,x} as the three independent
variables.

You can, of course, override these choices by entering something
different at the prompt.  If you leave some variables out of the list,
those variables must have stored values and those stored values will
be used as constants in the model.  (Stored values for the parameters
and independent variables are ignored by the @kbd{a F} command.)
If you list only independent variables, all the remaining variables
in the model formula will become parameters.

If there are @kbd{$} signs in the model you type, they will stand
for parameters and all other variables (in alphabetical order)
will be independent.  Use @kbd{$} for one parameter, @kbd{$$} for
another, and so on.  Thus @kbd{$ x + $$} is another way to describe
a linear model.

If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
Calc will take the model formula from the stack.  (The data must then
appear at the second stack level.)  The same conventions are used to
choose which variables in the formula are independent by default and
which are parameters.

Models taken from the stack can also be expressed as vectors of
two or three elements, @expr{[@var{model}, @var{vars}]} or
@expr{[@var{model}, @var{vars}, @var{params}]}.  Each of @var{vars}
and @var{params} may be either a variable or a vector of variables.
(If @var{params} is omitted, all variables in @var{model} except
those listed as @var{vars} are parameters.)

When you enter a model manually with @kbd{'}, Calc puts a 3-vector
describing the model in the trail so you can get it back if you wish.

@tex
\bigskip
@end tex

@vindex Model1
@vindex Model2
Finally, you can store a model in one of the Calc variables
@code{Model1} or @code{Model2}, then use this model by typing
@kbd{a F u} or @kbd{a F U} (respectively).  The value stored in
the variable can be any of the formats that @kbd{a F $} would
accept for a model on the stack.

@tex
\bigskip
@end tex

Calc uses the principal values of inverse functions like @code{ln}
and @code{arcsin} when doing fits.  For example, when you enter
the model @samp{y = sin(a t + b)} Calc actually uses the easier
form @samp{arcsin(y) = a t + b}.  The @code{arcsin} function always
returns results in the range from @mathit{-90} to 90 degrees (or the
equivalent range in radians).  Suppose you had data that you
believed to represent roughly three oscillations of a sine wave,
so that the argument of the sine might go from zero to 
@texline @math{3\times360}
@infoline @mathit{3*360} 
degrees.
The above model would appear to be a good way to determine the
true frequency and phase of the sine wave, but in practice it
would fail utterly.  The righthand side of the actual model
@samp{arcsin(y) = a t + b} will grow smoothly with @expr{t}, but
the lefthand side will bounce back and forth between @mathit{-90} and 90.
No values of @expr{a} and @expr{b} can make the two sides match,
even approximately.

There is no good solution to this problem at present.  You could
restrict your data to small enough ranges so that the above problem
doesn't occur (i.e., not straddling any peaks in the sine wave).
Or, in this case, you could use a totally different method such as
Fourier analysis, which is beyond the scope of the @kbd{a F} command.
(Unfortunately, Calc does not currently have any facilities for
taking Fourier and related transforms.)

@node Curve Fitting Details, Interpolation, Standard Nonlinear Models, Curve Fitting
@subsection Curve Fitting Details

@noindent
Calc's internal least-squares fitter can only handle multilinear
models.  More precisely, it can handle any model of the form
@expr{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
are the parameters and @expr{x,y,z} are the independent variables
(of course there can be any number of each, not just three).

In a simple multilinear or polynomial fit, it is easy to see how
to convert the model into this form.  For example, if the model
is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
and @expr{h(x) = x^2} are suitable functions.

For other models, Calc uses a variety of algebraic manipulations
to try to put the problem into the form

@smallexample
Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z)
@end smallexample

@noindent
where @expr{Y,A,B,C,F,G,H} are arbitrary functions.  It computes
@expr{Y}, @expr{F}, @expr{G}, and @expr{H} for all the data points,
does a standard linear fit to find the values of @expr{A}, @expr{B},
and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
in terms of @expr{A,B,C}.

A remarkable number of models can be cast into this general form.
We'll look at two examples here to see how it works.  The power-law
model @expr{y = a x^b} with two independent variables and two parameters
can be rewritten as follows:

@example
y = a x^b
y = a exp(b ln(x))
y = exp(ln(a) + b ln(x))
ln(y) = ln(a) + b ln(x)
@end example

@noindent
which matches the desired form with 
@texline @math{Y = \ln(y)},
@infoline @expr{Y = ln(y)}, 
@texline @math{A = \ln(a)},
@infoline @expr{A = ln(a)},
@expr{F = 1}, @expr{B = b}, and 
@texline @math{G = \ln(x)}.
@infoline @expr{G = ln(x)}.  
Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
does a linear fit for @expr{A} and @expr{B}, then solves to get 
@texline @math{a = \exp(A)} 
@infoline @expr{a = exp(A)} 
and @expr{b = B}.

Another interesting example is the ``quadratic'' model, which can
be handled by expanding according to the distributive law.

@example
y = a + b*(x - c)^2
y = a + b c^2 - 2 b c x + b x^2
@end example

@noindent
which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
@expr{B = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as easily
have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
@expr{H = x^2}.

The Gaussian model looks quite complicated, but a closer examination
shows that it's actually similar to the quadratic model but with an
exponential that can be brought to the top and moved into @expr{Y}.

An example of a model that cannot be put into general linear
form is a Gaussian with a constant background added on, i.e.,
@expr{d} + the regular Gaussian formula.  If you have a model like
this, your best bet is to replace enough of your parameters with
constants to make the model linearizable, then adjust the constants
manually by doing a series of fits.  You can compare the fits by
graphing them, by examining the goodness-of-fit measures returned by
@kbd{I a F}, or by some other method suitable to your application.
Note that some models can be linearized in several ways.  The
Gaussian-plus-@var{d} model can be linearized by setting @expr{d}
(the background) to a constant, or by setting @expr{b} (the standard
deviation) and @expr{c} (the mean) to constants.

To fit a model with constants substituted for some parameters, just
store suitable values in those parameter variables, then omit them
from the list of parameters when you answer the variables prompt.

@tex
\bigskip
@end tex

A last desperate step would be to use the general-purpose
@code{minimize} function rather than @code{fit}.  After all, both
functions solve the problem of minimizing an expression (the 
@texline @math{\chi^2}
@infoline @expr{chi^2}
sum) by adjusting certain parameters in the expression.  The @kbd{a F}
command is able to use a vastly more efficient algorithm due to its
special knowledge about linear chi-square sums, but the @kbd{a N}
command can do the same thing by brute force.

A compromise would be to pick out a few parameters without which the
fit is linearizable, and use @code{minimize} on a call to @code{fit}
which efficiently takes care of the rest of the parameters.  The thing
to be minimized would be the value of 
@texline @math{\chi^2}
@infoline @expr{chi^2} 
returned as the fifth result of the @code{xfit} function:

@smallexample
minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
@end smallexample

@noindent
where @code{gaus} represents the Gaussian model with background,
@code{data} represents the data matrix, and @code{guess} represents
the initial guess for @expr{d} that @code{minimize} requires.
This operation will only be, shall we say, extraordinarily slow
rather than astronomically slow (as would be the case if @code{minimize}
were used by itself to solve the problem).

@tex
\bigskip
@end tex

The @kbd{I a F} [@code{xfit}] command is somewhat trickier when
nonlinear models are used.  The second item in the result is the
vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}.  The
covariance matrix is written in terms of those raw parameters.
The fifth item is a vector of @dfn{filter} expressions.  This
is the empty vector @samp{[]} if the raw parameters were the same
as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
and so on (which is always true if the model is already linear
in the parameters as written, e.g., for polynomial fits).  If the
parameters had to be rearranged, the fifth item is instead a vector
of one formula per parameter in the original model.  The raw
parameters are expressed in these ``filter'' formulas as
@samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
and so on.

When Calc needs to modify the model to return the result, it replaces
@samp{fitdummy(1)} in all the filters with the first item in the raw
parameters list, and so on for the other raw parameters, then
evaluates the resulting filter formulas to get the actual parameter
values to be substituted into the original model.  In the case of
@kbd{H a F} and @kbd{I a F} where the parameters must be error forms,
Calc uses the square roots of the diagonal entries of the covariance
matrix as error values for the raw parameters, then lets Calc's
standard error-form arithmetic take it from there.

