emacs-diffs
[Top][All Lists]

## [Emacs-diffs] Changes to emacs/man/calc.texi [emacs-unicode-2]

 From: Miles Bader Subject: [Emacs-diffs] Changes to emacs/man/calc.texi [emacs-unicode-2] Date: Wed, 08 Dec 2004 02:04:00 -0500

Index: emacs/man/calc.texi
diff -c emacs/man/calc.texi:1.15.6.5 emacs/man/calc.texi:1.15.6.6
*** emacs/man/calc.texi:1.15.6.5        Fri Oct 22 10:13:37 2004
--- emacs/man/calc.texi Wed Dec  8 05:02:23 2004
***************
*** 7,39 ****
@setchapternewpage odd
@comment %**end of header (This is for running Texinfo on a region.)

@tex
- % Some special kludges to make TeX formatting prettier.
- % Because makeinfo.c exists, we can't just define new commands.
- % So instead, we take over little-used existing commands.
- %
% Suggested by Karl Berry <karl@@freefriends.org>
\gdef\!{\mskip-\thinmuskip}
- % Redefine @cite{text} to act like $text$ in regular TeX.
- % Info will typeset this same as @samp{text}.
- \gdef\goodtex{\tex \let\rm\goodrm \let\t\ttfont \turnoffactive}
- \gdef\goodrm{\fam0\tenrm}
- \gdef\cite{\goodtex$\citexxx} - \gdef\citexxx#1{#1$\Etex}
- \global\let\oldxrefX=\xrefX
- \gdef\xrefX[#1]{\begingroup\let\cite=\dfn\oldxrefX[#1]\endgroup}
-
- % Redefine @c{tex-stuff} \n @whatever{info-stuff}.
- \gdef\c{\futurelet\next\mycxxx}
- \gdef\mycxxx{%
-   \ifx\next\bgroup \goodtex\let\next\mycxxy
-   \else\ifx\next\mindex \let\next\relax
-   \else\ifx\next\kindex \let\next\relax
-   \else\ifx\next\starindex \let\next\relax \else \let\next\comment
-   \fi\fi\fi\fi \next
- }
- \gdef\mycxxy#1#2{#1\Etex\mycxxz}
- \gdef\mycxxz#1{}
@end tex

@c Fix some other things specifically for this manual.
--- 7,66 ----
@setchapternewpage odd
@comment %**end of header (This is for running Texinfo on a region.)

+ @c The following macros are used for conditional output for single lines.
+ @c @texline foo
+ @c    foo' will appear only in TeX output
+ @c @infoline foo
+ @c    foo' will appear only in non-TeX output
+
+ @c In TeX output, @tmath{expr} will typeset expr in  math mode.
+ @c In any output, @expr{expr} will typeset an expression;
+ @c $x$ in TeX, @samp{x} otherwise.
+
+ @iftex
+ @macro texline{stuff}
+ \stuff\
+ @end macro
+ @macro tmath{stuff}
+ @tex
+ $\stuff\$
+ @end tex
+ @end macro
+ @alias infoline=comment
+ @c @alias expr=math
+ @tex
+ \gdef\expr#1{\tex
+              \let\t\ttfont
+              \turnoffactive
+              $#1$
+              \endgroup}
+ @end tex
+ @macro cpi{}
+ @end macro
+ @macro cpiover{den}
+ @end macro
+ @end iftex
+
+ @ifnottex
+ @alias texline=comment
+ @macro infoline{stuff}
+ \stuff\
+ @end macro
+ @alias expr=samp
+ @macro cpi{}
+ @expr{pi}
+ @end macro
+ @macro cpiover{den}
+ @expr{pi/\den\}
+ @end macro
+ @end ifnottex
+
+
@tex
% Suggested by Karl Berry <karl@@freefriends.org>
\gdef\!{\mskip-\thinmuskip}
@end tex

@c Fix some other things specifically for this manual.
***************
*** 513,520 ****
@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.  @c{Each}
! @asis{In the printed manual, each}
paragraph that is referenced in the Key or Function Index is marked
in the margin with its index entry.

--- 540,548 ----
@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.

***************
*** 570,576 ****
be @samp{cos(@var{x})}.

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

@node Demonstration of Calc, Using Calc, Notations Used in This Manual,
Getting Started
@section A Demonstration of Calc
--- 598,604 ----
be @samp{cos(@var{x})}.

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

@node Demonstration of Calc, Using Calc, Notations Used in This Manual,
Getting Started
@section A Demonstration of Calc
***************
*** 595,606 ****
then the command to operate on the numbers.

@noindent
! Type @kbd{2 @key{RET} 3 + Q} to compute @c{$\sqrt{2+3} = 2.2360679775$}
! @asis{the square root of 2+3, which is 2.2360679775}.

@noindent
! Type @kbd{P 2 ^} to compute @c{$\pi^2 = 9.86960440109$}
! @asis{the value of pi' squared, 9.86960440109}.

@noindent
Type @key{TAB} to exchange the order of these two results.
--- 623,636 ----
then the command to operate on the numbers.

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

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

@noindent
Type @key{TAB} to exchange the order of these two results.
***************
*** 617,629 ****
use the apostrophe key.

@noindent
! Type @kbd{' sqrt(2+3) @key{RET}} to compute @c{$\sqrt{2+3}$}
! @asis{the square root of 2+3}.

@noindent
! Type @kbd{' pi^2 @key{RET}} to enter @c{$\pi^2$}
! @asis{pi' squared}.  To evaluate this symbolic
! formula as a number, type @kbd{=}.

@noindent
Type @kbd{' arcsinh($- $$) @key{RET}} to subtract the second-most-recent --- 647,661 ---- use the apostrophe key. @noindent ! Type @kbd{' sqrt(2+3) @key{RET}} to compute ! @texline @tmath{\sqrt{2+3}}. ! @infoline the square root of 2+3. @noindent ! Type @kbd{' pi^2 @key{RET}} to enter ! @texline @tmath{\pi^2}. ! @infoline pi' squared. ! To evaluate this symbolic formula as a number, type @kbd{=}. @noindent Type @kbd{' arcsinh( -$$) @key{RET}} to subtract the second-most-recent *************** *** 680,691 **** the lower-right @samp{8} and press @kbd{M-# r}. @noindent ! Type @kbd{v t} to transpose this @c{$3\times2$} ! @asis{3x2} matrix into a @c{$2\times3$} ! @asis{2x3} matrix. Type ! @address@hidden u}} to unpack the rows into two separate vectors. Now type ! @address@hidden 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{M-# :}.) @strong{Units conversion.} Units are entered algebraically. Type @address@hidden' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour. --- 712,727 ---- the lower-right @samp{8} and press @kbd{M-# r}. @noindent ! Type @kbd{v t} to transpose this ! @texline @tmath{3\times2} ! @infoline 3x2 ! matrix into a ! @texline @tmath{2\times3} ! @infoline 2x3 ! matrix. Type @address@hidden u}} to unpack the rows into two separate ! vectors. Now type @address@hidden 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{M-# :}.) @strong{Units conversion.} Units are entered algebraically. Type @address@hidden' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour. *************** *** 701,707 **** 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 @address@hidden S x,y @key{RET}}} to solve ! these equations for the variables @cite{x} and @address@hidden @noindent Type @kbd{d B} to view the solutions in more readable notation. --- 737,743 ---- 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 @address@hidden 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. *************** *** 710,716 **** Type @kbd{d N} to return to normal notation. @noindent ! Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @cite{a = 7.5} in these formulas. (That's a letter @kbd{l}, not a numeral @kbd{1}.) @iftex --- 746,752 ---- 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 *************** *** 739,750 **** different kinds of tasks. As well as Calc's standard interface, there are Quick Mode, Keypad Mode, and Embedded Mode. - @c [fix-ref Installation] - Calc must be @dfn{installed} before it can be used. @xref{Installation}, - for instructions on setting up and installing Calc. We will assume - you or someone on your system has already installed Calc as described - there. - @menu * Starting Calc:: * The Standard Interface:: --- 775,780 ---- *************** *** 769,775 **** @key{ESC} first, then @kbd{#}, to accomplish the same thing. If you don't even have an @key{ESC} key, you can fake it by holding down Control or @key{CTRL} while typing a left square bracket ! (that's @kbd{C-[} in Emacs notation)address@hidden @kbd{M-#} is a @dfn{prefix key}; when you press it, Emacs waits for you to press a second key to complete the command. In this case, --- 799,805 ---- @key{ESC} first, then @kbd{#}, to accomplish the same thing. If you don't even have an @key{ESC} key, you can fake it by holding down Control or @key{CTRL} while typing a left square bracket ! (that's @kbd{C-[} in Emacs notation). @kbd{M-#} is a @dfn{prefix key}; when you press it, Emacs waits for you to press a second key to complete the command. In this case, *************** *** 853,859 **** 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 @cite{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 address@hidden language mode.'' --- 883,889 ---- 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 address@hidden language mode.'' *************** *** 1317,1329 **** @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 @c{$2^{32}$} ! @cite{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 --- 1347,1361 ---- @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 @tmath{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 *************** *** 1373,1390 **** 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; @c{Fran\c cois} ! @asis{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 --- 1405,1423 ---- 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 cois ! @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 *************** *** 1545,1552 **** @cindex Operators @cindex Operands ! In an operation like @cite{2+3}, the 2 and 3 are called the @dfn{operands} ! and the @cite{+} 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 --- 1578,1585 ---- @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 *************** *** 1560,1566 **** 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:@refill @smallexample @group --- 1593,1599 ---- 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 *************** *** 1581,1591 **** @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 ! @cite{x}, @cite{y}, @cite{z}, and @cite{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 address@hidden @c [fix-ref Truncating the Stack] The Stack buffer is just an Emacs buffer, and you can move around in --- 1614,1624 ---- @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 *************** *** 1602,1608 **** @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 address@hidden Examples in this tutorial will often omit @key{RET} even when the stack displays shown would only happen if you did press @key{RET}: --- 1635,1641 ---- @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}: *************** *** 1636,1644 **** 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 @c{$(2\times4) + (7\times9.4) +
{5\over4}$} ! @cite{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 --- 1669,1678 ---- 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 @tmath{(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 *************** *** 1690,1696 **** 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 @cite{20 / (2+3)}. @smallexample @group --- 1724,1730 ---- 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 *************** *** 1798,1808 **** @end smallexample @noindent ! The notation @cite{(a, b)} represents a complex number. ! Complex numbers are more traditionally written @c{$a + b i$} ! @cite{a + b i}; Calc can display in this format, too, but for now we'll stick to the ! @cite{(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 --- 1832,1841 ---- @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 *************** *** 1861,1867 **** 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 @cite{(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.) --- 1894,1900 ---- 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.) *************** *** 1921,1927 **** 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 @cite{2+3} notation. 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{'}) --- 1954,1960 ---- 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. 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{'}) *************** *** 2004,2010 **** the notation @samp{sqrt(@var{x})}. Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should ! be @cite{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} --- 2037,2043 ---- 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} *************** *** 2030,2036 **** 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 @cite{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 --- 2063,2069 ---- 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 *************** *** 2050,2057 **** 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 @c{$\sqrt{2\times4+1}$} ! @cite{sqrt(2*4+1)}, which on a traditional calculator would be done by pressing @kbd{2 * 4 + 1 =} and then the square-root key. --- 2083,2091 ---- 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 @tmath{\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. *************** *** 2190,2196 **** @noindent Notice that the instant we stored a new value in @code{a}, all ! @samp{=>} operators already on the stack that referred to @cite{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. --- 2224,2230 ---- @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. *************** *** 2257,2263 **** @end smallexample @noindent ! It was not possible to redo past the @cite{6}, since that was placed there by something other than an undo command. @cindex Time travel --- 2291,2297 ---- @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 *************** *** 2265,2271 **** @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 @cite{6} even though there was an earlier undo command. You can always recall an earlier result using the Trail. We've ignored --- 2299,2305 ---- @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 *************** *** 2273,2285 **** 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 @cite{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 @address@hidden p}} to move the arrow onto the line containing ! @cite{8}, and press @address@hidden y}} to yank'' that number back onto the stack. If you press @kbd{t ]} again, you will see that even our Yank command --- 2307,2319 ---- 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 @address@hidden p}} to move the arrow onto the line containing ! @expr{8}, and press @address@hidden y}} to yank'' that number back onto the stack. If you press @kbd{t ]} again, you will see that even our Yank command *************** *** 2340,2347 **** to edit a stack entry. Try entering @samp{3.141439} now. If this is supposed to represent ! @c{$\pi$} ! @cite{pi}, 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 --- 2374,2380 ---- 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 *************** *** 2370,2376 **** 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 @cite{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}, --- 2403,2409 ---- 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}, *************** *** 2387,2400 **** 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 @cite{1/7} is an infinitely repeating decimal; Calc has to stop somewhere. Of course, calculations are slower the more digits you request. Press @address@hidden 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 @cite{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 @address@hidden +}}. Notice that the @key{RET} --- 2420,2433 ---- 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 @address@hidden 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 @address@hidden +}}. Notice that the @key{RET} *************** *** 2412,2418 **** @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 @cite{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 --- 2445,2451 ---- @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 *************** *** 2463,2469 **** @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 @cite{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 --- 2496,2502 ---- @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 *************** *** 2564,2570 **** 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 @cite{12345} was not affected by any of the float formats. Integers are integers, and are always displayed exactly. --- 2597,2603 ---- 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. *************** *** 2712,2724 **** @noindent The address@hidden command computes the sine of an angle. The sine ! of 45 degrees is @c{$\sqrt{2}/2$} ! @cite{sqrt(2)/2}; squaring this yields @cite{2/4 = 0.5}. ! However, there has been a slight roundoff error because the ! representation of @c{$\sqrt{2}/2$} ! @cite{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 --- 2745,2759 ---- @noindent The address@hidden command computes the sine of an angle. The sine ! of 45 degrees is ! @texline @tmath{\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 @tmath{\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 *************** *** 2729,2737 **** To do this calculation in radians, we would type @kbd{m r} first. (The indicator changes to @samp{Rad}.) 45 degrees corresponds to ! @c{$\pi\over4$} ! @cite{pi/4} radians. To get @c{$\pi$} ! @cite{pi}, press the @kbd{P} key. (Once again, this is a shifted capital @kbd{P}. Remember, unshifted @kbd{p} sets the precision.) --- 2764,2770 ---- 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.) *************** *** 2757,2765 **** @end smallexample @noindent ! Here we compute the Inverse Sine of @c{$\sqrt{0.5}$} ! @cite{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. --- 2790,2799 ---- @end smallexample @noindent ! Here we compute the Inverse Sine of ! @texline @tmath{\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. *************** *** 2939,2953 **** 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} (@cite{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 @c{$\sin^2x + \cos^2x = 1$} ! @cite{sin(x)^2 + cos(x)^2 = 1}. We'll ! arbitrarily pick @i{-64} degrees as a good value for @cite{x}. With ! the angular mode set to degrees (type @address@hidden d}}), do: @smallexample @group --- 2973,2988 ---- 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 @tmath{\sin^2x + \cos^2x = 1}. ! @infoline @expr{sin(x)^2 + cos(x)^2 = 1}. ! We'll arbitrarily pick @i{-64} degrees as a good value for @expr{x}. ! With the angular mode set to degrees (type @address@hidden d}}), do: @smallexample @group *************** *** 2966,2973 **** Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum of squares, command. ! Another identity is @c{$\displaystyle\tan x = {\sin x \over \cos x}$} ! @cite{tan(x) = sin(x) / cos(x)}. @smallexample @group --- 3001,3009 ---- Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum of squares, command. ! Another identity is ! @texline @tmath{\displaystyle\tan x = {\sin x \over \cos x}}. ! @infoline @expr{tan(x) = sin(x) / cos(x)}. @smallexample @group *************** *** 2980,2986 **** @end smallexample A physical interpretation of this calculation is that if you move ! @cite{0.89879} units downward and @cite{0.43837} units to the right, your direction of motion is @i{-64} degrees from horizontal. Suppose we move in the opposite direction, up and to the left: --- 3016,3022 ---- @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 @i{-64} degrees from horizontal. Suppose we move in the opposite direction, up and to the left: *************** *** 3029,3037 **** A similar identity is supposed to hold for hyperbolic sines and cosines, except that it is the @emph{difference} ! @c{$\cosh^2x - \sinh^2x$} ! @cite{cosh(x)^2 - sinh(x)^2} that always equals one. ! Let's try to verify this address@hidden @smallexample @group --- 3065,3073 ---- A similar identity is supposed to hold for hyperbolic sines and cosines, except that it is the @emph{difference} ! @texline @tmath{\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 *************** *** 3057,3068 **** 0.99999, reasonably close to 1. Of course, a more reasonable way to verify the identity is to use ! a more reasonable value for @cite{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 ! @cite{e} normally but use base-10 instead if you use the Hyperbolic prefix. @smallexample --- 3093,3104 ---- 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 *************** *** 3095,3101 **** 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 @cite{e} onto the stack. You may have noticed that both times we took the base-10 logarithm --- 3131,3137 ---- 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 *************** *** 3136,3143 **** If you take the factorial of a non-integer, Calc uses a generalized factorial function defined in terms of Euler's Gamma function ! @c{$\Gamma(n)$} ! @cite{gamma(n)} (which is itself available as the @kbd{f g} command). @smallexample --- 3172,3179 ---- If you take the factorial of a non-integer, Calc uses a generalized factorial function defined in terms of Euler's Gamma function ! @texline @tmath{\Gamma(n)} ! @infoline @expr{gamma(n)} (which is itself available as the @kbd{f g} command). @smallexample *************** *** 3152,3168 **** @end smallexample @noindent ! Here we verify the identity @c{$n! = \Gamma(n+1)$} ! @address@hidden@: = gamma(@var{n}+1)}. ! The binomial coefficient @address@hidden@c{ or$\displaystyle {n \choose m}$} ! @asis{} is defined by ! @c{$\displaystyle {n! \over m! \, (n-m)!}$} ! @cite{n!@: / m!@: (n-m)!} for all reals @cite{n} and ! @cite{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 --- 3188,3206 ---- @end smallexample @noindent ! Here we verify the identity ! @texline @tmath{n! = \Gamma(n+1)}. ! @infoline @address@hidden@: = gamma(@var{n}+1)}. ! The binomial coefficient @address@hidden ! @texline or @tmath{\displaystyle {n \choose m}} ! is defined by ! @texline @tmath{\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 *************** *** 3266,3272 **** @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 @cite{(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. --- 3304,3310 ---- @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. *************** *** 3379,3386 **** When two matrices are multiplied, the lefthand matrix must have the same number of columns as the righthand matrix has rows. ! Row @cite{i}, column @cite{j} of the result is effectively the ! dot product of row @cite{i} of the left matrix by column @cite{j} of the right matrix. If we try to duplicate this matrix and multiply it by itself, --- 3417,3424 ---- 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, *************** *** 3441,3449 **** vector. (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows ! of the above @c{$2\times3$} ! @asis{2x3} matrix to get @cite{[6, 15]}. Now use @samp{*} to ! sum along the columns to get @cite{[5, 7, 9]}. @xref{Matrix Answer 1, 1}. (@bullet{}) @cindex Identity matrix --- 3479,3489 ---- vector. (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows ! of the above ! @texline @tmath{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 *************** *** 3566,3572 **** @end smallexample @noindent ! The result is the @cite{[a, b, c]} vector that solves the equations. (Dividing by a square matrix is equivalent to multiplying by its inverse.) --- 3606,3612 ---- @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.) *************** *** 3590,3605 **** 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 @c{$N\times1$} ! @asis{Nx1} or @c{$1\times N$} ! @asis{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 @cite{x} and @cite{y} ! in terms of @cite{a} and @cite{b}. @ifinfo @group --- 3630,3648 ---- 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 @tmath{N\times1} ! @infoline Nx1 ! or ! @texline @tmath{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 *************** *** 3628,3637 **** 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 @cite{A X = B} directly because the matrix @cite{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 @cite{A}: @ifinfo @samp{trn(A)*A*X = trn(A)*B}. @end ifinfo --- 3671,3680 ---- 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 *************** *** 3639,3650 **** \turnoffactive$A^T A \, X = A^T B$, where$A^T$is the transpose \samp{trn(A)}. @end tex ! Now @c{$A^T A$} ! @cite{trn(A)*A} is a square matrix so a solution is possible. It ! turns out that the @cite{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:@refill @ifinfo @group --- 3682,3695 ---- \turnoffactive$A^T A \, X = A^T B$, where$A^T$is the transpose \samp{trn(A)}. @end tex ! Now ! @texline @tmath{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 *************** *** 3733,3740 **** of each element. (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two ! from @c{$2^{-4}$} ! @cite{2^-4} to @cite{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 --- 3778,3787 ---- of each element. (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two ! from ! @texline @tmath{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 *************** *** 3868,3880 **** @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 @cite{1.34}. Press @kbd{C-@@} to set the mark. (On your system this may be @kbd{C-2}, @address@hidden, or @kbd{NUL}.) ! Now position the cursor to the lower-right, just after the @cite{1.354}. You have now defined this region as an Emacs rectangle.'' Still in the Info buffer, type @kbd{M-# 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 address@hidden @smallexample @group --- 3915,3927 ---- @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}, @address@hidden, 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{M-# 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 *************** *** 3919,3925 **** (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 @cite{m} is given by the formula @ifinfo @example --- 3966,3972 ---- (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 *************** *** 3935,3945 **** @end tex @noindent ! where @c{$\sum x$} ! @cite{sum(x)} represents the sum of all the values of @cite{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 --- 3982,3994 ---- @end tex @noindent ! where ! @texline @tmath{\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 *************** *** 3973,3979 ****$\sum x y$.) @end tex ! Finally, we also need @cite{N}, the number of data points. This is just the length of either of our lists. @smallexample --- 4022,4028 ----$\sum x y$.) @end tex ! Finally, we also need @expr{N}, the number of data points. This is just the length of either of our lists. @smallexample *************** *** 4012,4018 **** @end group @end smallexample ! That gives us the slope @cite{m}. The y-intercept @cite{b} can now be found with the simple formula, @ifinfo --- 4061,4067 ---- @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 *************** *** 4038,4045 **** @end group @end smallexample ! Let's plot'' this straight line approximation, @c{$y \approx m x + b$} ! @cite{m x + b}, and compare it with the original address@hidden @smallexample @group --- 4087,4096 ---- @end group @end smallexample ! Let's plot'' this straight line approximation, ! @texline @tmath{y \approx m x + b}, ! @infoline @expr{m x + b}, ! and compare it with the original data. @smallexample @group *************** *** 4056,4062 **** 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 @cite{y} vector to get a feel for the error of our fit. Let's find the maximum error: @smallexample --- 4107,4113 ---- 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 *************** *** 4131,4139 **** (@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 @cite{y_i = m x_i + b} for ! different pairs of @cite{(x_i,y_i)}. Use the matrix-transpose method ! to solve for @cite{m} and @cite{b}, duplicating the above result. @xref{List Answer 2, 2}. (@bullet{}) @cindex Geometric mean --- 4182,4190 ---- (@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 *************** *** 4161,4167 **** @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so on up to @address@hidden, always comes out to zero. Let's verify this ! for @address@hidden @end ifinfo @tex As another example, a theorem about binomial coefficients tells --- 4212,4218 ---- @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so on up to @address@hidden, 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 *************** *** 4232,4244 **** and @kbd{v c} instead delete one row, column, or vector element. @cindex Divisor functions ! (@bullet{}) @strong{Exercise 4.