If you use @kbd{I a F} with a nonlinear model, be sure to remember
that the covariance matrix is in terms of the raw parameters,
@emph{not} the actual requested parameters.  It's up to you to
figure out how to interpret the covariances in the presence of
nontrivial filter functions.

Things are also complicated when the input contains error forms.
Suppose there are three independent and dependent variables, @expr{x},
@expr{y}, and @expr{z}, one or more of which are error forms in the
data.  Calc combines all the error values by taking the square root
of the sum of the squares of the errors.  It then changes @expr{x}
and @expr{y} to be plain numbers, and makes @expr{z} into an error
form with this combined error.  The @expr{Y(x,y,z)} part of the
linearized model is evaluated, and the result should be an error
form.  The error part of that result is used for 
@texline @math{\sigma_i}
@infoline @expr{sigma_i} 
for the data point.  If for some reason @expr{Y(x,y,z)} does not return 
an error form, the combined error from @expr{z} is used directly for 
@texline @math{\sigma_i}.
@infoline @expr{sigma_i}.  
Finally, @expr{z} is also stripped of its error
for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
the righthand side of the linearized model is computed in regular
arithmetic with no error forms.

(While these rules may seem complicated, they are designed to do
the most reasonable thing in the typical case that @expr{Y(x,y,z)}
depends only on the dependent variable @expr{z}, and in fact is
often simply equal to @expr{z}.  For common cases like polynomials
and multilinear models, the combined error is simply used as the
@texline @math{\sigma}
@infoline @expr{sigma} 
for the data point with no further ado.)

@tex
\bigskip
@end tex

@vindex FitRules
It may be the case that the model you wish to use is linearizable,
but Calc's built-in rules are unable to figure it out.  Calc uses
its algebraic rewrite mechanism to linearize a model.  The rewrite
rules are kept in the variable @code{FitRules}.  You can edit this
variable using the @kbd{s e FitRules} command; in fact, there is
a special @kbd{s F} command just for editing @code{FitRules}.
@xref{Operations on Variables}.

@xref{Rewrite Rules}, for a discussion of rewrite rules.

@ignore
@starindex
@end ignore
@tindex fitvar
@ignore
@starindex
@end ignore
@ignore
@mindex @idots
@end ignore
@tindex fitparam
@ignore
@starindex
@end ignore
@ignore
@mindex @null
@end ignore
@tindex fitmodel
@ignore
@starindex
@end ignore
@ignore
@mindex @null
@end ignore
@tindex fitsystem
@ignore
@starindex
@end ignore
@ignore
@mindex @null
@end ignore
@tindex fitdummy
Calc uses @code{FitRules} as follows.  First, it converts the model
to an equation if necessary and encloses the model equation in a
call to the function @code{fitmodel} (which is not actually a defined
function in Calc; it is only used as a placeholder by the rewrite rules).
Parameter variables are renamed to function calls @samp{fitparam(1)},
@samp{fitparam(2)}, and so on, and independent variables are renamed
to @samp{fitvar(1)}, @samp{fitvar(2)}, etc.  The dependent variable
is the highest-numbered @code{fitvar}.  For example, the power law
model @expr{a x^b} is converted to @expr{y = a x^b}, then to

@smallexample
@group
fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
@end group
@end smallexample

Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
(The zero prefix means that rewriting should continue until no further
changes are possible.)

When rewriting is complete, the @code{fitmodel} call should have
been replaced by a @code{fitsystem} call that looks like this:

@example
fitsystem(@var{Y}, @var{FGH}, @var{abc})
@end example

@noindent
where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
@var{FGH} is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
and @var{abc} is the vector of parameter filters which refer to the
raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
for @expr{B}, etc.  While the number of raw parameters (the length of
the @var{FGH} vector) is usually the same as the number of original
parameters (the length of the @var{abc} vector), this is not required.

The power law model eventually boils down to

@smallexample
@group
fitsystem(ln(fitvar(2)),
          [1, ln(fitvar(1))],
          [exp(fitdummy(1)), fitdummy(2)])
@end group
@end smallexample

The actual implementation of @code{FitRules} is complicated; it
proceeds in four phases.  First, common rearrangements are done
to try to bring linear terms together and to isolate functions like
@code{exp} and @code{ln} either all the way ``out'' (so that they
can be put into @var{Y}) or all the way ``in'' (so that they can
be put into @var{abc} or @var{FGH}).  In particular, all
non-constant powers are converted to logs-and-exponentials form,
and the distributive law is used to expand products of sums.
Quotients are rewritten to use the @samp{fitinv} function, where
@samp{fitinv(x)} represents @expr{1/x} while the @code{FitRules}
are operating.  (The use of @code{fitinv} makes recognition of
linear-looking forms easier.)  If you modify @code{FitRules}, you
will probably only need to modify the rules for this phase.

Phase two, whose rules can actually also apply during phases one
and three, first rewrites @code{fitmodel} to a two-argument
form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
initially zero and @var{model} has been changed from @expr{a=b}
to @expr{a-b} form.  It then tries to peel off invertible functions
from the outside of @var{model} and put them into @var{Y} instead,
calling the equation solver to invert the functions.  Finally, when
this is no longer possible, the @code{fitmodel} is changed to a
four-argument @code{fitsystem}, where the fourth argument is
@var{model} and the @var{FGH} and @var{abc} vectors are initially
empty.  (The last vector is really @var{ABC}, corresponding to
raw parameters, for now.)

Phase three converts a sum of items in the @var{model} to a sum
of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
terms @samp{@var{a}*@var{b}*@var{c}} of the sum, where @var{a}
is all factors that do not involve any variables, @var{b} is all
factors that involve only parameters, and @var{c} is the factors
that involve only independent variables.  (If this decomposition
is not possible, the rule set will not complete and Calc will
complain that the model is too complex.)  Then @code{fitpart}s
with equal @var{b} or @var{c} components are merged back together
using the distributive law in order to minimize the number of
raw parameters needed.

Phase four moves the @code{fitpart} terms into the @var{FGH} and
@var{ABC} vectors.  Also, some of the algebraic expansions that
were done in phase 1 are undone now to make the formulas more
computationally efficient.  Finally, it calls the solver one more
time to convert the @var{ABC} vector to an @var{abc} vector, and
removes the fourth @var{model} argument (which by now will be zero)
to obtain the three-argument @code{fitsystem} that the linear
least-squares solver wants to see.

@ignore
@starindex
@end ignore
@ignore
@mindex hasfit@idots
@end ignore
@tindex hasfitparams
@ignore
@starindex
@end ignore
@ignore
@mindex @null
@end ignore
@tindex hasfitvars
Two functions which are useful in connection with @code{FitRules}
are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
whether @expr{x} refers to any parameters or independent variables,
respectively.  Specifically, these functions return ``true'' if the
argument contains any @code{fitparam} (or @code{fitvar}) function
calls, and ``false'' otherwise.  (Recall that ``true'' means a
nonzero number, and ``false'' means zero.  The actual nonzero number
returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)

@tex
\bigskip
@end tex

The @code{fit} function in algebraic notation normally takes four
arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
where @var{model} is the model formula as it would be typed after
@kbd{a F '}, @var{vars} is the independent variable or a vector of
independent variables, @var{params} likewise gives the parameter(s),
and @var{data} is the data matrix.  Note that the length of @var{vars}
must be equal to the number of rows in @var{data} if @var{model} is
an equation, or one less than the number of rows if @var{model} is
a plain formula.  (Actually, a name for the dependent variable is
allowed but will be ignored in the plain-formula case.)

If @var{params} is omitted, the parameters are all variables in
@var{model} except those that appear in @var{vars}.  If @var{vars}
is also omitted, Calc sorts all the variables that appear in
@var{model} alphabetically and uses the higher ones for @var{vars}
and the lower ones for @var{params}.

Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
where @var{modelvec} is a 2- or 3-vector describing the model
and variables, as discussed previously.

If Calc is unable to do the fit, the @code{fit} function is left
in symbolic form, ordinarily with an explanatory message.  The
message will be ``Model expression is too complex'' if the
linearizer was unable to put the model into the required form.

The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
(for @kbd{I a F}) functions are completely analogous.