} The @cite{k}th @dfn{divisor function} @tex$\sigma_k(n)$@end tex ! is the sum of the @cite{k}th powers of all the divisors of an ! integer @cite{n}. Figure out a method for computing the divisor ! function for reasonably small values of @cite{n}. As a test, the 0th and 1st divisor functions of 30 are 8 and 72, respectively. @xref{List Answer 4, 4}. (@bullet{}) --- 4283,4295 ---- 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{}) *************** *** 4290,4312 **** @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 ! @c{$J_1(x)$} ! @cite{J1} function @samp{besJ(1,x)} for @cite{x} from 0 to 5 ! in steps of 0.25. ! Find the value of @cite{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{})@refill @cindex Digits, vectors of (@bullet{}) @strong{Exercise 9.} You are given an integer in the range ! @c{$0 \le N < 10^m$} ! @cite{0 <= N < 10^m} for @cite{m=12} (i.e., an integer of less than ! twelve digits). Convert this integer into a vector of @cite{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 @cite{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{}) --- 4341,4364 ---- @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 @tmath{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 @tmath{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{}) *************** *** 4316,4355 **** 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 @c{$\pi$} ! @cite{pi}. The area of the @c{$2\times2$} ! @asis{2x2} square that encloses that ! circle is 4. So if we throw @var{n} darts at random points in the square, ! about @c{$\pi/4$} ! @cite{pi/4} of them will land inside the circle. This gives us ! an entertaining way to estimate the value of @c{$\pi$} ! @cite{pi}. The @address@hidden r}} command picks a random number between zero and the value on the stack. We could get a random floating-point number between @i{-1} and 1 by typing ! @address@hidden k r 1 -}}. Build a vector of 100 random @cite{(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 @c{$\pi$} ! @cite{pi}. 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 @c{$2/\pi$} ! @cite{2/pi}. Toss 100 matchsticks to estimate ! @c{$\pi$} ! @cite{pi}. (If you want still more fun, the probability that the GCD ! (@address@hidden g}}) of two large integers is one turns out to be @c{$6/\pi^2$} ! @cite{6/pi^2}. ! That provides yet another way to estimate @c{$\pi$} ! @cite{pi}.) @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, @cite{[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 --- 4368,4406 ---- 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 @tmath{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 @address@hidden r}} command picks a random number between zero and the value on the stack. We could get a random floating-point number between @i{-1} and 1 by typing ! @address@hidden 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 @tmath{2/\pi}. ! @infoline @expr{2/pi}. ! Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun, ! the probability that the GCD (@address@hidden g}}) of two large integers is ! one turns out to be ! @texline @tmath{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 *************** *** 4358,4366 **** 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 @cite{h = 0}. ! Then, for each character from the string in turn, set @cite{h = 3h + c_i} ! where @cite{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 --- 4409,4417 ---- 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 *************** *** 4372,4379 **** 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 ! @cite{(0,0)}; then take one step a random distance between @i{-1} and 1 ! in both @cite{x} and @cite{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 --- 4423,4430 ---- 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 @i{-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 *************** *** 4436,4443 **** (@bullet{}) @strong{Exercise 1.} A calculation has produced the result 1.26508260337. You suspect it is the square root of the ! product of @c{$\pi$} ! @cite{pi} 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. --- 4487,4493 ---- (@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. *************** *** 4453,4459 **** @noindent The square root of @i{-9} is by default rendered in rectangular form ! (@address@hidden + 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. --- 4503,4509 ---- @noindent The square root of @i{-9} is by default rendered in rectangular form ! (@address@hidden + 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. *************** *** 4509,4517 **** 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 ! @cite{1 / x} around @address@hidden = 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 @cite{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 --- 4559,4567 ---- 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 @address@hidden = 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 *************** *** 4641,4650 **** @cindex Torus, volume of (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is ! @c{$2 \pi^2 R r^2$} ! @address@hidden pi^2 R r^2}} where @cite{R} is the radius of the circle that ! defines the center of the tube and @cite{r} is the radius of the tube ! itself. Suppose @cite{R} is 20 cm and @cite{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{}) --- 4691,4701 ---- @cindex Torus, volume of (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is ! @texline @tmath{2 \pi^2 R r^2} ! @infoline @address@hidden 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{}) *************** *** 4739,4752 **** @cindex Fermat, primality test of (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat ! says that @c{\w{$x^{n-1} \bmod n = 1$}} ! @cite{x^(n-1) mod n = 1} if @cite{n} is a prime number ! and @cite{x} is an integer less than @cite{n}. If @cite{n} is ! @emph{not} a prime number, this will @emph{not} be true for most ! values of @cite{x}. Thus we can test informally if a number is ! prime by trying this formula for several values of @cite{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. --- 4790,4804 ---- @cindex Fermat, primality test of (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat ! says that ! @texline @address@hidden \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. *************** *** 4766,4774 **** 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 @c{$\pi \times 10^7$} ! @address@hidden * 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}. --- 4818,4828 ---- 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 @tmath{\pi \times 10^7} ! @infoline @address@hidden * 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}. *************** *** 4949,4957 **** @noindent First we expand'' using the distributive law, then we collect'' ! terms involving like powers of @cite{x}. ! Let's find the value of this expression when @cite{x} is 2 and @cite{y} is one-half. @smallexample --- 5003,5011 ---- @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 *************** *** 4976,4986 **** properly.) @cindex Maximum of a function using Calculus ! Let's find the maximum value of our original expression when @cite{y} ! is one-half and @cite{x} ranges over all possible values. We can ! do this by taking the derivative with respect to @cite{x} and examining ! values of @cite{x} for which the derivative is zero. If the second ! derivative of the function at that value of @cite{x} is negative, the function has a local maximum there. @smallexample --- 5030,5040 ---- 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 *************** *** 4993,5000 **** @end smallexample @noindent ! Well, the derivative is clearly zero when @cite{x} is zero. To find ! the other root(s), let's divide through by @cite{x} and then solve: @smallexample @group --- 5047,5054 ---- @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 *************** *** 5020,5026 **** 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 @cite{x}: @smallexample @group --- 5074,5080 ---- 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 *************** *** 5050,5063 **** @noindent The first of these second derivatives is negative, so we know the function ! has a maximum value at @cite{x = 1.19023}. (The function also has a ! local @emph{minimum} at @cite{x = 0}.) ! When we solved for @cite{x}, we got only one value even though ! @cite{34 - 24 x^2 = 0} is a quadratic equation that ought to have two solutions. The reason is that @address@hidden 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 @cite{+}. 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}. --- 5104,5117 ---- @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 @address@hidden 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}. *************** *** 5075,5083 **** it is supposed to be either @i{+1} or @i{-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 @cite{x = -1.19023} is also a maximum. ! To find the actual maximum value, we must plug our two values of @cite{x} into the original formula. @smallexample --- 5129,5137 ---- it is supposed to be either @i{+1} or @i{-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 *************** *** 5127,5133 **** @address@hidden 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 ! @cite{[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.) --- 5181,5187 ---- @address@hidden 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.) *************** *** 5215,5221 **** are shown in normal mode.) @cindex Area under a curve ! What is the area under the portion of this curve from @cite{x = 1} to @cite{2}? This is simply the integral of the function: @smallexample --- 5269,5275 ---- 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 *************** *** 5228,5234 **** @end smallexample @noindent ! We want to evaluate this at our two values for @cite{x} and subtract. One way to do it is again with vector mapping and reduction: @smallexample --- 5282,5288 ---- @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 *************** *** 5240,5259 **** @end group @end smallexample ! (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @cite{y} ! of @c{$x \sin \pi x$} ! @address@hidden sin(pi x)}} (where the sine is calculated in radians). ! Find the values of the integral for integers @cite{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 @c{$\sin x \ln x$} ! @cite{sin(x) ln(x)} over the same range of @cite{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 --- 5294,5316 ---- @end group @end smallexample ! (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y} ! of ! @texline @tmath{x \sin \pi x} ! @infoline @address@hidden 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 @tmath{\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 *************** *** 5322,5328 **** @noindent Here we have computed the Taylor series expansion of the function ! about the point @cite{x=1}. We can now integrate this polynomial approximation, since polynomials are easy to integrate. @smallexample --- 5379,5385 ---- @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 *************** *** 5339,5346 **** 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 @cite{x=2} or @cite{x=1.5} instead ! of @cite{x=1}.) @cindex Simpson's rule @cindex Integration by Simpson's rule --- 5396,5403 ---- 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 *************** *** 5370,5376 **** @end tex @noindent ! where @cite{n} (which must be even) is the number of slices and @cite{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: --- 5427,5433 ---- @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: *************** *** 5389,5397 **** \afterdisplay @end tex ! Compute the integral from 1 to 2 of @c{$\sin x \ln x$} ! @cite{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 --- 5446,5456 ---- \afterdisplay @end tex ! Compute the integral from 1 to 2 of ! @texline @tmath{\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 *************** *** 5639,5645 **** 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 @address@hidden @cindex Fibonacci numbers @ignore --- 5698,5704 ---- 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 *************** *** 5789,5798 **** @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 @cite{a}, ! divide @cite{a} by two if it is even, otherwise compute @cite{3 a + 1}. Now repeat this step over and over. A famous unproved conjecture ! is that for any starting @cite{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. --- 5848,5857 ---- @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. *************** *** 5801,5827 **** Now make the result be a vector of values in the sequence, from @var{a} to 1. (The formula @address@hidden|@var{y}} appends the vectors @var{x} and @var{y}.) For example, rewriting @samp{seq(6)} should yield the ! vector @cite{[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 @cite{2 - 3 (x + y) + x y} is a sum of three terms.) @xref{Rewrites Answer 5, 5}. (@bullet{}) ! (@bullet{}) @strong{Exercise 6.} Calc considers the form @cite{0^0} to be indeterminate,'' and leaves it unevaluated (assuming infinite ! mode is not enabled). Some people prefer to define @cite{0^0 = 1}, ! so that the identity @cite{x^0 = 1} can safely be used for all @cite{x}. Find a way to make Calc follow this convention. What happens if you now type @kbd{m i} to turn on infinite mode? @xref{Rewrites Answer 6, 6}. (@bullet{}) (@bullet{}) @strong{Exercise 7.} A Taylor series for a function is an infinite series that exactly equals the value of that function at ! values of @cite{x} near zero. @ifinfo @example --- 5860,5886 ---- Now make the result be a vector of values in the sequence, from @var{a} to 1. (The formula @address@hidden|@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.} Calc considers the form @expr{0^0} to be indeterminate,'' and leaves it unevaluated (assuming infinite ! mode is not enabled). Some people prefer to define @expr{0^0 = 1}, ! so that the identity @expr{x^0 = 1} can safely be used for all @expr{x}. Find a way to make Calc follow this convention. What happens if you now type @kbd{m i} to turn on infinite mode? @xref{Rewrites Answer 6, 6}. (@bullet{}) (@bullet{}) @strong{Exercise 7.} 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 *************** *** 5829,5843 **** @end example @end ifinfo @tex ! \turnoffactive \let\rm\goodrm \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, @cite{x^2}. ! Calc represents the truncated Taylor series as a polynomial in @cite{x}. Mathematicians often write a truncated series using a big-O'' notation that records what was the lowest term that was truncated. --- 5888,5902 ---- @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. *************** *** 5847,5861 **** @end example @end ifinfo @tex ! \turnoffactive \let\rm\goodrm \beforedisplay $$\cos x = 1 - {x^2 \over 2!} + O(x^3)$$ \afterdisplay @end tex @noindent ! The meaning of @cite{O(x^3)} is a quantity which is negligibly small ! if @cite{x^3} is considered negligibly small as @cite{x} goes to zero.'' The exercise is to create rewrite rules that simplify sums and products of power series represented as @address@hidden + O(@address@hidden)}. --- 5906,5920 ---- @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 @address@hidden + O(@address@hidden)}. *************** *** 5934,5942 **** @end ignore @tindex Si (@bullet{}) @strong{Exercise 1.} The sine integral'' function ! @c{${\rm Si}(x)$} ! @cite{Si(x)} is defined as the integral of @samp{sin(t)/t} for ! @cite{t = 0} to @cite{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, --- 5993,6002 ---- @end ignore @tindex Si (@bullet{}) @strong{Exercise 1.} The sine integral'' function ! @texline @tmath{{\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, *************** *** 6010,6021 **** @enumerate @item ! Compute @c{$\displaystyle{\sin x \over x}$} ! @cite{sin(x) / x}, where @cite{x} is the number on the ! top of the stack. @item ! Compute the address@hidden logarithm, just like the @kbd{B} key except the arguments are taken in the opposite order. @item --- 6070,6082 ---- @enumerate @item ! Compute ! @texline @tmath{\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 address@hidden logarithm, just like the @kbd{B} key except the arguments are taken in the opposite order. @item *************** *** 6044,6050 **** @end smallexample @noindent ! Here we have computed the fourth derivative of @cite{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. --- 6105,6111 ---- @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. *************** *** 6074,6087 **** @cindex Golden ratio @cindex Phi, golden ratio ! A fascinating property of the Fibonacci numbers is that the @cite{n}th ! Fibonacci number can be found directly by computing @c{$\phi^n / \sqrt{5}$} ! @cite{phi^n / sqrt(5)} ! and then rounding to the nearest integer, where @c{$\phi$(phi'')} ! @cite{phi}, the ! golden ratio,'' is @c{$(1 + \sqrt{5}) / 2$} ! @cite{(1 + sqrt(5)) / 2}. (For convenience, this constant is available ! from the @code{phi} variable, or the @kbd{I H P} command.) @smallexample @group --- 6135,6152 ---- @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 @tmath{\phi^n / \sqrt{5}} ! @infoline @expr{phi^n / sqrt(5)} ! and then rounding to the nearest integer, where ! @texline @tmath{\phi} (phi''), ! @infoline @expr{phi}, ! the golden ratio,'' is ! @texline @tmath{(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 *************** *** 6094,6115 **** @cindex Continued fractions (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction} ! representation of @c{$\phi$} ! @cite{phi} is @c{$1 + 1/(1 + 1/(1 + 1/( \ldots )))$} ! @cite{1 + 1/(1 + 1/(1 + 1/( ...@: )))}. We can compute an approximate value by carrying this however far ! and then replacing the innermost @c{$1/( \ldots )$} ! @cite{1/( ...@: )} by 1. Approximate ! @c{$\phi$} ! @cite{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 @address@hidden, b]}} determine a matrix which, when multiplied by this ! vector, produces the vector @cite{[b, c]}, where @cite{a}, @cite{b} and ! @cite{c} are three successive Fibonacci numbers. Now write a program ! that, given an integer @cite{n}, computes the @cite{n}th Fibonacci number using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{}) @cindex Harmonic numbers --- 6159,6186 ---- @cindex Continued fractions (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction} ! representation of ! @texline @tmath{\phi} ! @infoline @expr{phi} ! is ! @texline @tmath{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 @tmath{1/( \ldots )} ! @infoline @expr{1/( ...@: )} ! by 1. Approximate ! @texline @tmath{\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 @address@hidden, 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 *************** *** 6198,6209 **** 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 @c{$\displaystyle{2 n! \over (2 \pi)^n}$} ! @cite{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 @cite{n} since the higher ! Bernoulli numbers are very large fractions.) @smallexample @group --- 6269,6281 ---- 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 @tmath{\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 *************** *** 6223,6229 **** 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 @cite{5/66}. @smallexample @group --- 6295,6301 ---- 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 *************** *** 6354,6361 **** (@bullet{}) @strong{Exercise 8.} A general algorithm for solving equations numerically is @dfn{Newton's Method}. Given the equation ! @cite{f(x) = 0} for any function @cite{f}, and an initial guess ! @cite{x_0} which is reasonably close to the desired solution, apply this formula over and over: @ifinfo --- 6426,6433 ---- (@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 *************** *** 6365,6396 **** @end ifinfo @tex \beforedisplay ! $$x_{\goodrm new} = x - {f(x) \over f'(x)}$$ \afterdisplay @end tex @noindent ! where @cite{f'(x)} is the derivative of @cite{f}. The @cite{x} values will quickly converge to a solution, i.e., eventually ! @c{$x_{\rm new}$} ! @cite{new_x} and @cite{x} will be equal to within the limits of the current precision. Write a program which takes a formula ! involving the variable @cite{x}, and an initial guess @cite{x_0}, ! on the stack, and produces a value of @cite{x} for which the formula ! is zero. Use it to find a solution of @c{$\sin(\cos x) = 0.5$} ! @cite{sin(cos(x)) = 0.5} ! near @cite{x = 4.5}. (Use angles measured in radians.) Note that the built-in @address@hidden 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 @c{$\psi(z)$(psi'')} ! @cite{psi(z)} ! is defined as the derivative of @c{$\ln \Gamma(z)$} ! @cite{ln(gamma(z))}. For large ! values of @cite{z}, it can be approximated by the infinite sum @ifinfo @example --- 6437,6472 ---- @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 @tmath{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 @tmath{\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 @address@hidden 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 @tmath{\psi(z) (psi'')} ! @infoline @expr{psi(z)} ! is defined as the derivative of ! @texline @tmath{\ln \Gamma(z)}. ! @infoline @expr{ln(gamma(z))}. ! For large values of @expr{z}, it can be approximated by the infinite sum @ifinfo @example *************** *** 6398,6404 **** @end example @end ifinfo @tex - \let\rm\goodrm \beforedisplay $$\psi(z) \approx \ln z - {1\over2z} - \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}} --- 6474,6479 ---- *************** *** 6407,6443 **** @end tex @noindent ! where @c{\sum} ! @cite{sum} represents the sum over @cite{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, ! @c{\gamma = -\psi(1)} ! @cite{gamma = -psi(1)}. Unfortunately, 1 isn't a large enough argument ! for the above formula to work (5 is a much safer value for @cite{z}). ! Fortunately, we can compute @c{\psi(1)} ! @cite{psi(1)} from @c{\psi(5)} ! @cite{psi(5)} using ! the recurrence @c{\psi(z+1) = \psi(z) + {1 \over z}} ! @cite{psi(z+1) = psi(z) + 1/z}. Your task: Develop ! a program to compute @c{\psi(z)} ! @cite{psi(z)}; it should pump up'' @cite{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 @c{\gamma} ! @cite{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 @cite{x} and ! a number @cite{m} on the stack, where the polynomial is of degree ! @cite{m} or less (i.e., does not have any terms higher than @cite{x^m}), write a program to convert the polynomial into a list-of-coefficients ! notation. For example, @cite{5 x^4 + (x + 1)^2} with @cite{m = 6} ! should produce the list @cite{[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{}) --- 6482,6529 ---- @end tex @noindent ! where ! @texline @tmath{\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 @tmath{\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 @tmath{\psi(1)} ! @infoline @expr{psi(1)} ! from ! @texline @tmath{\psi(5)} ! @infoline @expr{psi(5)} ! using the recurrence ! @texline @tmath{\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 @tmath{\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 @tmath{\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{}) *************** *** 6478,6486 **** the complete recursive program. (Another way is to use @address@hidden E}} or @kbd{M-# 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 @cite{n} and ! @cite{m} on the stack. Test it with @emph{small} inputs like ! @cite{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{}) --- 6564,6572 ---- the complete recursive program. (Another way is to use @address@hidden E}} or @kbd{M-# 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{}) *************** *** 6492,6498 **** (@bullet{}) @strong{Exercise 12.} Write another program for computing Stirling numbers of the first kind, this time using ! rewrite rules. Once again, @cite{n} and @cite{m} should be taken from the stack. @xref{Programming Answer 12, 12}. (@bullet{}) @example --- 6578,6584 ---- (@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 *************** *** 6601,6621 **** @noindent @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -} ! The result is @c{1 - (2 \times (3 + 4)) = -13} ! @cite{1 - (2 * (3 + 4)) = -13}. @node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises @subsection RPN Tutorial Exercise 2 @noindent ! @c{2\times4 + 7\times9.5 + {5\over4} = 75.75} ! @cite{2*4 + 7*9.5 + 5/4 = 75.75} ! After computing the intermediate term @c{2\times4 = 8} ! @cite{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 --- 6687,6709 ---- @noindent @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -} ! The result is ! @texline @tmath{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 @tmath{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 @tmath{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 *************** *** 6739,6746 **** 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 @cite{3.0} whereas ! @samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @cite{3}.) @node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises @subsection Algebraic Entry Tutorial Exercise 2 --- 6827,6834 ---- 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 *************** *** 6755,6768 **** @subsection Algebraic Entry Tutorial Exercise 3 @noindent ! The result from @kbd{1 @key{RET} 0 /} will be the formula @cite{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 @cite{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 --- 6843,6856 ---- @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 *************** *** 6852,6858 **** @subsection Modes Tutorial Exercise 3 @noindent ! The answer he got was @cite{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. --- 6940,6946 ---- @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. *************** *** 6917,6932 **** @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 @cite{123456789 / 2} when the current ! precision is 6 digits. The true answer is @cite{61728394.5}, but ! with a precision of 6 this will be rounded to @c{12345700.0/2.0 = 61728500.0} ! @cite{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 @cite{123456789 / 2} ! produces the exact fraction @cite{123456789:2}, which can be rounded down by the @kbd{F} command without ever switching to floating-point format. --- 7005,7021 ---- @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 @tmath{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. *************** *** 6934,6941 **** @subsection Arithmetic Tutorial Exercise 2 @noindent ! @kbd{27 @key{RET} 9 B} could give the exact result @cite{3:2}, but it ! does a floating-point calculation instead and produces @cite{1.5}. Calc will find an exact result for a logarithm if the result is an integer or the reciprocal of an integer. But there is no efficient way to search --- 7023,7030 ---- @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 the reciprocal of an integer. But there is no efficient way to search *************** *** 7029,7042 **** @subsection Matrix Tutorial Exercise 3 @noindent ! To solve @c{A^T A \, X = A^T B} ! @cite{trn(A) * A * X = trn(A) * B}, first we compute ! @c{A' = A^T A} ! @cite{A2 = trn(A) * A} and @c{B' = A^T B} ! @cite{B2 = trn(A) * B}; now, we have a ! system @c{A' X = B'} ! @cite{A2 * X = B2} which we can solve using Calc's @samp{/} ! command. @ifinfo @example --- 7118,7136 ---- @subsection Matrix Tutorial Exercise 3 @noindent ! To solve ! @texline @tmath{A^T A \, X = A^T B}, ! @infoline @expr{trn(A) * A * X = trn(A) * B}, ! first we compute ! @texline @tmath{A' = A^T A} ! @infoline @expr{A2 = trn(A) * A} ! and ! @texline @tmath{B' = A^T B}; ! @infoline @expr{B2 = trn(A) * B}; ! now, we have a system ! @texline @tmath{A' X = B'} ! @infoline @expr{A2 * X = B2} ! which we can solve using Calc's @samp{/} command. @ifinfo @example *************** *** 7066,7073 **** 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 ! @c{B'} ! @cite{B2} vector. @smallexample @group --- 7160,7168 ---- 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 @tmath{B'} ! @infoline @expr{B2} ! vector. @smallexample @group *************** *** 7082,7089 **** @end smallexample @noindent ! Now we compute the matrix @c{A'} ! @cite{A2} and divide. @smallexample @group --- 7177,7186 ---- @end smallexample @noindent ! Now we compute the matrix ! @texline @tmath{A'} ! @infoline @expr{A2} ! and divide. @smallexample @group *************** *** 7101,7114 **** (The actual computed answer will be slightly inexact due to round-off error.) ! Notice that the answers are similar to those for the @c{3\times3} ! @asis{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 @c{4\times3} ! @asis{4x3} system would be equivalent to the original @c{3\times3} ! @asis{3x3} system.) Since the first and fourth equations aren't quite equivalent, they --- 7198,7215 ---- (The actual computed answer will be slightly inexact due to round-off error.) ! Notice that the answers are similar to those for the ! @texline @tmath{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 @tmath{4\times3} ! @infoline 4x3 ! system would be equivalent to the original ! @texline @tmath{3\times3} ! @infoline 3x3 system.) Since the first and fourth equations aren't quite equivalent, they *************** *** 7129,7136 **** @end smallexample @noindent ! This is reasonably close to our original @cite{B} vector, ! @cite{[6, 2, 3, 11]}. @node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises @subsection List Tutorial Exercise 1 --- 7230,7237 ---- @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 *************** *** 7168,7174 **** @subsection List Tutorial Exercise 2 @noindent ! Given @cite{x} and @cite{y} vectors in quick variables 1 and 2 as before, the first job is to form the matrix that describes the problem. @ifinfo --- 7269,7275 ---- @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 *************** *** 7183,7192 **** \afterdisplay @end tex ! Thus we want a @c{19\times2} ! @asis{19x2} matrix with our @cite{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 @cite{A} matrix. @smallexample @group --- 7284,7295 ---- \afterdisplay @end tex ! Thus we want a ! @texline @tmath{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 *************** *** 7200,7208 **** @end smallexample @noindent ! Now we compute @c{A^T y} ! @cite{trn(A) * y} and @c{A^T A} ! @cite{trn(A) * A} and divide. @smallexample @group --- 7303,7315 ---- @end smallexample @noindent ! Now we compute ! @texline @tmath{A^T y} ! @infoline @expr{trn(A) * y} ! and ! @texline @tmath{A^T A} ! @infoline @expr{trn(A) * A} ! and divide. @smallexample @group *************** *** 7227,7236 **** @end group @end smallexample ! Since we were solving equations of the form @c{m \times x + b \times 1 = y} ! @cite{m*x + b*1 = y}, these ! numbers should be @cite{m} and @cite{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 --- 7334,7345 ---- @end group @end smallexample ! Since we were solving equations of the form ! @texline @tmath{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 *************** *** 7288,7296 **** @subsection List Tutorial Exercise 4 @noindent ! A number @cite{j} is a divisor of @cite{n} if @c{n \mathbin{\hbox{\code{\%}}} j = 0} ! @samp{n % j = 0}. The first ! step is to get a vector that identifies the divisors. @smallexample @group --- 7397,7406 ---- @subsection List Tutorial Exercise 4 @noindent ! A number @expr{j} is a divisor of @expr{n} if ! @texline @tmath{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 *************** *** 7358,7367 **** zero, so adding zeros on the left and right is safe. From then on the job is pretty straightforward. ! Incidentally, Calc provides the @c{\dfn{M\"obius} \mu} ! @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 --- 7468,7478 ---- 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} @tmath{\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 *************** *** 7389,7398 **** @end smallexample The numbers down the lefthand edge of the list we desire are called ! the triangular numbers'' (now you know why!). The @cite{n}th ! triangular number is the sum of the integers from 1 to @cite{n}, and ! can be computed directly by the formula @c{n (n+1) \over 2} ! @cite{n * (n+1) / 2}. @smallexample @group --- 7500,7510 ---- @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 @tmath{n (n+1) \over 2}. ! @infoline @expr{n * (n+1) / 2}. @smallexample @group *************** *** 7446,7452 **** @subsection List Tutorial Exercise 8 @noindent ! The first step is to build a list of values of @cite{x}. @smallexample @group --- 7558,7564 ---- @subsection List Tutorial Exercise 8 @noindent ! The first step is to build a list of values of @expr{x}. @smallexample @group *************** *** 7486,7497 **** @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 @c{\sin x} ! @cite{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 @cite{x}. Now it is a simple matter to convert this back into the corresponding value itself. @smallexample --- 7598,7610 ---- @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 @tmath{\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 *************** *** 7504,7515 **** @end group @end smallexample ! If @kbd{a =} had produced more than one @cite{1} value, this method ! would have given the sum of all maximum @cite{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 @cite{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} --- 7617,7628 ---- @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} *************** *** 7526,7532 **** @end smallexample @noindent ! The output from @kbd{a X} is a vector containing the value of @cite{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. --- 7639,7645 ---- @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. *************** *** 7656,7665 **** @subsection List Tutorial Exercise 10 @noindent ! For the list @cite{[a, b, c, d]}, the result is @cite{((a = b) = c) = d}, ! which will compare @cite{a} and @cite{b} to produce a 1 or 0, which is ! then compared with @cite{c} to produce another 1 or 0, which is then ! compared with @cite{d}. This is not at all what Joe wanted. Here's a more correct method: --- 7769,7778 ---- @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: *************** *** 7687,7695 **** @subsection List Tutorial Exercise 11 @noindent ! The circle of unit radius consists of those points @cite{(x,y)} for which ! @cite{x^2 + y^2 < 1}. We start by generating a vector of @cite{x^2} ! and a vector of @cite{y^2}. We can make this go a bit faster by using the @kbd{v .} and @kbd{t .} commands. --- 7800,7808 ---- @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. *************** *** 7715,7721 **** @end group @end smallexample ! Now we sum the @cite{x^2} and @cite{y^2} values, compare with 1 to get a vector of 1/0 truth values, then sum the truth values. @smallexample --- 7828,7834 ---- @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 *************** *** 7728,7735 **** @end smallexample @noindent ! The ratio @cite{84/100} should approximate the ratio @c{\pi/4} ! @cite{pi/4}. @smallexample @group --- 7841,7847 ---- @end smallexample @noindent ! The ratio @expr{84/100} should approximate the ratio @cpiover{4}. @smallexample @group *************** *** 7757,7786 **** @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 ! @cite{y} axis can be ignored altogether. Just pick a random @cite{x} ! component for one end of the match, pick a random direction @c{\theta} ! @cite{theta}, ! and see if @cite{x} and @c{x + \cos \theta} ! @cite{x + cos(theta)} (which is the @cite{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 @cite{x}. Then @cite{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 @c{\pi/2} ! @cite{pi/2} radians while trying ! to estimate @c{\pi} ! @cite{pi}!) 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 @cite{x} and @c{\theta} ! @cite{theta}, compute @c{x + \cos \theta} ! @cite{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 .} --- 7869,7901 ---- @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 @tmath{\theta}, ! @infoline @expr{theta}, ! and see if @expr{x} and ! @texline @tmath{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 @tmath{\theta}, ! @infoline @expr{theta}, ! compute ! @texline @tmath{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 .