@node Interpolation,  , Curve Fitting Details, Curve Fitting
@subsection Polynomial Interpolation

@kindex a p
@pindex calc-poly-interp
@tindex polint
The @kbd{a p} (@code{calc-poly-interp}) [@code{polint}] command does
a polynomial interpolation at a particular @expr{x} value.  It takes
two arguments from the stack:  A data matrix of the sort used by
@kbd{a F}, and a single number which represents the desired @expr{x}
value.  Calc effectively does an exact polynomial fit as if by @kbd{a F i},
then substitutes the @expr{x} value into the result in order to get an
approximate @expr{y} value based on the fit.  (Calc does not actually
use @kbd{a F i}, however; it uses a direct method which is both more
efficient and more numerically stable.)

The result of @kbd{a p} is actually a vector of two values:  The @expr{y}
value approximation, and an error measure @expr{dy} that reflects Calc's
estimation of the probable error of the approximation at that value of
@expr{x}.  If the input @expr{x} is equal to any of the @expr{x} values
in the data matrix, the output @expr{y} will be the corresponding @expr{y}
value from the matrix, and the output @expr{dy} will be exactly zero.

A prefix argument of 2 causes @kbd{a p} to take separate x- and
y-vectors from the stack instead of one data matrix.

If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
interpolated results for each of those @expr{x} values.  (The matrix will
have two columns, the @expr{y} values and the @expr{dy} values.)
If @expr{x} is a formula instead of a number, the @code{polint} function
remains in symbolic form; use the @kbd{a "} command to expand it out to
a formula that describes the fit in symbolic terms.

In all cases, the @kbd{a p} command leaves the data vectors or matrix
on the stack.  Only the @expr{x} value is replaced by the result.

@kindex H a p
@tindex ratint
The @kbd{H a p} [@code{ratint}] command does a rational function
interpolation.  It is used exactly like @kbd{a p}, except that it
uses as its model the quotient of two polynomials.  If there are
@expr{N} data points, the numerator and denominator polynomials will
each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
have degree one higher than the numerator).

Rational approximations have the advantage that they can accurately
describe functions that have poles (points at which the function's value
goes to infinity, so that the denominator polynomial of the approximation
goes to zero).  If @expr{x} corresponds to a pole of the fitted rational
function, then the result will be a division by zero.  If Infinite mode
is enabled, the result will be @samp{[uinf, uinf]}.

There is no way to get the actual coefficients of the rational function
used by @kbd{H a p}.  (The algorithm never generates these coefficients
explicitly, and quotients of polynomials are beyond @w{@kbd{a F}}'s
capabilities to fit.)

@node Summations, Logical Operations, Curve Fitting, Algebra
@section Summations

@noindent
@cindex Summation of a series
@kindex a +
@pindex calc-summation
@tindex sum
The @kbd{a +} (@code{calc-summation}) [@code{sum}] command computes
the sum of a formula over a certain range of index values.  The formula
is taken from the top of the stack; the command prompts for the
name of the summation index variable, the lower limit of the
sum (any formula), and the upper limit of the sum.  If you
enter a blank line at any of these prompts, that prompt and
any later ones are answered by reading additional elements from
the stack.  Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
produces the result 55.
@tex
\turnoffactive
$$ \sum_{k=1}^5 k^2 = 55 $$
@end tex

The choice of index variable is arbitrary, but it's best not to
use a variable with a stored value.  In particular, while
@code{i} is often a favorite index variable, it should be avoided
in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
as a value.  If you pressed @kbd{=} on a sum over @code{i}, it would
be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
If you really want to use @code{i} as an index variable, use
@w{@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
(@xref{Storing Variables}.)

A numeric prefix argument steps the index by that amount rather
than by one.  Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0 @key{RET}}
yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}.  A prefix
argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
step value, in which case you can enter any formula or enter
a blank line to take the step value from the stack.  With the
@kbd{C-u} prefix, @kbd{a +} can take up to five arguments from
the stack:  The formula, the variable, the lower limit, the
upper limit, and (at the top of the stack), the step value.

Calc knows how to do certain sums in closed form.  For example,
@samp{sum(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}.  In particular,
this is possible if the formula being summed is polynomial or
exponential in the index variable.  Sums of logarithms are
transformed into logarithms of products.  Sums of trigonometric
and hyperbolic functions are transformed to sums of exponentials
and then done in closed form.  Also, of course, sums in which the
lower and upper limits are both numbers can always be evaluated
just by grinding them out, although Calc will use closed forms
whenever it can for the sake of efficiency.

The notation for sums in algebraic formulas is
@samp{sum(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
If @var{step} is omitted, it defaults to one.  If @var{high} is
omitted, @var{low} is actually the upper limit and the lower limit
is one.  If @var{low} is also omitted, the limits are @samp{-inf}
and @samp{inf}, respectively.

Infinite sums can sometimes be evaluated:  @samp{sum(.5^k, k, 1, inf)}
returns @expr{1}.  This is done by evaluating the sum in closed
form (to @samp{1. - 0.5^n} in this case), then evaluating this
formula with @code{n} set to @code{inf}.  Calc's usual rules
for ``infinite'' arithmetic can find the answer from there.  If
infinite arithmetic yields a @samp{nan}, or if the sum cannot be
solved in closed form, Calc leaves the @code{sum} function in
symbolic form.  @xref{Infinities}.

As a special feature, if the limits are infinite (or omitted, as
described above) but the formula includes vectors subscripted by
expressions that involve the iteration variable, Calc narrows
the limits to include only the range of integers which result in
valid subscripts for the vector.  For example, the sum
@samp{sum(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.

The limits of a sum do not need to be integers.  For example,
@samp{sum(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
Calc computes the number of iterations using the formula
@samp{1 + (@var{high} - @var{low}) / @var{step}}, which must,
after simplification as if by @kbd{a s}, evaluate to an integer.

If the number of iterations according to the above formula does
not come out to an integer, the sum is invalid and will be left
in symbolic form.  However, closed forms are still supplied, and
you are on your honor not to misuse the resulting formulas by
substituting mismatched bounds into them.  For example,
@samp{sum(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
evaluate the closed form solution for the limits 1 and 10 to get
the rather dubious answer, 29.25.

If the lower limit is greater than the upper limit (assuming a
positive step size), the result is generally zero.  However,
Calc only guarantees a zero result when the upper limit is
exactly one step less than the lower limit, i.e., if the number
of iterations is @mathit{-1}.  Thus @samp{sum(f(k), k, n, n-1)} is zero
but the sum from @samp{n} to @samp{n-2} may report a nonzero value
if Calc used a closed form solution.

Calc's logical predicates like @expr{a < b} return 1 for ``true''
and 0 for ``false.''  @xref{Logical Operations}.  This can be
used to advantage for building conditional sums.  For example,
@samp{sum(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
its argument is prime and 0 otherwise.  You can read this expression
as ``the sum of @expr{k^2}, where @expr{k} is prime.''  Indeed,
@samp{sum(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
squared, since the limits default to plus and minus infinity, but
there are no such sums that Calc's built-in rules can do in
closed form.

As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
one value @expr{k_0}.  Slightly more tricky is the summand
@samp{(k != k_0) / (k - k_0)}, which is an attempt to describe
the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
this would be a division by zero.  But at @expr{k = k_0}, this
formula works out to the indeterminate form @expr{0 / 0}, which
Calc will not assume is zero.  Better would be to use
@samp{(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
an ``if-then-else'' test:  This expression says, ``if 
@texline @math{k \ne k_0},
@infoline @expr{k != k_0},
then @expr{1/(k-k_0)}, else zero.''  Now the formula @expr{1/(k-k_0)}
will not even be evaluated by Calc when @expr{k = k_0}.

@cindex Alternating sums
@kindex a -
@pindex calc-alt-summation
@tindex asum
The @kbd{a -} (@code{calc-alt-summation}) [@code{asum}] command
computes an alternating sum.  Successive terms of the sequence
are given alternating signs, with the first term (corresponding
to the lower index value) being positive.  Alternating sums
are converted to normal sums with an extra term of the form
@samp{(-1)^(k-@var{low})}.  This formula is adjusted appropriately
if the step value is other than one.  For example, the Taylor
series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
(Calc cannot evaluate this infinite series, but it can approximate
it if you replace @code{inf} with any particular odd number.)
Calc converts this series to a regular sum with a step of one,
namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.

@cindex Product of a sequence
@kindex a *
@pindex calc-product
@tindex prod
The @kbd{a *} (@code{calc-product}) [@code{prod}] command is
the analogous way to take a product of many terms.  Calc also knows
some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
or @samp{prod(prime(k) ? k : 1, k, 1, n)}.

@kindex a T
@pindex calc-tabulate
@tindex table
The @kbd{a T} (@code{calc-tabulate}) [@code{table}] command
evaluates a formula at a series of iterated index values, just
like @code{sum} and @code{prod}, but its result is simply a
vector of the results.  For example, @samp{table(a_i, i, 1, 7, 2)}
produces @samp{[a_1, a_3, a_5, a_7]}.

@node Logical Operations, Rewrite Rules, Summations, Algebra
@section Logical Operations

@noindent
The following commands and algebraic functions return true/false values,
where 1 represents ``true'' and 0 represents ``false.''  In cases where
a truth value is required (such as for the condition part of a rewrite
rule, or as the condition for a @w{@kbd{Z [ Z ]}} control structure), any
nonzero value is accepted to mean ``true.''  (Specifically, anything
for which @code{dnonzero} returns 1 is ``true,'' and anything for
which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
Note that this means that @w{@kbd{Z [ Z ]}} will execute the ``then''
portion if its condition is provably true, but it will execute the
``else'' portion for any condition like @expr{a = b} that is not
provably true, even if it might be true.  Algebraic functions that
have conditions as arguments, like @code{? :} and @code{&&}, remain
unevaluated if the condition is neither provably true nor provably
false.  @xref{Declarations}.)

@kindex a =
@pindex calc-equal-to
@tindex eq
@tindex =
@tindex ==
The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
(which can also be written @samp{a = b} or @samp{a == b} in an algebraic
formula) is true if @expr{a} and @expr{b} are equal, either because they
are identical expressions, or because they are numbers which are
numerically equal.  (Thus the integer 1 is considered equal to the float
1.0.)  If the equality of @expr{a} and @expr{b} cannot be determined,
the comparison is left in symbolic form.  Note that as a command, this
operation pops two values from the stack and pushes back either a 1 or
a 0, or a formula @samp{a = b} if the values' equality cannot be determined.

Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
an equation to solve for a given variable.  The @kbd{a M}
(@code{calc-map-equation}) command can be used to apply any
function to both sides of an equation; for example, @kbd{2 a M *}
multiplies both sides of the equation by two.  Note that just
@kbd{2 *} would not do the same thing; it would produce the formula
@samp{2 (a = b)} which represents 2 if the equality is true or
zero if not.

The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
or @samp{a = b = c}) tests if all of its arguments are equal.  In
algebraic notation, the @samp{=} operator is unusual in that it is
neither left- nor right-associative:  @samp{a = b = c} is not the
same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
one variable with the 1 or 0 that results from comparing two other
variables).

@kindex a #
@pindex calc-not-equal-to
@tindex neq
@tindex !=
The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
@samp{a != b} function, is true if @expr{a} and @expr{b} are not equal.
This also works with more than two arguments; @samp{a != b != c != d}
tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
distinct numbers.

@kindex a <
@tindex lt
@ignore
@mindex @idots
@end ignore
@kindex a >
@ignore
@mindex @null
@end ignore
@kindex a [
@ignore
@mindex @null
@end ignore
@kindex a ]
@pindex calc-less-than
@pindex calc-greater-than
@pindex calc-less-equal
@pindex calc-greater-equal
@ignore
@mindex @null
@end ignore
@tindex gt
@ignore
@mindex @null
@end ignore
@tindex leq
@ignore
@mindex @null
@end ignore
@tindex geq
@ignore
@mindex @null
@end ignore
@tindex <
@ignore
@mindex @null
@end ignore
@tindex >
@ignore
@mindex @null
@end ignore
@tindex <=
@ignore
@mindex @null
@end ignore
@tindex >=
The @kbd{a <} (@code{calc-less-than}) [@samp{lt(a,b)} or @samp{a < b}]
operation is true if @expr{a} is less than @expr{b}.  Similar functions
are @kbd{a >} (@code{calc-greater-than}) [@samp{gt(a,b)} or @samp{a > b}],
@kbd{a [} (@code{calc-less-equal}) [@samp{leq(a,b)} or @samp{a <= b}], and
@kbd{a ]} (@code{calc-greater-equal}) [@samp{geq(a,b)} or @samp{a >= b}].

While the inequality functions like @code{lt} do not accept more
than two arguments, the syntax @w{@samp{a <= b < c}} is translated to an
equivalent expression involving intervals: @samp{b in [a .. c)}.
(See the description of @code{in} below.)  All four combinations
of @samp{<} and @samp{<=} are allowed, or any of the four combinations
of @samp{>} and @samp{>=}.  Four-argument constructions like
@samp{a < b < c < d}, and mixtures like @w{@samp{a < b = c}} that
involve both equalities and inequalities, are not allowed.

@kindex a .
@pindex calc-remove-equal
@tindex rmeq
The @kbd{a .} (@code{calc-remove-equal}) [@code{rmeq}] command extracts
the righthand side of the equation or inequality on the top of the
stack.  It also works elementwise on vectors.  For example, if
@samp{[x = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
@samp{[2.34, z / 2]}.  As a special case, if the righthand side is a
variable and the lefthand side is a number (as in @samp{2.34 = x}), then
Calc keeps the lefthand side instead.  Finally, this command works with
assignments @samp{x := 2.34} as well as equations, always taking the
righthand side, and for @samp{=>} (evaluates-to) operators, always
taking the lefthand side.

@kindex a &
@pindex calc-logical-and
@tindex land
@tindex &&
The @kbd{a &} (@code{calc-logical-and}) [@samp{land(a,b)} or @samp{a && b}]
function is true if both of its arguments are true, i.e., are
non-zero numbers.  In this case, the result will be either @expr{a} or
@expr{b}, chosen arbitrarily.  If either argument is zero, the result is
zero.  Otherwise, the formula is left in symbolic form.

@kindex a |
@pindex calc-logical-or
@tindex lor
@tindex ||
The @kbd{a |} (@code{calc-logical-or}) [@samp{lor(a,b)} or @samp{a || b}]
function is true if either or both of its arguments are true (nonzero).
The result is whichever argument was nonzero, choosing arbitrarily if both
are nonzero.  If both @expr{a} and @expr{b} are zero, the result is
zero.

@kindex a !
@pindex calc-logical-not
@tindex lnot
@tindex !
The @kbd{a !} (@code{calc-logical-not}) [@samp{lnot(a)} or @samp{!@: a}]
function is true if @expr{a} is false (zero), or false if @expr{a} is
true (nonzero).  It is left in symbolic form if @expr{a} is not a
number.

@kindex a :
@pindex calc-logical-if
@tindex if
@ignore
@mindex ? :
@end ignore
@tindex ?
@ignore
@mindex @null
@end ignore
@tindex :
@cindex Arguments, not evaluated
The @kbd{a :} (@code{calc-logical-if}) [@samp{if(a,b,c)} or @samp{a ? b :@: c}]
function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
number or zero, respectively.  If @expr{a} is not a number, the test is
left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
any way.  In algebraic formulas, this is one of the few Calc functions
whose arguments are not automatically evaluated when the function itself
is evaluated.  The others are @code{lambda}, @code{quote}, and
@code{condition}.

One minor surprise to watch out for is that the formula @samp{a?3:4}
will not work because the @samp{3:4} is parsed as a fraction instead of
as three separate symbols.  Type something like @samp{a ? 3 : 4} or
@samp{a?(3):4} instead.

As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
and @expr{c} are evaluated; the result is a vector of the same length
as @expr{a} whose elements are chosen from corresponding elements of
@expr{b} and @expr{c} according to whether each element of @expr{a}
is zero or nonzero.  Each of @expr{b} and @expr{c} must be either a
vector of the same length as @expr{a}, or a non-vector which is matched
with all elements of @expr{a}.