} *************** *** 7881,7888 **** 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 @cite{[a, b, c, d]}, then the hash code is ! @cite{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 --- 7996,8003 ---- 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 *************** *** 7944,7950 **** @end smallexample Why does this work? Think about a two-step computation: ! @address@hidden (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, --- 8059,8065 ---- @end smallexample Why does this work? Think about a two-step computation: ! @address@hidden (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, *************** *** 7961,7967 **** @end tex @noindent ! for some suitable integers @cite{m} and @cite{n}. Expanding out by the distributive law yields @ifinfo --- 8076,8082 ---- @end tex @noindent ! for some suitable integers @expr{m} and @expr{n}. Expanding out by the distributive law yields @ifinfo *************** *** 7977,7986 **** @end tex @noindent ! The @cite{m} term in the latter formula is redundant because any ! contribution it makes could just as easily be made by the @cite{n} term. So we can take it out to get an equivalent formula with ! @cite{n' = 3m + n}, @ifinfo @example --- 8092,8101 ---- @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 *************** *** 8006,8012 **** @subsection List Tutorial Exercise 14 We want to use @kbd{H V U} to nest a function which adds a random ! step to an @cite{(x,y)} coordinate. The function is a bit long, but otherwise the problem is quite straightforward. @smallexample --- 8121,8127 ---- @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 *************** *** 8024,8032 **** notation to keep the two @code{random} calls from being evaluated before nesting even begins. ! We now have a vector of @cite{[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, @cite{x} and @cite{y}, suitable for graphing. @smallexample @group --- 8139,8147 ---- 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 *************** *** 8038,8049 **** @end group @end smallexample ! Incidentally, because the @cite{x} and @cite{y} are completely independent in this case, we could have done two separate commands ! to create our @cite{x} and @cite{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 @cite{[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. --- 8153,8164 ---- @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. *************** *** 8071,8080 **** @subsection Types Tutorial Exercise 1 @noindent ! If the number is the square root of @c{\pi} ! @cite{pi} times a rational number, ! then its square, divided by @c{\pi} ! @cite{pi}, should be a rational number. @smallexample @group --- 8186,8193 ---- @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 *************** *** 8106,8113 **** @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 ! @c{\sqrt{27 \pi / 53}} ! @cite{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 --- 8219,8226 ---- @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 @tmath{\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 *************** *** 8123,8139 **** @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 @cite{x} goes to infinity, @cite{e^x} also goes ! to infinity, but the fact the @cite{e^x} grows much faster than ! @cite{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 @cite{(0, 1)} ! represents the imaginary number @cite{i}. Here's a derivation: @samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}. ! The first part is, by definition, @cite{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 --- 8236,8252 ---- @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 *************** *** 8141,8151 **** 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 @cite{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 @cite{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. --- 8254,8264 ---- 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. *************** *** 8154,8162 **** @noindent We can make @samp{inf - inf} be any real number we like, say, ! @cite{a}, just by claiming that we added @cite{a} to the first infinity but not to the second. This is just as true for complex ! values of @cite{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}). --- 8267,8275 ---- @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}). *************** *** 8403,8409 **** @subsection Types Tutorial Exercise 10 @noindent ! Testing the first number, we might arbitrarily choose 17 for @cite{x}. @smallexample @group --- 8516,8522 ---- @subsection Types Tutorial Exercise 10 @noindent ! Testing the first number, we might arbitrarily choose 17 for @expr{x}. @smallexample @group *************** *** 8435,8441 **** @end smallexample @noindent ! The result is three ones (modulo @cite{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{$$^(-1) % } --- 8548,8554 ---- @end smallexample @noindent ! The result is three ones (modulo @expr{n}), so it's very probable that 15485863 is prime. (In fact, this number is the millionth prime.) Note that the functions @samp{($$^($-1)) mod } or @samp{$$^(-1) % } *************** *** 8604,8623 **** @noindent @c [fix-ref Declarations] ! The result @samp{sqrt(x)^2} is simplified back to @cite{x} by the Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens ! if @address@hidden = -4}}.) If @cite{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 @cite{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 @cite{[a, b, c]}. We want a polynomial which ! is zero when @cite{x} is any of these values. The trivial polynomial ! @cite{x-a} is zero when @cite{x=a}, so the product @cite{(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. --- 8717,8736 ---- @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 @address@hidden = -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. *************** *** 8808,8814 **** @end smallexample @noindent ! Multiplying by the conjugate helps because @cite{(a+b) (a-b) = a^2 - b^2}. @smallexample @group --- 8921,8927 ---- @end smallexample @noindent ! Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}. @smallexample @group *************** *** 8858,8868 **** 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 @cite{x} and @cite{y} are two consecutive Fibonacci numbers, ! then @cite{y} and @cite{x+y} are the next (overlapping) pair of Fibonacci numbers. ! Notice that because the number @cite{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 --- 8971,8981 ---- 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 *************** *** 8959,8966 **** @noindent Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}. ! Notice that the @cite{n > 1} guard is no longer necessary on the last ! rule since the @cite{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. --- 9072,9079 ---- @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. *************** *** 8978,8985 **** @starindex @end ignore @tindex nterms ! If @cite{x} is the sum @cite{a + b}, then address@hidden(address@hidden@t{)}' must ! be address@hidden(address@hidden@t{)}' plus address@hidden(address@hidden@t{)}'. If @cite{x} is not a sum, then address@hidden(address@hidden@t{)}' = 1. @smallexample --- 9091,9098 ---- @starindex @end ignore @tindex nterms ! If @expr{x} is the sum @expr{a + b}, then address@hidden(address@hidden@t{)}' must ! be address@hidden(address@hidden@t{)}' plus address@hidden(address@hidden@t{)}'. If @expr{x} is not a sum, then address@hidden(address@hidden@t{)}' = 1. @smallexample *************** *** 9026,9032 **** Perhaps more surprisingly, this rule still works with infinite mode turned on. Calc tries @code{EvalRules} before any built-in rules for a function. This allows you to override the default behavior of any ! Calc feature: Even though Calc now wants to evaluate @cite{0^0} to @code{nan}, your rule gets there first and evaluates it to 1 instead. Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}. --- 9139,9145 ---- Perhaps more surprisingly, this rule still works with infinite mode turned on. Calc tries @code{EvalRules} before any built-in rules for a function. This allows you to override the default behavior of any ! Calc feature: Even though Calc now wants to evaluate @expr{0^0} to @code{nan}, your rule gets there first and evaluates it to 1 instead. Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}. *************** *** 9085,9091 **** 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 @cite{x^0}, @cite{x^1}, @cite{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 --- 9198,9204 ---- 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 *************** *** 9115,9121 **** 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 @cite{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}. --- 9228,9234 ---- 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}. *************** *** 9146,9153 **** algebraic entry, whichever way you prefer: @noindent ! Computing @c{\displaystyle{\sin x \over x}} ! @cite{sin(x) / x}: Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}. --- 9259,9267 ---- algebraic entry, whichever way you prefer: @noindent ! Computing ! @texline @tmath{\displaystyle{\sin x \over x}}: ! @infoline @expr{sin(x) / x}: Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}. *************** *** 9207,9214 **** @end example @noindent ! Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @cite{n+1} ! and @cite{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 ) --- 9321,9328 ---- @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 ) *************** *** 9216,9223 **** @noindent This program is quite efficient because Calc knows how to raise a ! matrix (or other value) to the power @cite{n} in only @c{\log_2 n} ! @cite{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 --- 9330,9338 ---- @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 @tmath{\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 *************** *** 9268,9276 **** @subsection Programming Tutorial Exercise 8 @noindent ! The first step is to compute the derivative @cite{f'(x)} and thus ! the formula @c{\displaystyle{x - {f(x) \over f'(x)}}} ! @cite{x - f(x)/f'(x)}. (Because this definition is long, it will be repeated in concise form below. You can use @address@hidden m}} to load it from there. While you are --- 9383,9392 ---- @subsection Programming Tutorial Exercise 8 @noindent ! The first step is to compute the derivative @expr{f'(x)} and thus ! the formula ! @texline @tmath{\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 @address@hidden m}} to load it from there. While you are *************** *** 9317,9323 **** @end group @end smallexample ! This is the new guess for @cite{x}. Now we compare it with the old one to see if we've converged. @smallexample --- 9433,9439 ---- @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 *************** *** 9383,9395 **** @subsection Programming Tutorial Exercise 9 @noindent ! The first step is to adjust @cite{z} to be greater than 5. A simple ! for'' loop will do the job here. If @cite{z} is less than 5, we ! reduce the problem using @c{\psi(z) = \psi(z+1) - 1/z} ! @cite{psi(z) = psi(z+1) - 1/z}. We go ! on to compute @c{\psi(z+1)} ! @cite{psi(z+1)}, and remember to add back a factor of ! @cite{-1/z} when we're done. This step is repeated until @cite{z > 5}. (Because this definition is long, it will be repeated in concise form below. You can use @address@hidden m}} to load it from there. While you are --- 9499,9514 ---- @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 @tmath{\psi(z) = \psi(z+1) - 1/z}. ! @infoline @expr{psi(z) = psi(z+1) - 1/z}. We go ! on to compute ! @texline @tmath{\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 @address@hidden m}} to load it from there. While you are *************** *** 9407,9414 **** @end group @end smallexample ! Here, variable 1 holds @cite{z} and variable 2 holds the adjustment ! factor. If @cite{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, --- 9526,9533 ---- @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, *************** *** 9426,9433 **** @end group @end smallexample ! Now we compute the initial part of the sum: @c{\ln z - {1 \over 2z}} ! @cite{ln(z) - 1/2z} minus the adjustment factor. @smallexample --- 9545,9553 ---- @end group @end smallexample ! Now we compute the initial part of the sum: ! @texline @tmath{\ln z - {1 \over 2z}} ! @infoline @expr{ln(z) - 1/2z} minus the adjustment factor. @smallexample *************** *** 9441,9447 **** @end smallexample Now we evaluate the series. We'll use another for'' loop counting ! up the value of @cite{2 n}. (Calc does have a summation command, @kbd{a +}, but we'll use loops just to get more practice with them.) @smallexample --- 9561,9567 ---- @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 *************** *** 9468,9476 **** @end group @end smallexample ! This is the value of @c{-\gamma} ! @cite{- gamma}, with a slight bit of roundoff error. ! To get a full 12 digits, let's use a higher precision: @smallexample @group --- 9588,9598 ---- @end group @end smallexample ! This is the value of ! @texline @tmath{-\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 *************** *** 9500,9511 **** @subsection Programming Tutorial Exercise 10 @noindent ! Taking the derivative of a term of the form @cite{x^n} will produce ! a term like @c{n x^{n-1}} ! @cite{n x^(n-1)}. Taking the derivative of a constant ! produces zero. From this it is easy to see that the @cite{n}th ! derivative of a polynomial, evaluated at @cite{x = 0}, will equal the ! coefficient on the @cite{x^n} term times @cite{n!}. (Because this definition is long, it will be repeated in concise form below. You can use @address@hidden m}} to load it from there. While you are --- 9622,9635 ---- @subsection Programming Tutorial Exercise 10 @noindent ! Taking the derivative of a term of the form @expr{x^n} will produce ! a term like ! @texline @tmath{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 @address@hidden m}} to load it from there. While you are *************** *** 9554,9560 **** @end smallexample To convert back, a simple method is just to map the coefficients ! against a table of powers of @cite{x}. @smallexample @group --- 9678,9684 ---- @end smallexample To convert back, a simple method is just to map the coefficients ! against a table of powers of @expr{x}. @smallexample @group *************** *** 9614,9620 **** 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 @cite{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{M-# m} to load it from there.) --- 9738,9744 ---- 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{M-# m} to load it from there.) *************** *** 9783,9789 **** 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 address@hidden @kindex M-# c @kindex M-# M-# --- 9907,9913 ---- 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 M-# c @kindex M-# M-# *************** *** 9805,9811 **** 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 @address@hidden @cindex Extensions module @cindex @file{calc-ext} module --- 9929,9935 ---- 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 *************** *** 9818,9831 **** 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{M-# L} (@code{calc-load-everything}) ! command. @xref{Mode address@hidden If you type @kbd{M-x calc} or @kbd{M-# 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 address@hidden @kindex M-# b @pindex full-calc --- 9942,9955 ---- 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{M-# L} (@code{calc-load-everything}) ! command. @xref{Mode Settings}. If you type @kbd{M-x calc} or @kbd{M-# 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 M-# b @pindex full-calc *************** *** 9872,9878 **** contents of the stack intact. Typing @kbd{M-# c} or @kbd{M-# M-#} again from inside the Calculator buffer is equivalent to executing @code{calc-quit}; you can think of @kbd{M-# M-#} as toggling the ! Calculator on and address@hidden @kindex M-# x The @kbd{M-# x} command also turns the Calculator off, no matter which --- 9996,10002 ---- contents of the stack intact. Typing @kbd{M-# c} or @kbd{M-# M-#} again from inside the Calculator buffer is equivalent to executing @code{calc-quit}; you can think of @kbd{M-# M-#} as toggling the ! Calculator on and off. @kindex M-# x The @kbd{M-# x} command also turns the Calculator off, no matter which *************** *** 9904,9910 **** @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.)@refill @kindex @{ @kindex @} --- 10028,10034 ---- @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 @} *************** *** 9913,9919 **** @cindex Vertical scrolling The @address@hidden and @address@hidden 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 address@hidden @kindex M-# 0 @pindex calc-reset --- 10037,10043 ---- @cindex Vertical scrolling The @address@hidden and @address@hidden 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 M-# 0 @pindex calc-reset *************** *** 10017,10023 **** @noindent which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for} ! takes a value @cite{a} from the stack, prompts for a value @cite{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. --- 10141,10147 ---- @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. *************** *** 10074,10080 **** @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 (@i{-2})address@hidden Note that the top'' of the stack actually appears at the @emph{bottom} of the buffer. A line containing a single @samp{.} character signifies --- 10198,10204 ---- @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 (@i{-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 *************** *** 10103,10114 **** (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. @address@hidden *} is thus a handy way to square a address@hidden 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 ! address@hidden @node Numeric Entry, Algebraic Entry, Stack Basics, Introduction @section Numeric Entry --- 10227,10238 ---- (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. @address@hidden *} 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 *************** *** 10129,10136 **** @cindex Negative numbers, entering @kindex _ There are three different concepts corresponding to the word minus,'' ! typified by @cite{a-b} (subtraction), @cite{-x} ! (change-sign), and @cite{-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 --- 10253,10260 ---- @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 *************** *** 10138,10144 **** 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 @i{-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 _ @address@hidden Some other keys are active during numeric entry, such as @kbd{#} for non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms. --- 10262,10268 ---- 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 @i{-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. *************** *** 10158,10168 **** Calculations can also be entered in algebraic form. This is accomplished by typing the apostrophe key, @kbd{'}, followed by the expression in standard format: @address@hidden'} 2+3*4 @key{RET}} computes ! @c{2+(3\times4) = 14} ! @cite{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 address@hidden 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 --- 10282,10293 ---- Calculations can also be entered in algebraic form. This is accomplished by typing the apostrophe key, @kbd{'}, followed by the expression in standard format: @address@hidden'} 2+3*4 @key{RET}} computes ! @texline @tmath{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 *************** *** 10185,10191 **** 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 @address@hidden @cindex Incomplete algebraic mode If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a} --- 10310,10316 ---- 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} *************** *** 10221,10227 **** 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 address@hidden Higher stack elements can be accessed from an entered formula with the symbols @kbd{$$}, @kbd{$$}, and so on. The number of stack elements --- 10346,10352 ---- stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{*2 @key{RET}} replaces it with 6. Note that the @kbd{} key always initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{} is the ! first character in the new formula. Higher stack elements can be accessed from an entered formula with the symbols @kbd{$$}, @kbd{$$}, and so on. The number of stack elements *************** *** 10229,10235 **** signs in the longest such symbol in the formula. For example, @samp{$$+$$} adds the second and third stack elements, replacing the top three elements with the answer. (All information about the top stack element is thus lost ! since no single @samp{} appears in this formula.)@refill A slightly different way to refer to stack elements is with a dollar sign followed by a number: @samp{1}, @samp{2}, and so on are much --- 10354,10360 ---- signs in the longest such symbol in the formula. For example, @samp{$$+$$} 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 *************** *** 10254,10260 **** 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 @cite{1+2}; you might then press @kbd{=} when it is time to evaluate this formula. @node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction --- 10379,10385 ---- 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 *************** *** 10333,10339 **** @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 address@hidden As a rule, stack-manipulation commands accept a numeric prefix argument which is interpreted as an index into the stack. A positive argument --- 10458,10464 ---- @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 *************** *** 10360,10366 **** 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) address@hidden @kindex ~ @pindex calc-num-prefix --- 10485,10491 ---- 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 *************** *** 10368,10374 **** 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 address@hidden Conversely, if you have typed a numeric prefix argument the @kbd{~} key pushes it onto the stack in the form of an integer. --- 10493,10499 ---- 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. *************** *** 10416,10422 **** The @address@hidden 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 @cite{n}th most recent command which removed items from the stack; it pushes those items back onto the stack. --- 10541,10547 ---- The @address@hidden 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. *************** *** 10438,10444 **** @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 @cite{1 / 0}; @address@hidden L}} pushes the formula @samp{ln(0)}. Floating-point overflow and underflow are also reasons for this to happen. --- 10563,10569 ---- @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}; @address@hidden L}} pushes the formula @samp{ln(0)}. Floating-point overflow and underflow are also reasons for this to happen. *************** *** 10543,10549 **** 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.)@refill 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 --- 10668,10674 ---- 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 *************** *** 10561,10576 **** @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 @c{\pi} ! @cite{pi} to about 20 decimal places; if the current precision ! is greater than this, it will recompute @c{\pi} ! @cite{pi} 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 ! @c{\pi \over 4} ! @cite{pi/4} and @c{\ln 2} ! @cite{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 --- 10686,10700 ---- @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 @tmath{\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 *************** *** 10643,10649 **** 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.)