@kindex a @{
@pindex calc-in-set
@tindex in
The @kbd{a @{} (@code{calc-in-set}) [@samp{in(a,b)}] function is true if
the number @expr{a} is in the set of numbers represented by @expr{b}.
If @expr{b} is an interval form, @expr{a} must be one of the values
encompassed by the interval.  If @expr{b} is a vector, @expr{a} must be
equal to one of the elements of the vector.  (If any vector elements are
intervals, @expr{a} must be in any of the intervals.)  If @expr{b} is a
plain number, @expr{a} must be numerically equal to @expr{b}.
@xref{Set Operations}, for a group of commands that manipulate sets
of this sort.

@ignore
@starindex
@end ignore
@tindex typeof
The @samp{typeof(a)} function produces an integer or variable which
characterizes @expr{a}.  If @expr{a} is a number, vector, or variable,
the result will be one of the following numbers:

@example
 1   Integer
 2   Fraction
 3   Floating-point number
 4   HMS form
 5   Rectangular complex number
 6   Polar complex number
 7   Error form
 8   Interval form
 9   Modulo form
10   Date-only form
11   Date/time form
12   Infinity (inf, uinf, or nan)
100  Variable
101  Vector (but not a matrix)
102  Matrix
@end example

Otherwise, @expr{a} is a formula, and the result is a variable which
represents the name of the top-level function call.

@ignore
@starindex
@end ignore
@tindex integer
@ignore
@starindex
@end ignore
@tindex real
@ignore
@starindex
@end ignore
@tindex constant
The @samp{integer(a)} function returns true if @expr{a} is an integer.
The @samp{real(a)} function
is true if @expr{a} is a real number, either integer, fraction, or
float.  The @samp{constant(a)} function returns true if @expr{a} is
any of the objects for which @code{typeof} would produce an integer
code result except for variables, and provided that the components of
an object like a vector or error form are themselves constant.
Note that infinities do not satisfy any of these tests, nor do
special constants like @code{pi} and @code{e}.

@xref{Declarations}, for a set of similar functions that recognize
formulas as well as actual numbers.  For example, @samp{dint(floor(x))}
is true because @samp{floor(x)} is provably integer-valued, but
@samp{integer(floor(x))} does not because @samp{floor(x)} is not
literally an integer constant.

@ignore
@starindex
@end ignore
@tindex refers
The @samp{refers(a,b)} function is true if the variable (or sub-expression)
@expr{b} appears in @expr{a}, or false otherwise.  Unlike the other
tests described here, this function returns a definite ``no'' answer
even if its arguments are still in symbolic form.  The only case where
@code{refers} will be left unevaluated is if @expr{a} is a plain
variable (different from @expr{b}).

@ignore
@starindex
@end ignore
@tindex negative
The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
because it is a negative number, because it is of the form @expr{-x},
or because it is a product or quotient with a term that looks negative.
This is most useful in rewrite rules.  Beware that @samp{negative(a)}
evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
be stored in a formula if the default simplifications are turned off
first with @kbd{m O} (or if it appears in an unevaluated context such
as a rewrite rule condition).

@ignore
@starindex
@end ignore
@tindex variable
The @samp{variable(a)} function is true if @expr{a} is a variable,
or false if not.  If @expr{a} is a function call, this test is left
in symbolic form.  Built-in variables like @code{pi} and @code{inf}
are considered variables like any others by this test.

@ignore
@starindex
@end ignore
@tindex nonvar
The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
If its argument is a variable it is left unsimplified; it never
actually returns zero.  However, since Calc's condition-testing
commands consider ``false'' anything not provably true, this is
often good enough.

@ignore
@starindex
@end ignore
@tindex lin
@ignore
@starindex
@end ignore
@tindex linnt
@ignore
@starindex
@end ignore
@tindex islin
@ignore
@starindex
@end ignore
@tindex islinnt
@cindex Linearity testing
The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
check if an expression is ``linear,'' i.e., can be written in the form
@expr{a + b x} for some constants @expr{a} and @expr{b}, and some
variable or subformula @expr{x}.  The function @samp{islin(f,x)} checks
if formula @expr{f} is linear in @expr{x}, returning 1 if so.  For
example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
@samp{islin(x y / 3 - 2, x)} all return 1.  The @samp{lin(f,x)} function
is similar, except that instead of returning 1 it returns the vector
@expr{[a, b, x]}.  For the above examples, this vector would be
@expr{[0, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
@expr{[-2, y/3, x]}, respectively.  Both @code{lin} and @code{islin}
generally remain unevaluated for expressions which are not linear,
e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}.  The second
argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
returns true.

The @code{linnt} and @code{islinnt} functions perform a similar check,
but require a ``non-trivial'' linear form, which means that the
@expr{b} coefficient must be non-zero.  For example, @samp{lin(2,x)}
returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
(in other words, these formulas are considered to be only ``trivially''
linear in @expr{x}).

All four linearity-testing functions allow you to omit the second
argument, in which case the input may be linear in any non-constant
formula.  Here, the @expr{a=0}, @expr{b=1} case is also considered
trivial, and only constant values for @expr{a} and @expr{b} are
recognized.  Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
@samp{lin(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
returns @expr{[0, 1, x y]}.  The @code{linnt} function would allow the
first two cases but not the third.  Also, neither @code{lin} nor
@code{linnt} accept plain constants as linear in the one-argument
case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.

@ignore
@starindex
@end ignore
@tindex istrue
The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
number or provably nonzero formula, or 0 if @expr{a} is anything else.
Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
used to make sure they are not evaluated prematurely.  (Note that
declarations are used when deciding whether a formula is true;
@code{istrue} returns 1 when @code{dnonzero} would return 1, and
it returns 0 when @code{dnonzero} would return 0 or leave itself
in symbolic form.)

@node Rewrite Rules,  , Logical Operations, Algebra
@section Rewrite Rules

@noindent
@cindex Rewrite rules
@cindex Transformations
@cindex Pattern matching
@kindex a r
@pindex calc-rewrite
@tindex rewrite
The @kbd{a r} (@code{calc-rewrite}) [@code{rewrite}] command makes
substitutions in a formula according to a specified pattern or patterns
known as @dfn{rewrite rules}.  Whereas @kbd{a b} (@code{calc-substitute})
matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
matches only the @code{sin} function applied to the variable @code{x},
rewrite rules match general kinds of formulas; rewriting using the rule
@samp{sin(x) := cos(x)} matches @code{sin} of any argument and replaces
it with @code{cos} of that same argument.  The only significance of the
name @code{x} is that the same name is used on both sides of the rule.

Rewrite rules rearrange formulas already in Calc's memory.
@xref{Syntax Tables}, to read about @dfn{syntax rules}, which are
similar to algebraic rewrite rules but operate when new algebraic
entries are being parsed, converting strings of characters into
Calc formulas.

@menu
* Entering Rewrite Rules::
* Basic Rewrite Rules::
* Conditional Rewrite Rules::
* Algebraic Properties of Rewrite Rules::
* Other Features of Rewrite Rules::
* Composing Patterns in Rewrite Rules::
* Nested Formulas with Rewrite Rules::
* Multi-Phase Rewrite Rules::
* Selections with Rewrite Rules::
* Matching Commands::
* Automatic Rewrites::
* Debugging Rewrites::
* Examples of Rewrite Rules::
@end menu

@node Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules, Rewrite Rules
@subsection Entering Rewrite Rules

@noindent
Rewrite rules normally use the ``assignment'' operator
@samp{@var{old} := @var{new}}.
This operator is equivalent to the function call @samp{assign(old, new)}.
The @code{assign} function is undefined by itself in Calc, so an
assignment formula such as a rewrite rule will be left alone by ordinary
Calc commands.  But certain commands, like the rewrite system, interpret
assignments in special ways.

For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
every occurrence of the sine of something, squared, with one minus the
square of the cosine of that same thing.  All by itself as a formula
on the stack it does nothing, but when given to the @kbd{a r} command
it turns that command into a sine-squared-to-cosine-squared converter.

To specify a set of rules to be applied all at once, make a vector of
rules.

When @kbd{a r} prompts you to enter the rewrite rules, you can answer
in several ways:

@enumerate
@item
With a rule:  @kbd{f(x) := g(x) @key{RET}}.
@item
With a vector of rules:  @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
(You can omit the enclosing square brackets if you wish.)
@item
With the name of a variable that contains the rule or rules vector:
@kbd{myrules @key{RET}}.
@item
With any formula except a rule, a vector, or a variable name; this
will be interpreted as the @var{old} half of a rewrite rule,
and you will be prompted a second time for the @var{new} half:
@kbd{f(x) @key{RET} g(x) @key{RET}}.
@item
With a blank line, in which case the rule, rules vector, or variable
will be taken from the top of the stack (and the formula to be
rewritten will come from the second-to-top position).
@end enumerate

If you enter the rules directly (as opposed to using rules stored
in a variable), those rules will be put into the Trail so that you
can retrieve them later.  @xref{Trail Commands}.

It is most convenient to store rules you use often in a variable and
invoke them by giving the variable name.  The @kbd{s e}
(@code{calc-edit-variable}) command is an easy way to create or edit a
rule set stored in a variable.  You may also wish to use @kbd{s p}
(@code{calc-permanent-variable}) to save your rules permanently;
@pxref{Operations on Variables}.

Rewrite rules are compiled into a special internal form for faster
matching.  If you enter a rule set directly it must be recompiled
every time.  If you store the rules in a variable and refer to them
through that variable, they will be compiled once and saved away
along with the variable for later reference.  This is another good
reason to store your rules in a variable.

Calc also accepts an obsolete notation for rules, as vectors
@samp{[@var{old}, @var{new}]}.  But because it is easily confused with a
vector of two rules, the use of this notation is no longer recommended.

@node Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite Rules, Rewrite Rules
@subsection Basic Rewrite Rules

@noindent
To match a particular formula @expr{x} with a particular rewrite rule
@samp{@var{old} := @var{new}}, Calc compares the structure of @expr{x} with
the structure of @var{old}.  Variables that appear in @var{old} are
treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
may contain any sub-formulas.  For example, the pattern @samp{f(x,y)}
would match the expression @samp{f(12, a+1)} with the meta-variable
@samp{x} corresponding to 12 and with @samp{y} corresponding to
@samp{a+1}.  However, this pattern would not match @samp{f(12)} or
@samp{g(12, a+1)}, since there is no assignment of the meta-variables
that will make the pattern match these expressions.  Notice that if
the pattern is a single meta-variable, it will match any expression.

If a given meta-variable appears more than once in @var{old}, the
corresponding sub-formulas of @expr{x} must be identical.  Thus
the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
@samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
(@xref{Conditional Rewrite Rules}, for a way to match the latter.)

Things other than variables must match exactly between the pattern
and the target formula.  To match a particular variable exactly, use
the pseudo-function @samp{quote(v)} in the pattern.  For example, the
pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
@samp{sin(a)+y}.

The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
@samp{gamma}, @samp{inf}, @samp{uinf}, and @samp{nan} always match
literally.  Thus the pattern @samp{sin(d + e + f)} acts exactly like
@samp{sin(d + quote(e) + f)}.

If the @var{old} pattern is found to match a given formula, that
formula is replaced by @var{new}, where any occurrences in @var{new}
of meta-variables from the pattern are replaced with the sub-formulas
that they matched.  Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.

The normal @kbd{a r} command applies rewrite rules over and over
throughout the target formula until no further changes are possible
(up to a limit of 100 times).  Use @kbd{C-u 1 a r} to make only one
change at a time.

@node Conditional Rewrite Rules, Algebraic Properties of Rewrite Rules, Basic Rewrite Rules, Rewrite Rules
@subsection Conditional Rewrite Rules

@noindent
A rewrite rule can also be @dfn{conditional}, written in the form
@samp{@var{old} := @var{new} :: @var{cond}}.  (There is also the obsolete
form @samp{[@var{old}, @var{new}, @var{cond}]}.)  If a @var{cond} part
is present in the
rule, this is an additional condition that must be satisfied before
the rule is accepted.  Once @var{old} has been successfully matched
to the target expression, @var{cond} is evaluated (with all the
meta-variables substituted for the values they matched) and simplified
with @kbd{a s} (@code{calc-simplify}).  If the result is a nonzero
number or any other object known to be nonzero (@pxref{Declarations}),
the rule is accepted.  If the result is zero or if it is a symbolic
formula that is not known to be nonzero, the rule is rejected.
@xref{Logical Operations}, for a number of functions that return
1 or 0 according to the results of various tests.

For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
is replaced by a positive or nonpositive number, respectively (or if
@expr{n} has been declared to be positive or nonpositive).  Thus,
the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
@samp{f(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
(assuming no outstanding declarations for @expr{a}).  In the case of
@samp{f(-3, 2)}, the condition can be shown not to be satisfied; in
the case of @samp{f(12, a+1)}, the condition merely cannot be shown
to be satisfied, but that is enough to reject the rule.

While Calc will use declarations to reason about variables in the
formula being rewritten, declarations do not apply to meta-variables.
For example, the rule @samp{f(a) := g(a+1)} will match for any values
of @samp{a}, such as complex numbers, vectors, or formulas, even if
@samp{a} has been declared to be real or scalar.  If you want the
meta-variable @samp{a} to match only literal real numbers, use
@samp{f(a) := g(a+1) :: real(a)}.  If you want @samp{a} to match only
reals and formulas which are provably real, use @samp{dreal(a)} as
the condition.

The @samp{::} operator is a shorthand for the @code{condition}
function; @samp{@var{old} := @var{new} :: @var{cond}} is equivalent to
the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.

If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
or @samp{... :: c1 && c2 && c3}.  The two are entirely equivalent.

It is also possible to embed conditions inside the pattern:
@samp{f(x :: x>0, y) := g(y+x, x)}.  This is purely a notational
convenience, though; where a condition appears in a rule has no
effect on when it is tested.  The rewrite-rule compiler automatically
decides when it is best to test each condition while a rule is being
matched.

Certain conditions are handled as special cases by the rewrite rule
system and are tested very efficiently:  Where @expr{x} is any
meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
@samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
is either a constant or another meta-variable and @samp{>=} may be
replaced by any of the six relational operators, and @samp{x % a = b}
where @expr{a} and @expr{b} are constants.  Other conditions, like
@samp{x >= y+1} or @samp{dreal(x)}, will be less efficient to check
since Calc must bring the whole evaluator and simplifier into play.

An interesting property of @samp{::} is that neither of its arguments
will be touched by Calc's default simplifications.  This is important
because conditions often are expressions that cannot safely be
evaluated early.  For example, the @code{typeof} function never
remains in symbolic form; entering @samp{typeof(a)} will put the
number 100 (the type code for variables like @samp{a}) on the stack.
But putting the condition @samp{... :: typeof(a) = 6} on the stack
is safe since @samp{::} prevents the @code{typeof} from being
evaluated until the condition is actually used by the rewrite system.

Since @samp{::} protects its lefthand side, too, you can use a dummy
condition to protect a rule that must itself not evaluate early.
For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
where the meta-variable-ness of @code{f} on the righthand side has been
lost.  But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
the condition @samp{1} is always true (nonzero) so it has no effect on
the functioning of the rule.  (The rewrite compiler will ensure that
it doesn't even impact the speed of matching the rule.)

@node Algebraic Properties of Rewrite Rules, Other Features of Rewrite Rules, Conditional Rewrite Rules, Rewrite Rules
@subsection Algebraic Properties of Rewrite Rules

@noindent
The rewrite mechanism understands the algebraic properties of functions
like @samp{+} and @samp{*}.  In particular, pattern matching takes
the associativity and commutativity of the following functions into
account:

@smallexample
+ - *  = !=  && ||  and or xor  vint vunion vxor  gcd lcm  max min  beta
@end smallexample

For example, the rewrite rule:

@example
a x + b x  :=  (a + b) x
@end example

@noindent
will match formulas of the form,

@example
a x + b x,  x a + x b,  a x + x b,  x a + b x
@end example

Rewrites also understand the relationship between the @samp{+} and @samp{-}
operators.  The above rewrite rule will also match the formulas,

@example
a x - b x,  x a - x b,  a x - x b,  x a - b x
@end example

@noindent
by matching @samp{b} in the pattern to @samp{-b} from the formula.

Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
pattern will check all pairs of terms for possible matches.  The rewrite
will take whichever suitable pair it discovers first.