@refill Integers, fractions, and floats are various ways of describing real numbers. HMS forms also for many purposes act as real numbers. These --- 10767,10773 ---- 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 *************** *** 10686,10692 **** 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 address@hidden @kindex # A non-decimal integer is represented as an optional sign, a radix --- 10810,10816 ---- 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 *************** *** 10695,10701 **** 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 address@hidden @node Fractions, Floats, Integers, Data Types @section Fractions --- 10819,10825 ---- 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 *************** *** 10708,10722 **** @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 address@hidden 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 address@hidden Non-decimal fractions are entered and displayed as @address@hidden@var{num}:@var{denom}} (or in the analogous three-part ! form). The numerator and denominator always use the same address@hidden @node Floats, Complex Numbers, Fractions, Data Types @section Floats --- 10832,10846 ---- @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 @address@hidden@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 *************** *** 10726,10737 **** 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 @c{10^{-3999999}} ! @cite{10^-3999999} (inclusive) ! to @c{10^{4000000}} ! @cite{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 --- 10850,10862 ---- 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 @tmath{10^{-3999999}} ! @infoline @expr{10^-3999999} ! (inclusive) to ! @texline @tmath{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 *************** *** 10772,10778 **** 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 ! address@hidden While floats are always @emph{stored} in decimal, they can be entered and displayed in any radix just like integers and fractions. The --- 10897,10903 ---- 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. The *************** *** 10798,10813 **** @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 @address@hidden@var{b}i} ! notation; @pxref{Complex address@hidden ! Polar complex numbers are displayed in the form address@hidden(address@hidden@t{;address@hidden ! @address@hidden)}' ! where @var{r} is the nonnegative magnitude and @c{\theta} ! @var{theta} is the argument ! or phase angle. The range of @c{\theta} ! @var{theta} depends on the current angular ! mode (@pxref{Angular Modes}); it is generally between @i{-180} and ! @i{+180} degrees or the equivalent range in address@hidden Complex numbers are entered in stages using incomplete objects. @xref{Incomplete Objects}. --- 10923,10942 ---- @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 @address@hidden@var{b}i} ! notation; @pxref{Complex Formats}. ! Polar complex numbers are displayed in the form ! @texline address@hidden(address@hidden@t{;address@hidden@t{)}' ! @infoline address@hidden(address@hidden@t{;address@hidden@t{)}' ! where @var{r} is the nonnegative magnitude and ! @texline @tmath{\theta} ! @infoline @var{theta} ! is the argument or phase angle. The range of ! @texline @tmath{\theta} ! @infoline @var{theta} ! depends on the current angular mode (@pxref{Angular Modes}); it is ! generally between @i{-180} and @i{+180} degrees or the equivalent range ! in radians. Complex numbers are entered in stages using incomplete objects. @xref{Incomplete Objects}. *************** *** 10819,10826 **** type. @xref{Polar Mode}. A complex result in which the imaginary part is zero (or the phase angle ! is 0 or 180 degrees or @c{\pi} ! @cite{pi} radians) is automatically converted to a real number. @node Infinities, Vectors and Matrices, Complex Numbers, Data Types --- 10948,10954 ---- 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 *************** *** 10845,10868 **** 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 @cite{1 / x}, as @cite{x} becomes larger and larger, becomes arbitrarily close to zero. So you can imagine ! that if @cite{x} got all the way to infinity,'' then @cite{1 / x} would go all the way to zero. Similarly, when they say that ! @samp{exp(inf) = inf}, they mean that @c{e^x} ! @cite{exp(x)} grows without ! bound as @cite{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 @cite{1 / 0}. We ! really want the value that @cite{1 / x} approaches as @cite{x} ! approaches zero. But if all we have is @cite{1 / 0}, we can't ! tell which direction @cite{x} was coming from. If @cite{x} was positive and decreasing toward zero, then we should say that ! @samp{1 / 0 = inf}. But if @cite{x} was negative and increasing ! toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @cite{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 --- 10973,10997 ---- 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 @tmath{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 *************** *** 10870,10879 **** Calc actually has three modes that say how infinities are handled. Normally, infinities never arise from calculations that didn't ! already have them. Thus, @cite{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 ! @cite{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 --- 10999,11008 ---- 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 *************** *** 10892,10900 **** 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 @cite{x / x} (giving a result of one), ! or @cite{2 x / x} (giving two), or @cite{x^2 / x} (giving @code{inf}), ! or @cite{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 --- 11021,11029 ---- 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 *************** *** 10938,10952 **** @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 address@hidden @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 @c{n\times m} ! @address@hidden@var{m}} matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers from 1 to @samp{n}. --- 11067,11082 ---- @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 @tmath{n\times m} ! @infoline @address@hidden matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers from 1 to @samp{n}. *************** *** 11077,11083 **** (exclusive). A positive HMS form is interpreted as @var{hours} + @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted as @i{- @var{hours}} @i{-} @var{mins}/60 @i{-} @var{secs}/3600. ! Display format for HMS forms is quite flexible. @xref{HMS address@hidden HMS forms can be added and subtracted. When they are added to numbers, the numbers are interpreted according to the current angular mode. HMS --- 11207,11213 ---- (exclusive). A positive HMS form is interpreted as @var{hours} + @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted as @i{- @var{hours}} @i{-} @var{mins}/60 @i{-} @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 *************** *** 11216,11225 **** often arises in number theory. Modulo forms are written address@hidden @t{mod} @var{M}', where @var{a} and @var{M} are real numbers or HMS forms, and ! @c{0 \le a < M} ! @cite{0 <= a < @var{M}}. ! In many applications @cite{a} and @cite{M} will be ! integers but this is not address@hidden 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 --- 11346,11355 ---- often arises in number theory. Modulo forms are written address@hidden @t{mod} @var{M}', where @var{a} and @var{M} are real numbers or HMS forms, and ! @texline @tmath{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. 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 *************** *** 11228,11255 **** 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 @cite{M}'s are added or multiplied, the Calculator simply adds or multiplies the values, then reduces modulo ! @cite{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 @cite{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 address@hidden @t{mod} @var{M}' and address@hidden @t{mod} @var{M}' ! can be divided if @cite{a}, @cite{b}, and @cite{M} are all integers. The result is the modulo form which, when multiplied by address@hidden @t{mod} @var{M}', produces address@hidden @t{mod} @var{M}'. If there is no solution to this equation (which can happen only when ! @cite{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 @address@hidden(address@hidden @t{mod} @address@hidden)^.5}'} will compute a modulo square root'' ! in the sense of reducing @c{\sqrt a} ! @cite{sqrt(a)} modulo @cite{M}, this is not a ! useful definition from the number-theoretical point of view.)@refill @ignore @mindex M --- 11358,11387 ---- 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 address@hidden @t{mod} @var{M}' and address@hidden @t{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 address@hidden @t{mod} @var{M}', produces address@hidden @t{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 @address@hidden(address@hidden @t{mod} @address@hidden)^.5}'} will compute a modulo square root'' ! in the sense of reducing ! @texline @tmath{\sqrt a} ! @infoline @expr{sqrt(a)} ! modulo @expr{M}, this is not a useful definition from the ! number-theoretical point of view.) @ignore @mindex M *************** *** 11261,11277 **** @tindex mod (operator) To create a modulo form during numeric entry, press the address@hidden key to enter the word @samp{mod}. As a special convenience, pressing ! address@hidden a second time automatically enters the value of @cite{M} that was most recently used before. During algebraic entry, either type @samp{mod} by hand or press @kbd{M-m} (that's @address@hidden). ! Once again, pressing this a second time enters the current address@hidden You can also use @kbd{v p} and @kbd{%} to modify modulo forms. @xref{Building Vectors}. @xref{Basic Arithmetic}. 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 @cite{M} also be an HMS form eliminates troubles that would arise if the angular mode were inadvertently set to Radians, in which case @address@hidden@@ 0' 0" mod 24}} would be interpreted as two degrees modulo --- 11393,11409 ---- @tindex mod (operator) To create a modulo form during numeric entry, press the address@hidden key to enter the word @samp{mod}. As a special convenience, pressing ! address@hidden 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 @address@hidden). ! Once again, pressing this a second time enters the current modulo. You can also use @kbd{v p} and @kbd{%} to modify modulo forms. @xref{Building Vectors}. @xref{Basic Arithmetic}. 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 @address@hidden@@ 0' 0" mod 24}} would be interpreted as two degrees modulo *************** *** 11296,11319 **** @cindex Standard deviations An @dfn{error form} is a number with an associated standard deviation, as in @samp{2.3 +/- 0.12}. The notation ! address@hidden @t{+/-} @c{\sigma} ! @asis{sigma}' stands for an uncertain value which follows a normal or ! Gaussian distribution of mean @cite{x} and standard deviation or ! error'' @c{\sigma} ! @cite{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 address@hidden 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 @cite{f(x)} (such as @c{\sin x} ! @cite{sin(x)}) ! is defined by the error of @cite{x} times the derivative of @cite{f} ! evaluated at the mean value of @cite{x}. For a two-argument function ! @cite{f(x,y)} (such as addition) the error is the square root of the sum ! of the squares of the errors due to @cite{x} and @cite{y}. @tex$$ \eqalign{ f(x \hbox{\code{ +/- }} \sigma) --- 11428,11455 ---- @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 address@hidden @t{+/-} @tmath{\sigma}' ! @infoline address@hidden @t{+/-} sigma' ! stands for an uncertain value which follows ! a normal or Gaussian distribution of mean @expr{x} and standard ! deviation or error'' ! @texline @tmath{\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 @tmath{\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) *************** *** 11327,11364 **** } @end tex Note that this ! definition assumes the errors in @cite{x} and @cite{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 address@hidden 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 ! address@hidden(address@hidden @t{+/-} @c{\sigma$} ! @address@hidden)}' is ! address@hidden ! @var{sigma} @t{abs(cos(address@hidden@t{))}'. When @cite{x} is close to zero, ! @c{$\cos x$} ! @cite{cos(x)} is ! close to one so the error in the sine is close to @c{$\sigma$} ! @cite{sigma}; this makes sense, since @c{$\sin x$} ! @cite{sin(x)} is approximately @cite{x} near zero, so a given ! error in @cite{x} will produce about the same error in the sine. Likewise, ! near 90 degrees @c{$\cos x$} ! @cite{cos(x)} is nearly zero and so the computed error is ! small: The sine curve is nearly flat in that region, so an error in @cite{x} ! has relatively little effect on the value of @c{$\sin x$} ! @cite{sin(x)}. However, consider ! @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so Calc will report ! zero error! We get an obviously wrong result because we have violated ! the small-error approximation underlying the error analysis. If the error ! in @cite{x} had been small, the error in @c{$\sin x} ! @cite{sin(x)} would indeed have been address@hidden @ignore @mindex p --- 11463,11510 ---- } @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 address@hidden(address@hidden @t{+/-} @address@hidden)}' ! @infoline address@hidden(address@hidden @t{+/-} @address@hidden)}' ! is ! @texline address@hidden @t{abs(cos(address@hidden@t{))}'. ! @infoline address@hidden @t{abs(cos(address@hidden@t{))}'. ! When @expr{x} is close to zero, ! @texline @tmath{\cos x} ! @infoline @expr{cos(x)} ! is close to one so the error in the sine is close to ! @texline @tmath{\sigma}; ! @infoline @expr{sigma}; ! this makes sense, since ! @texline @tmath{\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 @tmath{\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 @tmath{\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 @tmath{\sin x} ! @infoline @expr{sin(x)} ! would indeed have been negligible. @ignore @mindex p *************** *** 11411,11420 **** @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 @cite{2 <= x <= 4}, whereas ! @samp{[2 ..@: 4)} represents @cite{2 <= x < 4}, ! @samp{(2 ..@: 4]} represents @cite{2 < x <= 4}, and ! @samp{(2 ..@: 4)} represents @cite{2 < x < address@hidden @end ifinfo @tex Calc supports several varieties of intervals, including \dfn{closed} --- 11557,11566 ---- @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} *************** *** 11464,11477 **** While it may seem that intervals and error forms are similar, they are based on entirely different concepts of inexact quantities. An error ! form address@hidden @t{+/-} @c{\sigma} ! @var{sigma}' means a variable is random, and its value could ! be anything but is probably'' within one @c{\sigma} ! @var{sigma} of the mean value @cite{x}. ! An interval address@hidden@var{a} @t{..@:} @address@hidden' 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 address@hidden Intervals may not contain complex numbers, but they may contain HMS forms or date forms. --- 11610,11628 ---- 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 address@hidden @t{+/-} @tmath{\sigma}' ! @infoline address@hidden @t{+/-} @var{sigma}' ! means a variable is random, and its value could ! be anything but is probably'' within one ! @texline @tmath{\sigma} ! @infoline @var{sigma} ! of the mean value @expr{x}. An interval ! address@hidden@var{a} @t{..@:} @address@hidden' 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. *************** *** 11537,11543 **** 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 @address@hidden As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or @kbd{,} adds a zero or duplicates the preceding value in the list being --- 11688,11694 ---- 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 *************** *** 11575,11588 **** The Calc variable @code{foo} corresponds to the Emacs Lisp variable @code{var-foo}. Commands like @kbd{s s} (@code{calc-store}) that operate on variables can be made to use any arbitrary Lisp variable simply by ! backspacing over the @samp{var-} prefix in the address@hidden 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{var-q0} through @code{var-q9}. For example, @kbd{3 s s 2} stores the number 3 in variable @code{var-q2}, and @address@hidden s s foo @key{RET}}} stores that number in variable ! @address@hidden 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 --- 11726,11739 ---- The Calc variable @code{foo} corresponds to the Emacs Lisp variable @code{var-foo}. Commands like @kbd{s s} (@code{calc-store}) that operate on variables can be made to use any arbitrary Lisp variable simply by ! backspacing over the @samp{var-} prefix in the minibuffer. 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{var-q0} through @code{var-q9}. For example, @kbd{3 s s 2} stores the number 3 in variable @code{var-q2}, and @address@hidden s s foo @key{RET}}} stores that number in variable ! @code{var-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 *************** *** 11623,11629 **** (@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 @address@hidden The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for infinite or indeterminate values. It's best not to use them as --- 11774,11780 ---- (@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 *************** *** 11697,11704 **** @samp{=>} address@hidden Note that, unlike in usual computer notation, multiplication binds more ! strongly than division: @samp{a*b/c*d} is equivalent to @c{a b \over c d} ! @cite{(a*b)/(c*d)}. @cindex Multiplication, implicit @cindex Implicit multiplication --- 11848,11856 ---- @samp{=>} address@hidden Note that, unlike in usual computer notation, multiplication binds more ! strongly than division: @samp{a*b/c*d} is equivalent to ! @texline @tmath{a b \over c d}. ! @infoline @expr{(a*b)/(c*d)}. @cindex Multiplication, implicit @cindex Implicit multiplication *************** *** 11712,11718 **** 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 address@hidden @cindex Implicit comma in vectors The rules are slightly different for vectors written with square brackets. --- 11864,11870 ---- 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. *************** *** 11724,11730 **** 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 address@hidden 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 --- 11876,11882 ---- 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 *************** *** 11753,11764 **** 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 address@hidden In the documentation for particular commands, the notation @kbd{H S} (@code{calc-sinh}) address@hidden 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 address@hidden Commands that interpret (parse'') text as algebraic formulas include algebraic entry (@kbd{'}), editing commands like @kbd{} which parse --- 11905,11916 ---- 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}) address@hidden 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 *************** *** 11823,11829 **** @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 address@hidden @kindex @key{LFD} @pindex calc-over --- 11975,11981 ---- @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 *************** *** 11833,11839 **** 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 address@hidden @kindex @key{DEL} @kindex C-d --- 11985,11991 ---- 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 *************** *** 11851,11857 **** @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 address@hidden @kindex address@hidden @pindex calc-pop-above --- 12003,12009 ---- @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 address@hidden @pindex calc-pop-above *************** *** 11874,11880 **** @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 address@hidden @kindex address@hidden @pindex calc-roll-up --- 12026,12032 ---- @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 address@hidden @pindex calc-roll-up *************** *** 11885,11891 **** @address@hidden creates @samp{10 20 40 50 30}, @kbd{C-u 4 address@hidden creates @samp{10 30 40 50 20}, @kbd{C-u - 2 address@hidden creates @samp{30 40 50 10 20}, and ! @kbd{C-u 0 address@hidden creates @samp{50 40 30 20 address@hidden A good way to view the operation of @key{TAB} and @address@hidden is in terms of moving a particular element to a new position in the stack. --- 12037,12043 ---- @address@hidden creates @samp{10 20 40 50 30}, @kbd{C-u 4 address@hidden creates @samp{10 30 40 50 20}, @kbd{C-u - 2 address@hidden creates @samp{30 40 50 10 20}, and ! @kbd{C-u 0 address@hidden creates @samp{50 40 30 20 10}. A good way to view the operation of @key{TAB} and @address@hidden is in terms of moving a particular element to a new position in the stack. *************** *** 11988,11994 **** 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 address@hidden @cindex Trail pointer There is a @dfn{trail pointer} which selects some entry of the trail at --- 12140,12146 ---- 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 *************** *** 12011,12017 **** @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 address@hidden @kindex t n @pindex calc-trail-next --- 12163,12169 ---- @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 *************** *** 12026,12032 **** 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 address@hidden @kindex t [ @pindex calc-trail-first --- 12178,12184 ---- 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 *************** *** 12038,12044 **** (@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 address@hidden @kindex t s @pindex calc-trail-isearch-forward --- 12190,12196 ---- (@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 *************** *** 12050,12056 **** 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 address@hidden @end ifinfo @tex The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r} --- 12202,12208 ---- 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} *************** *** 12248,12254 **** 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 address@hidden @cindex Accuracy of calculations It is important to distinguish the concepts of @dfn{precision} and --- 12400,12406 ---- 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 *************** *** 12291,12297 **** 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 address@hidden @kindex H @pindex calc-hyperbolic --- 12443,12449 ---- 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 *************** *** 12300,12306 **** 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 address@hidden, logarithm.)