In general, a pattern using an associative operator like @samp{a + b}
will try @var{2 n} different ways to match a sum of @var{n} terms
like @samp{x + y + z - w}.  First, @samp{a} is matched against each
of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
If none of these succeed, then @samp{b} is matched against each of the
four terms with @samp{a} matching the remainder.  Half-and-half matches,
like @samp{(x + y) + (z - w)}, are not tried.

Note that @samp{*} is not commutative when applied to matrices, but
rewrite rules pretend that it is.  If you type @kbd{m v} to enable
Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
literally, ignoring its usual commutativity property.  (In the
current implementation, the associativity also vanishes---it is as
if the pattern had been enclosed in a @code{plain} marker; see below.)
If you are applying rewrites to formulas with matrices, it's best to
enable Matrix mode first to prevent algebraically incorrect rewrites
from occurring.

The pattern @samp{-x} will actually match any expression.  For example,
the rule

@example
f(-x)  :=  -f(x)
@end example

@noindent
will rewrite @samp{f(a)} to @samp{-f(-a)}.  To avoid this, either use
a @code{plain} marker as described below, or add a @samp{negative(x)}
condition.  The @code{negative} function is true if its argument
``looks'' negative, for example, because it is a negative number or
because it is a formula like @samp{-x}.  The new rule using this
condition is:

@example
f(x)  :=  -f(-x)  :: negative(x)    @r{or, equivalently,}
f(-x)  :=  -f(x)  :: negative(-x)
@end example

In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
by matching @samp{y} to @samp{-b}.

The pattern @samp{a b} will also match the formula @samp{x/y} if
@samp{y} is a number.  Thus the rule @samp{a x + @w{b x} := (a+b) x}
will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
@samp{(a + 1:2) x}, depending on the current fraction mode).

Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
@samp{^}.  For example, the pattern @samp{f(a b)} will not match
@samp{f(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
though conceivably these patterns could match with @samp{a = b = x}.
Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
constant, even though it could be considered to match with @samp{a = x}
and @samp{b = 1/y}.  The reasons are partly for efficiency, and partly
because while few mathematical operations are substantively different
for addition and subtraction, often it is preferable to treat the cases
of multiplication, division, and integer powers separately.

Even more subtle is the rule set

@example
[ f(a) + f(b) := f(a + b),  -f(a) := f(-a) ]
@end example

@noindent
attempting to match @samp{f(x) - f(y)}.  You might think that Calc
will view this subtraction as @samp{f(x) + (-f(y))} and then apply
the above two rules in turn, but actually this will not work because
Calc only does this when considering rules for @samp{+} (like the
first rule in this set).  So it will see first that @samp{f(x) + (-f(y))}
does not match @samp{f(a) + f(b)} for any assignments of the
meta-variables, and then it will see that @samp{f(x) - f(y)} does
not match @samp{-f(a)} for any assignment of @samp{a}.  Because Calc
tries only one rule at a time, it will not be able to rewrite
@samp{f(x) - f(y)} with this rule set.  An explicit @samp{f(a) - f(b)}
rule will have to be added.

Another thing patterns will @emph{not} do is break up complex numbers.
The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
it will not match actual complex numbers like @samp{(3, -4)}.  A version
of the above rule for complex numbers would be

@example
myconj(a)  :=  re(a) - im(a) (0,1)  :: im(a) != 0
@end example

@noindent
(Because the @code{re} and @code{im} functions understand the properties
of the special constant @samp{i}, this rule will also work for
@samp{3 - 4 i}.  In fact, this particular rule would probably be better
without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
righthand side of the rule will still give the correct answer for the
conjugate of a real number.)

It is also possible to specify optional arguments in patterns.  The rule

@example
opt(a) x + opt(b) (x^opt(c) + opt(d))  :=  f(a, b, c, d)
@end example

@noindent
will match the formula

@example
5 (x^2 - 4) + 3 x
@end example

@noindent
in a fairly straightforward manner, but it will also match reduced
formulas like

@example
x + x^2,    2(x + 1) - x,    x + x
@end example

@noindent
producing, respectively,

@example
f(1, 1, 2, 0),   f(-1, 2, 1, 1),   f(1, 1, 1, 0)
@end example

(The latter two formulas can be entered only if default simplifications
have been turned off with @kbd{m O}.)

The default value for a term of a sum is zero.  The default value
for a part of a product, for a power, or for the denominator of a
quotient, is one.  Also, @samp{-x} matches the pattern @samp{opt(a) b}
with @samp{a = -1}.

In particular, the distributive-law rule can be refined to

@example
opt(a) x + opt(b) x  :=  (a + b) x
@end example

@noindent
so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.

The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
are linear in @samp{x}.  You can also use the @code{lin} and @code{islin}
functions with rewrite conditions to test for this; @pxref{Logical
Operations}.  These functions are not as convenient to use in rewrite
rules, but they recognize more kinds of formulas as linear:
@samp{x/z} is considered linear with @expr{b = 1/z} by @code{lin},
but it will not match the above pattern because that pattern calls
for a multiplication, not a division.

As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
by 1,

@example
sin(x)^2 + cos(x)^2  :=  1
@end example

@noindent
misses many cases because the sine and cosine may both be multiplied by
an equal factor.  Here's a more successful rule:

@example
opt(a) sin(x)^2 + opt(a) cos(x)^2  :=  a
@end example

Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
because one @expr{a} would have ``matched'' 1 while the other matched 6.

Calc automatically converts a rule like

@example
f(x-1, x)  :=  g(x)
@end example

@noindent
into the form

@example
f(temp, x)  :=  g(x)  :: temp = x-1
@end example

@noindent
(where @code{temp} stands for a new, invented meta-variable that
doesn't actually have a name).  This modified rule will successfully
match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
respectively, then verifying that they differ by one even though
@samp{6} does not superficially look like @samp{x-1}.

However, Calc does not solve equations to interpret a rule.  The
following rule,

@example
f(x-1, x+1)  :=  g(x)
@end example

@noindent
will not work.  That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
but not @samp{f(6, 8)}.  Calc always interprets at least one occurrence
of a variable by literal matching.  If the variable appears ``isolated''
then Calc is smart enough to use it for literal matching.  But in this
last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
:= g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
actual ``something-minus-one'' in the target formula.

A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
You could make this resemble the original form more closely by using
@code{let} notation, which is described in the next section:

@example
f(xm1, x+1)  :=  g(x)  :: let(x := xm1+1)
@end example

Calc does this rewriting or ``conditionalizing'' for any sub-pattern
which involves only the functions in the following list, operating
only on constants and meta-variables which have already been matched
elsewhere in the pattern.  When matching a function call, Calc is
careful to match arguments which are plain variables before arguments
which are calls to any of the functions below, so that a pattern like
@samp{f(x-1, x)} can be conditionalized even though the isolated
@samp{x} comes after the @samp{x-1}.

@smallexample
+ - * / \ % ^  abs sign  round rounde roundu trunc floor ceil
max min  re im conj arg
@end smallexample

You can suppress all of the special treatments described in this
section by surrounding a function call with a @code{plain} marker.
This marker causes the function call which is its argument to be
matched literally, without regard to commutativity, associativity,
negation, or conditionalization.  When you use @code{plain}, the
``deep structure'' of the formula being matched can show through.
For example,

@example
plain(a - a b)  :=  f(a, b)
@end example

@noindent
will match only literal subtractions.  However, the @code{plain}
marker does not affect its arguments' arguments.  In this case,
commutativity and associativity is still considered while matching
the @w{@samp{a b}} sub-pattern, so the whole pattern will match
@samp{x - y x} as well as @samp{x - x y}.  We could go still
further and use

@example
plain(a - plain(a b))  :=  f(a, b)
@end example

@noindent
which would do a completely strict match for the pattern.

By contrast, the @code{quote} marker means that not only the
function name but also the arguments must be literally the same.
The above pattern will match @samp{x - x y} but

@example
quote(a - a b)  :=  f(a, b)
@end example

@noindent
will match only the single formula @samp{a - a b}.  Also,

@example
quote(a - quote(a b))  :=  f(a, b)
@end example

@noindent
will match only @samp{a - quote(a b)}---probably not the desired
effect!

A certain amount of algebra is also done when substituting the
meta-variables on the righthand side of a rule.  For example,
in the rule

@example
a + f(b)  :=  f(a + b)
@end example

@noindent
matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
taken literally, but the rewrite mechanism will simplify the
righthand side to @samp{f(x - y)} automatically.  (Of course,
the default simplifications would do this anyway, so this
special simplification is only noticeable if you have turned the
default simplifications off.)  This rewriting is done only when
a meta-variable expands to a ``negative-looking'' expression.
If this simplification is not desirable, you can use a @code{plain}
marker on the righthand side:

@example
a + f(b)  :=  f(plain(a + b))
@end example

@noindent
In this example, we are still allowing the pattern-matcher to
use all the algebra it can muster, but the righthand side will
always simplify to a literal addition like @samp{f((-y) + x)}.

@node Other Features of Rewrite Rules, Composing Patterns in Rewrite Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
@subsection Other Features of Rewrite Rules

@noindent
Certain ``function names'' serve as markers in rewrite rules.
Here is a complete list of these markers.  First are listed the
markers that work inside a pattern; then come the markers that
work in the righthand side of a rule.

@ignore
@starindex
@end ignore
@tindex import
One kind of marker, @samp{import(x)}, takes the place of a whole
rule.  Here @expr{x} is the name of a variable containing another
rule set; those rules are ``spliced into'' the rule set that
imports them.  For example, if @samp{[f(a+b) := f(a) + f(b),
f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
all three rules.  It is possible to modify the imported rules
slightly:  @samp{import(x, v1, x1, v2, x2, @dots{})} imports
the rule set @expr{x} with all occurrences of 
@texline @math{v_1},
@infoline @expr{v1}, 
as either a variable name or a function name, replaced with 
@texline @math{x_1}
@infoline @expr{x1} 
and so on.  (If 
@texline @math{v_1}
@infoline @expr{v1} 
is used as a function name, then 
@texline @math{x_1}
@infoline @expr{x1}
must be either a function name itself or a @w{@samp{< >}} nameless
function; @pxref{Specifying Operators}.)  For example, @samp{[g(0) := 0,
import(linearF, f, g)]} applies the linearity rules to the function
@samp{g} instead of @samp{f}.  Imports can be nested, but the
import-with-renaming feature may fail to rename sub-imports properly.

The special functions allowed in patterns are:

@table @samp
@item quote(x)
@ignore
@starindex
@end ignore
@tindex quote
This pattern matches exactly @expr{x}; variable names in @expr{x} are
not interpreted as meta-variables.  The only flexibility is that
numbers are compared for numeric equality, so that the pattern
@samp{f(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
(Numbers are always treated this way by the rewrite mechanism:
The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
as a result in this case.)

@item plain(x)
@ignore
@starindex
@end ignore
@tindex plain
Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}.  This
pattern matches a call to function @expr{f} with the specified
argument patterns.  No special knowledge of the properties of the
function @expr{f} is used in this case; @samp{+} is not commutative or
associative.  Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
are treated as patterns.  If you wish them to be treated ``plainly''
as well, you must enclose them with more @code{plain} markers:
@samp{plain(plain(@w{-a}) + plain(b c))}.

@item opt(x,def)
@ignore
@starindex
@end ignore
@tindex opt
Here @expr{x} must be a variable name.  This must appear as an
argument to a function or an element of a vector; it specifies that
the argument or element is optional.
As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
or as the second argument to @samp{/} or @samp{^}, the value @var{def}
may be omitted.  The pattern @samp{x + opt(y)} matches a sum by
binding one summand to @expr{x} and the other to @expr{y}, and it
matches anything else by binding the whole expression to @expr{x} and
zero to @expr{y}.  The other operators above work similarly.

For general miscellaneous functions, the default value @code{def}
must be specified.  Optional arguments are dropped starting with
the rightmost one during matching.  For example, the pattern
@samp{f(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
or @samp{f(a,b,c)}.  Default values of zero and @expr{b} are
supplied in this example for the omitted arguments.  Note that
the literal variable @expr{b} will be the default in the latter
case, @emph{not} the value that matched the meta-variable @expr{b}.
In other words, the default @var{def} is effectively quoted.

@item condition(x,c)
@ignore
@starindex
@end ignore
@tindex condition
@tindex ::
This matches the pattern @expr{x}, with the attached condition
@expr{c}.  It is the same as @samp{x :: c}.

@item pand(x,y)
@ignore
@starindex
@end ignore
@tindex pand
@tindex &&&
This matches anything that matches both pattern @expr{x} and
pattern @expr{y}.  It is the same as @samp{x &&& y}.
@pxref{Composing Patterns in Rewrite Rules}.

@item por(x,y)
@ignore
@starindex
@end ignore
@tindex por
@tindex |||
This matches anything that matches either pattern @expr{x} or
pattern @expr{y}.  It is the same as @w{@samp{x ||| y}}.

@item pnot(x)
@ignore
@starindex
@end ignore
@tindex pnot
@tindex !!!
This matches anything that does not match pattern @expr{x}.
It is the same as @samp{!!! x}.

@item cons(h,t)
@ignore
@mindex cons
@end ignore
@tindex cons (rewrites)
This matches any vector of one or more elements.  The first
element is matched to @expr{h}; a vector of the remaining
elements is matched to @expr{t}.  Note that vectors of fixed
length can also be matched as actual vectors:  The rule
@samp{cons(a,cons(b,[])) := cons(a+b,[])} is equivalent
to the rule @samp{[a,b] := [a+b]}.

@item rcons(t,h)
@ignore
@mindex rcons
@end ignore
@tindex rcons (rewrites)
This is like @code{cons}, except that the @emph{last} element
is matched to @expr{h}, with the remaining elements matched
to @expr{t}.

@item apply(f,args)
@ignore
@mindex apply
@end ignore
@tindex apply (rewrites)
This matches any function call.  The name of the function, in
the form of a variable, is matched to @expr{f}.  The arguments
of the function, as a vector of zero or more objects, are
matched to @samp{args}.  Constants, variables, and vectors
do @emph{not} match an @code{apply} pattern.  For example,
@samp{apply(f,x)} matches any function call, @samp{apply(quote(f),x)}
matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
matches any function call with exactly two arguments, and
@samp{apply(quote(f), cons(a,cons(b,x)))} matches any call
to the function @samp{f} with two or more arguments.  Another
way to implement the latter, if the rest of the rule does not
need to refer to the first two arguments of @samp{f} by name,
would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
Here's a more interesting sample use of @code{apply}:

@example
apply(f,[x+n])  :=  n + apply(f,[x])
   :: in(f, [floor,ceil,round,trunc]) :: integer(n)
@end example

Note, however, that this will be slower to match than a rule
set with four separate rules.  The reason is that Calc sorts
the rules of a rule set according to top-level function name;
if the top-level function is @code{apply}, Calc must try the
rule for every single formula and sub-formula.  If the top-level
function in the pattern is, say, @code{floor}, then Calc invokes
the rule only for sub-formulas which are calls to @code{floor}.

Formulas normally written with operators like @code{+} are still
considered function calls:  @code{apply(f,x)} matches @samp{a+b}
with @samp{f = add}, @samp{x = [a,b]}.

You must use @code{apply} for meta-variables with function names
on both sides of a rewrite rule:  @samp{apply(f, [x]) := f(x+1)}
is @emph{not} correct, because it rewrites @samp{spam(6)} into
@samp{f(7)}.  The righthand side should be @samp{apply(f, [x+1])}.
Also note that you will have to use No-Simplify mode (@kbd{m O})
when entering this rule so that the @code{apply} isn't
evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
Or, use @kbd{s e} to enter the rule without going through the stack,
or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
@xref{Conditional Rewrite Rules}.

@item select(x)
@ignore
@starindex
@end ignore
@tindex select
This is used for applying rules to formulas with selections;
@pxref{Selections with Rewrite Rules}.
@end table

Special functions for the righthand sides of rules are:

@table @samp
@item quote(x)
The notation @samp{quote(x)} is changed to @samp{x} when the
righthand side is used.  As far as the rewrite rule is concerned,
@code{quote} is invisible.  However, @code{quote} has the special
property in Calc that its argument is not evaluated.  Thus,
while it will not work to put the rule @samp{t(a) := typeof(a)}
on the stack because @samp{typeof(a)} is evaluated i