@refill 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 --- 12452,12458 ---- 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 address@hidden, 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 *************** *** 12356,12363 **** 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 @c{\pi} ! @cite{pi}.) @kindex m r @pindex calc-radians-mode --- 12508,12514 ---- 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 *************** *** 12368,12374 **** 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 address@hidden @node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes @subsection Polar Mode --- 12519,12525 ---- 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 *************** *** 12396,12411 **** 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 multiply fractions instead: @kbd{6 @key{RET} 1:4 *} produces @cite{3:2} ! even though @kbd{6 @key{RET} 4 /} produces @cite{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, @cite{8/4} produces @cite{2} in either mode, ! but @cite{6/4} produces @cite{3:2} in Fraction Mode, @cite{1.5} in ! Float address@hidden 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 --- 12547,12562 ---- 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 multiply fractions instead: @kbd{6 @key{RET} 1:4 *} 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 *************** *** 12416,12422 **** @noindent @cindex Infinite mode ! The Calculator normally treats results like @cite{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. --- 12567,12573 ---- @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. *************** *** 12432,12438 **** 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, @cite{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, --- 12583,12589 ---- 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, *************** *** 12477,12483 **** @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 address@hidden 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 --- 12628,12634 ---- @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 *************** *** 12504,12510 **** @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., @cite{a - a} and @cite{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 @address@hidden --- 12655,12661 ---- @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 @address@hidden *************** *** 12598,12604 **** 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 address@hidden While it may seem that the working'' messages will slow Calc down considerably, experiments have shown that their impact is actually --- 12749,12755 ---- 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 *************** *** 12613,12625 **** 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 @cite{a+a} (or @cite{2+3}), are done by default but can be turned off when necessary. ! When you press a key like @kbd{+} when @cite{2} and @cite{3} are on the stack, Calc pops these numbers, normalizes them, creates the formula ! @cite{2+3}, normalizes it, and pushes the result. Of course the standard ! rules for normalizing @cite{2+3} will produce the result @cite{5}. Simplification mode commands consist of the lower-case @kbd{m} prefix key followed by a shifted letter. --- 12764,12776 ---- 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. *************** *** 12627,12633 **** @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 @cite{2+3} alone. In fact, nothing except simple numbers are ever affected by normalization in this mode. --- 12778,12784 ---- @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. *************** *** 12635,12656 **** @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, @cite{1+2} is simplified to @cite{3}, and @cite{a+(2-2)} is ! simplified to @cite{a+0} but no further, since one argument of the sum ! is not a constant. Unfortunately, @cite{(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 @cite{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 address@hidden @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 @cite{a+0} to @cite{a}, and ! @cite{a + 2 a} to @cite{3 a}, as well as evaluating functions like ! @address@hidden(x^2, x)} to @cite{2 x}. @kindex m B @pindex calc-bin-simplify-mode --- 12786,12809 ---- @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 ! @texline @address@hidden(x^2,x)} ! @infoline @address@hidden(x^2, x)} ! to @expr{2 x}. @kindex m B @pindex calc-bin-simplify-mode *************** *** 12681,12693 **** (@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 address@hidden 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 address@hidden @node Declarations, Display Modes, Simplification Modes, Mode Settings @section Declarations --- 12834,12846 ---- (@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 *************** *** 12949,12956 **** Calc does not check the declarations for a variable when you store a value in it. However, storing @i{-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 @cite{3.5} ! if it substitutes the value first, or to @cite{-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} --- 13102,13109 ---- Calc does not check the declarations for a variable when you store a value in it. However, storing @i{-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} *************** *** 13018,13024 **** @end ignore @tindex dimag The @code{dimag} function checks if its argument is imaginary, ! i.e., is mathematically equal to a real number times @cite{i}. @ignore @starindex --- 13171,13177 ---- @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 *************** *** 13036,13042 **** 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 @cite{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. --- 13189,13195 ---- 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. *************** *** 13105,13111 **** (@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 address@hidden 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 --- 13258,13264 ---- (@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 *************** *** 13165,13171 **** 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 address@hidden @kindex d r @pindex calc-radix --- 13318,13324 ---- 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 *************** *** 13180,13189 **** 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 @cite{w}, all integers ! are displayed with at least enough digits to represent @c{2^w-1} ! @cite{(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 --- 13333,13344 ---- 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 @tmath{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 *************** *** 13202,13211 **** The @kbd{d g} command toggles grouping on and off. With a numerix 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 @cite{N} it enables grouping on every @cite{N} digits. For floating-point numbers, grouping normally occurs only ! before the decimal point. A negative prefix argument @cite{-N} enables ! grouping every @cite{N} digits both before and after the decimal address@hidden @kindex d , @pindex calc-group-char --- 13357,13366 ---- The @kbd{d g} command toggles grouping on and off. With a numerix 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 *************** *** 13298,13304 **** 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 address@hidden @kindex d i @pindex calc-i-notation --- 13453,13459 ---- 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 *************** *** 13307,13313 **** 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 address@hidden @cindex @code{i} variable @vindex i --- 13462,13468 ---- 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 *************** *** 13318,13324 **** 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.''@refill @node Fraction Formats, HMS Formats, Complex Formats, Display Modes @subsection Fraction Formats --- 13473,13479 ---- 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 *************** *** 13346,13357 **** 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 @cite{1:6} will be unaffected, but @cite{2:3} will be ! displayed as @cite{4:6}, @cite{1:2} will be displayed as @cite{3:6}, ! and @cite{1:8} will be displayed as @cite{3:24}. Integers are also ! affected by this mode: 3 is displayed as @cite{18:6}. Note that the format @samp{:1} writes fractions the same as @samp{:}, but it writes ! integers as @cite{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 --- 13501,13512 ---- 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 *************** *** 13710,13724 **** 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 address@hidden 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 @address@hidden t}}. ! With a positive numeric prefix argument @cite{n}, @kbd{d t} hides the ! bottom @cite{n} values in the buffer. With a negative argument, it hides ! all but the top @cite{n} values. With an argument of zero, it hides zero ! values, i.e., moves the @samp{.} all the way down to the address@hidden @kindex d [ @pindex calc-truncate-up --- 13865,13879 ---- 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 @address@hidden 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 *************** *** 13726,13732 **** @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)address@hidden @node Justification, Labels, Truncating the Stack, Display Modes @subsection Justification --- 13881,13887 ---- @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 *************** *** 13743,13749 **** @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 ! address@hidden 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 --- 13898,13904 ---- @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 *************** *** 13933,13942 **** @end example @noindent ! can represent either the negative rational number @cite{-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 ! @cite{-3:4} or @cite{-0.75}, so this ambiguity is not a problem in typical use. Non-decimal numbers are displayed with subscripts. Thus there is no --- 14088,14097 ---- @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 *************** *** 14061,14067 **** Math formulas are usually enclosed by @samp{ } signs in @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)address@hidden Complex numbers are displayed as in @samp{3 + 4i}. Fractions and quotients are written using @code{\over}; --- 14216,14222 ---- Math formulas are usually enclosed by @samp{ } signs in @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}; *************** *** 14073,14079 **** The words @code{\left} and @code{\right} are ignored when reading formulas in @TeX{} mode. Both @code{inf} and @code{uinf} are written as @code{\infty}; when read, @code{\infty} always translates to ! @address@hidden Function calls are written the usual way, with the function name followed by the arguments in parentheses. However, functions for which @TeX{} has --- 14228,14234 ---- The words @code{\left} and @code{\right} are ignored when reading formulas in @TeX{} mode. 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{} has *************** *** 14081,14089 **** 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 ! @c{\sin{2 x}} ! @cite{sin 2x} but @c{\sin(2 + x)} ! @cite{sin(2 + x)}. Function and variable names not treated specially by @TeX{} are simply written out as-is, which will cause them to come out in italic letters --- 14236,14246 ---- 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 @tmath{\sin{2 x}} ! @infoline @expr{sin 2x} ! but ! @texline @tmath{\sin(2 + x)}. ! @infoline @expr{sin(2 + x)}. Function and variable names not treated specially by @TeX{} are simply written out as-is, which will cause them to come out in italic letters *************** *** 14244,14250 **** @end group @end example @tex - \let\rm\goodrm \sin\left( a^2 \over b_i \right) $$@end tex @sp 1 --- 14401,14406 ---- *************** *** 14282,14288 **** @end group @end example @tex ! \turnoffactive\let\rm\goodrm$$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$@end tex @sp 2 --- 14438,14444 ---- @end group @end example @tex ! \turnoffactive$$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$@end tex @sp 2 *************** *** 14300,14306 **** @end group @end example @tex - \let\rm\goodrm$$ [f(a), foo(bar), \sin{\pi}]  [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}]  [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$--- 14456,14461 ---- *************** *** 14453,14459 **** 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: @address@hidden @node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes @subsection Maple Language Mode --- 14608,14614 ---- 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 *************** *** 14594,14602 **** => 40 @end example ! The general rule is that if an operator with precedence @cite{n} ! occurs as an argument to an operator with precedence @cite{m}, then ! the argument is enclosed in parentheses if @cite{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). --- 14749,14757 ---- => 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). *************** *** 15579,15586 **** @item Matrix/scalar mode. Default value is @i{-1}. Value is 0 for scalar ! mode, @i{-2} for matrix mode, or @var{N} for @c{N\times N} ! @address@hidden matrix mode. Command is @kbd{m v}. @item Simplification mode. Default is 1. Value is @i{-1} for off (@kbd{m O}), --- 15734,15743 ---- @item Matrix/scalar mode. Default value is @i{-1}. Value is 0 for scalar ! mode, @i{-2} for matrix mode, or @var{N} for ! @texline @tmath{N\times N} ! @infoline @address@hidden ! matrix mode. Command is @kbd{m v}. @item Simplification mode. Default is 1. Value is @i{-1} for off (@kbd{m O}), *************** *** 15906,15914 **** 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 @cite{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 @cite{M}, of the two values. If both arguments of @kbd{+} are intervals, the result is an interval --- 16063,16071 ---- 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 *************** *** 15964,15982 **** @end ignore @tindex / The @kbd{/} (@code{calc-divide}) command divides two numbers. When ! dividing a scalar @cite{B} by a square matrix @cite{A}, the computation ! performed is @cite{B} times the inverse of @cite{A}. This also occurs ! if @cite{B} is itself a vector or matrix, in which case the effect is ! to solve the set of linear equations represented by @cite{B}. If @cite{B} ! is a matrix with the same number of rows as @cite{A}, or a plain vector (which is interpreted here as a column vector), then the equation ! @cite{A X = B} is solved for the vector or matrix @cite{X}. Otherwise, ! if @cite{B} is a non-square matrix with the same number of @emph{columns} ! as @cite{A}, the equation @cite{X A = B} is solved. If you wish a vector ! @cite{B} to be interpreted as a row vector to be solved as @cite{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 @cite{B}, transpose @cite{A} and ! @cite{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 --- 16121,16139 ---- @end ignore @tindex / The @kbd{/} (@code{calc-divide}) command divides two numbers. 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 *************** *** 16026,16035 **** @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 @cite{a} and @cite{b} (except @cite{b=0}). For ! positive @cite{b}, the result will always be between 0 (inclusive) and ! @cite{b} (exclusive). Modulo does not work for HMS forms and error forms. ! If @cite{a} is a modulo form, its modulo is changed to @cite{b}, which must be positive real number. @kindex : --- 16183,16192 ---- @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 : *************** *** 16083,16089 **** @tindex inv @cindex Reciprocal The @kbd{&} (@code{calc-inv}) address@hidden command computes the ! reciprocal of a number, i.e., @cite{1 / x}. Operating on a square matrix, it computes the inverse of that matrix. @kindex Q --- 16240,16246 ---- @tindex inv @cindex Reciprocal The @kbd{&} (@code{calc-inv}) address@hidden 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 *************** *** 16098,16105 **** @tindex hypot The @kbd{f h} (@code{calc-hypot}) address@hidden 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 @cite{a} ! and @cite{b}. If the arguments are complex numbers, their squared magnitudes are used. @kindex f Q --- 16255,16262 ---- @tindex hypot The @kbd{f h} (@code{calc-hypot}) address@hidden 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 *************** *** 16124,16130 **** 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.)@refill @kindex f M @kindex f X --- 16281,16287 ---- 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 *************** *** 16133,16149 **** @pindex calc-xpon-part @tindex xpon The @kbd{f M} (@code{calc-mant-part}) address@hidden function extracts ! the mantissa'' part @cite{m} of its floating-point argument; @kbd{f X} (@code{calc-xpon-part}) address@hidden extracts the exponent'' part ! @cite{e}. The original number is equal to @c{m \times 10^e} ! @cite{m * 10^e}, ! where @cite{m} is in the interval @samp{[1.0 ..@: 10.0)} except that ! @cite{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 @cite{m=e=0} case)address@hidden @kindex f S @pindex calc-scale-float --- 16290,16307 ---- @pindex calc-xpon-part @tindex xpon The @kbd{f M} (@code{calc-mant-part}) address@hidden function extracts ! the mantissa'' part @expr{m} of its floating-point argument; @kbd{f X} (@code{calc-xpon-part}) address@hidden extracts the exponent'' part ! @expr{e}. The original number is equal to ! @texline @tmath{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 *************** *** 16152,16158 **** 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 address@hidden @kindex f [ @kindex f ] --- 16310,16316 ---- 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 ] *************** *** 16167,16176 **** 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 @c{10^{-p}} ! @cite{10^-p}, where @cite{p} is the current precision. These operations are defined only on integers and floats. ! With numeric prefix arguments, they change the number by @cite{n} units. Note that incrementing followed by decrementing, or vice-versa, will almost but not quite always cancel out. Suppose the precision is --- 16325,16336 ---- 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 @tmath{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 *************** *** 16207,16213 **** The @kbd{F} (@code{calc-floor}) address@hidden 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 ! @address@hidden @kindex I F @pindex calc-ceiling --- 16367,16373 ---- The @kbd{F} (@code{calc-floor}) address@hidden 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 ! @i{-4}. @kindex I F @pindex calc-ceiling *************** *** 16219,16225 **** @kindex H I F The @kbd{I F} (@code{calc-ceiling}) address@hidden or @code{fceil}] command truncates toward positive infinity. Thus @kbd{3.6 I F} produces ! 4, and @kbd{_3.6 I F} produces @address@hidden @kindex R @pindex calc-round --- 16379,16385 ---- @kindex H I F The @kbd{I F} (@code{calc-ceiling}) address@hidden or @code{fceil}] command truncates toward positive infinity. Thus @kbd{3.6 I F} produces ! 4, and @kbd{_3.6 I F} produces @i{-3}. @kindex R @pindex calc-round *************** *** 16233,16239 **** 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 @address@hidden @kindex I R @pindex calc-trunc --- 16393,16399 ---- 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 @i{-4}. @kindex I R @pindex calc-trunc *************** *** 16246,16252 **** The @kbd{I R} (@code{calc-trunc}) address@hidden 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 @address@hidden These functions may not be applied meaningfully to error forms, but they do work for intervals. As a convenience, applying @code{floor} to a --- 16406,16412 ---- The @kbd{I R} (@code{calc-trunc}) address@hidden 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 @i{-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 *************** *** 16284,16290 **** 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 @cite{2.5} (which @code{rounde} sees as an exact tie between 2 and 3). Each of these functions, when written in algebraic formulas, allows --- 16444,16450 ---- 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 *************** *** 16298,16304 **** @cindex Fractional part of a number To compute the fractional part of a number (i.e., the amount which, when added to address@hidden(address@hidden@t{)}', will produce @var{n}) just take @var{n} ! modulo 1 using the @code{%} address@hidden Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm), and @kbd{f Q} (integer square root) commands, which are analogous to --- 16458,16464 ---- @cindex Fractional part of a number To compute the fractional part of a number (i.e., the amount which, when added to address@hidden(address@hidden@t{)}', 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 *************** *** 16313,16320 **** @pindex calc-conj @tindex conj The @kbd{J} (@code{calc-conj}) address@hidden command computes the ! complex conjugate of a number. For complex number @cite{a+bi}, the ! complex conjugate is @cite{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. --- 16473,16480 ---- @pindex calc-conj @tindex conj The @kbd{J} (@code{calc-conj}) address@hidden 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. *************** *** 16324,16338 **** The @kbd{G} (@code{calc-argument}) address@hidden 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 ! address@hidden(address@hidden@t{;address@hidden ! @address@hidden)}'. The result is expressed according to the current angular mode and will be in the range @i{-180} degrees (exclusive) to @i{+180} degrees ! (inclusive), or the equivalent range in address@hidden @pindex calc-imaginary The @code{calc-imaginary} command multiplies the number on the ! top of the stack by the imaginary number @cite{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. --- 16484,16498 ---- The @kbd{G} (@code{calc-argument}) address@hidden 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 address@hidden(address@hidden@t{;address@hidden@t{)}'. ! @infoline address@hidden(address@hidden@t{;address@hidden@t{)}'. The result is expressed according to the current angular mode and will be in the range @i{-180} degrees (exclusive) to @i{+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. *************** *** 16342,16355 **** The @kbd{f r} (@code{calc-re}) address@hidden 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.)@refill @kindex f i @pindex calc-im @tindex im The @kbd{f i} (@code{calc-im}) address@hidden command replaces a complex number by its imaginary part; real numbers are converted to zero. With a vector ! or matrix argument, these functions operate address@hidden @ignore @mindex v p --- 16502,16515 ---- The @kbd{f r} (@code{calc-re}) address@hidden 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}) address@hidden 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 *************** *** 16383,16395 **** @tindex pfloat The @kbd{c f} (@code{calc-float}) address@hidden command converts the number on the top of the stack to floating-point form. For example, ! @cite{23} is converted to @cite{23.0}, @cite{3:2} is converted to ! @cite{1.5}, and @cite{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 address@hidden 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, --- 16543,16555 ---- @tindex pfloat The @kbd{c f} (@code{calc-float}) address@hidden 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, *************** *** 16442,16448 **** The @kbd{c d} (@code{calc-to-degrees}) address@hidden 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 address@hidden @kindex c r @pindex calc-to-radians --- 16602,16608 ---- The @kbd{c d} (@code{calc-to-degrees}) address@hidden 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 *************** *** 16475,16481 **** 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}.)@refill @kindex c c @pindex calc-clean --- 16635,16641 ---- 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 *************** *** 16488,16494 **** 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)address@hidden 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} --- 16648,16654 ---- 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} *************** *** 16809,16815 **** There is no @code{newday} function at all because @kbd{F} address@hidden serves this purpose. Similarly, instead of @code{incday} and ! @code{incweek} simply use @cite{d + n} or @cite{d + 7 n}. @xref{Basic Arithmetic}, for the @kbd{f ]} address@hidden command which can adjust a date/time form by a certain number of seconds. --- 16969,16975 ---- There is no @code{newday} function at all because @kbd{F} address@hidden 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 ]} address@hidden command which can adjust a date/time form by a certain number of seconds. *************** *** 16870,16881 **** @item Any Calc formula which evaluates to one of the above three things. ! If the formula involves the variable @cite{y}, it stands for a ! yearly repeating holiday; @cite{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 @cite{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. --- 17030,17041 ---- @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. *************** *** 17290,17296 **** 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 @cite{(50-40)/40}, converting the answer to percentage form as if by @kbd{c %}. @node Future Value, Present Value, Percentages, Financial Functions --- 17450,17456 ---- 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 *************** *** 17334,17340 **** 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})address@hidden To illustrate the relationships between these functions, we could do the @code{fvb} calculation by hand'' using @code{fvl}. The --- 17494,17500 ---- 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 *************** *** 17352,17358 **** 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, @cite{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 --- 17512,17518 ---- 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 *************** *** 17382,17388 **** 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, @cite{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. --- 17542,17548 ---- 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. *************** *** 17431,17437 **** 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 address@hidden @kindex I b N @tindex npvb --- 17591,17597 ---- 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 *************** *** 17453,17459 **** 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}) = @address@hidden @kindex I b M @tindex pmtb --- 17613,17619 ---- 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 *************** *** 17475,17488 **** 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 address@hidden @kindex I b # @tindex nperb The @kbd{I b #} address@hidden 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 address@hidden @kindex H b # @tindex nperl --- 17635,17648 ---- 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 #} address@hidden 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 *************** *** 17490,17496 **** 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 address@hidden @kindex b T @pindex calc-fin-rate --- 17650,17656 ---- 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 *************** *** 17499,17505 **** 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 @address@hidden @kindex I b T @kindex H b T --- 17659,17665 ---- 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 *************** *** 17511,17517 **** 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 address@hidden @kindex b I @pindex calc-fin-irr --- 17671,17677 ---- 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 *************** *** 17573,17579 **** 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 @cite{[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: --- 17733,17739 ---- 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: *************** *** 17594,17600 **** @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 @cite{[10000, 10000, 10000]}; the total depreciation in any method is (by definition) the difference between the cost and the salvage value. --- 17754,17760 ---- @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. *************** *** 17716,17722 **** @end tex @noindent ! In @code{pmt} and @code{pmtb}, @cite{x=0} if omitted. These functions accept any numeric objects, including error forms, intervals, and even (though not very usefully) complex numbers. The --- 17876,17882 ---- @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 *************** *** 17776,17806 **** zeros with @kbd{d z}. @xref{Radix Modes}. @cindex Word size for binary operations ! The Calculator maintains a current @dfn{word size} @cite{w}, an arbitrary positive or negative integer. For a positive word size, all ! of the binary operations described here operate modulo @cite{2^w}. In particular, negative arguments are converted to positive integers modulo ! @cite{2^w} by all binary address@hidden If the word size is negative, binary operations produce 2's complement ! integers from @c{-2^{-w-1}} ! @cite{-(2^(-w-1))} to @c{2^{-w-1}-1} ! @cite{2^(-w-1)-1} inclusive. Either ! mode accepts inputs in any range; the sign of @cite{w} affects only ! the results produced. @kindex b c @pindex calc-clip @tindex clip The @kbd{b c} (@code{calc-clip}) address@hidden command can be used to clip a number by reducing it modulo ! @cite{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 @i{-8} @kbd{b c} converts to the range @i{-128} to address@hidden @kindex b w @pindex calc-word-size --- 17936,17969 ---- 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 @tmath{-2^{-w-1}} ! @infoline @expr{-(2^(-w-1))} ! to ! @texline @tmath{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}) address@hidden 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 @i{-8} @kbd{b c} converts to the range @i{-128} to 127. @kindex b w @pindex calc-word-size *************** *** 17820,17826 **** 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 @cite{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. --- 17983,17989 ---- 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. *************** *** 17830,17836 **** @tindex and The @kbd{b a} (@code{calc-and}) address@hidden command computes the bitwise AND of the two numbers on the top of the stack. In other words, for each ! of the @cite{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}. --- 17993,17999 ---- @tindex and The @kbd{b a} (@code{calc-and}) address@hidden 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}. *************** *** 17965,17981 **** @cindex Phi, golden ratio @cindex Golden ratio One miscellaneous command is address@hidden (@code{calc-pi}), which pushes ! the value of @c{\pi} ! @cite{pi} (at the current precision) onto the stack. With the ! Hyperbolic flag, it pushes the value @cite{e}, the base of natural logarithms. ! With the Inverse flag, it pushes Euler's constant @c{\gamma} ! @cite{gamma} (about 0.5772). With both Inverse and Hyperbolic, it ! pushes the golden ratio'' @c{\phi} ! @cite{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 address@hidden @ignore @mindex Q --- 18128,18147 ---- @cindex Phi, golden ratio @cindex Golden ratio One miscellaneous command is address@hidden (@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 @tmath{\gamma} ! @infoline @expr{gamma} ! (about 0.5772). With both Inverse and Hyperbolic, it ! pushes the golden ratio'' ! @texline @tmath{\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 *************** *** 18027,18033 **** @end ignore @kindex I L The address@hidden (@code{calc-exp}) address@hidden command computes the ! exponential, i.e., @cite{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. --- 18193,18199 ---- @end ignore @kindex I L The address@hidden (@code{calc-exp}) address@hidden 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. *************** *** 18048,18055 **** (base-10) logarithm of a number. (With the Inverse flag address@hidden, 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 @c{\ln10} ! @cite{ln(10)}. @kindex B @kindex I B --- 18214,18222 ---- (base-10) logarithm of a number. (With the Inverse flag address@hidden, 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 @tmath{\ln10}. ! @infoline @expr{ln(10)}. @kindex B @kindex I B *************** *** 18058,18066 **** @tindex alog The @kbd{B} (@code{calc-log}) address@hidden command computes a logarithm to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since ! @c{2^{10} = 1024} ! @cite{2^10 = 1024}. In certain cases like @samp{log(3,9)}, the result ! will be either @cite{1:2} or @cite{0.5} depending on the current Fraction Mode setting. With the Inverse flag address@hidden, this command is similar to @kbd{^} except that the order of the arguments is reversed. --- 18225,18234 ---- @tindex alog The @kbd{B} (@code{calc-log}) address@hidden command computes a logarithm to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since ! @texline @tmath{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 address@hidden, this command is similar to @kbd{^} except that the order of the arguments is reversed. *************** *** 18070,18076 **** The @kbd{f I} (@code{calc-ilog}) address@hidden 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 @cite{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))}. --- 18238,18244 ---- The @kbd{f I} (@code{calc-ilog}) address@hidden 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))}. *************** *** 18079,18097 **** @pindex calc-expm1 @tindex expm1 The @kbd{f E} (@code{calc-expm1}) address@hidden command computes ! @c{e^x - 1} ! @cite{exp(x)-1}, but using an algorithm that produces a more accurate ! answer when the result is close to zero, i.e., when @c{e^x} ! @cite{exp(x)} is close ! to one. @kindex f L @pindex calc-lnp1 @tindex lnp1 The @kbd{f L} (@code{calc-lnp1}) address@hidden command computes ! @c{\ln(x+1)} ! @cite{ln(x+1)}, producing a more accurate answer when @cite{x} is close ! to zero. @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions @section Trigonometric/Hyperbolic Functions --- 18247,18267 ---- @pindex calc-expm1 @tindex expm1 The @kbd{f E} (@code{calc-expm1}) address@hidden command computes ! @texline @tmath{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 @tmath{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}) address@hidden command computes ! @texline @tmath{\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 *************** *** 18104,18110 **** 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 address@hidden Calc's units'' mechanism includes angular units like @code{deg}, @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated --- 18274,18280 ---- 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 *************** *** 18120,18134 **** @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 @cite{x} plus a multiple of @c{\pi/2} ! @cite{pi/2} are also simplified. ! Calc includes similar formulas for @code{cos} and @address@hidden The @kbd{a s} command knows all angles which are integer multiples of ! @c{\pi/12} ! @cite{pi/12}, @c{\pi/10} ! @cite{pi/10}, or @c{\pi/8} ! @cite{pi/8} radians. In degrees mode, analogous simplifications occur for integer multiples of 15 or 18 degrees, and for arguments plus multiples of 90 degrees. --- 18290,18300 ---- @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. *************** *** 18253,18259 **** @address@hidden, @var{sin}]}. With the Inverse flag address@hidden, 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.)@refill @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions @section Advanced Mathematical Functions --- 18419,18425 ---- @address@hidden, @var{sin}]}. With the Inverse flag address@hidden, 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.) @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions @section Advanced Mathematical Functions *************** *** 18278,18285 **** 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: @c{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt} ! @cite{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}. (The actual implementation uses far more efficient computational methods.) @kindex f G --- 18444,18452 ---- 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 @tmath{\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 *************** *** 18311,18332 **** @tindex gammaG The @kbd{f G} (@code{calc-inc-gamma}) address@hidden command computes the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by ! the integral, @c{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)} ! @cite{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}. ! This implies that @samp{gammaP(a,inf) = 1} for any @cite{a} (see the definition of the normal gamma function). Several other varieties of incomplete gamma function are defined. ! The complement of @cite{P(a,x)}, called @cite{Q(a,x) = 1-P(a,x)} by some authors, is computed by the @kbd{I f G} address@hidden command. You can think of this as taking the other half of the integral, from ! @cite{x} to infinity. @ifinfo ! The functions corresponding to the integrals that define @cite{P(a,x)} ! and @cite{Q(a,x)} but without the normalizing @cite{1/gamma(a)} ! factor are called @cite{g(a,x)} and @cite{G(a,x)}, respectively ! (where @cite{g} and @cite{G} represent the lower- and upper-case Greek letter gamma). You can obtain these using the @kbd{H f G} address@hidden and @kbd{H I f G} address@hidden commands. @end ifinfo --- 18478,18500 ---- @tindex gammaG The @kbd{f G} (@code{calc-inc-gamma}) address@hidden command computes the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by ! the integral, ! @texline @tmath{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} address@hidden 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} address@hidden and @kbd{H I f G} address@hidden commands. @end ifinfo *************** *** 18344,18353 **** @tindex beta The @kbd{f b} (@code{calc-beta}) address@hidden command computes the Euler beta function, which is defined in terms of the gamma function as ! @c{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)} ! @cite{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)}, or by ! @c{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt} ! @cite{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}. @kindex f B @kindex H f B --- 18512,18522 ---- @tindex beta The @kbd{f b} (@code{calc-beta}) address@hidden command computes the Euler beta function, which is defined in terms of the gamma function as ! @texline @tmath{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 @tmath{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 *************** *** 18355,18363 **** @tindex betaI @tindex betaB The @kbd{f B} (@code{calc-inc-beta}) address@hidden command computes ! the incomplete beta function @cite{I(x,a,b)}. It is defined by ! @c{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)} ! @cite{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 address@hidden --- 18524,18532 ---- @tindex betaI @tindex betaB The @kbd{f B} (@code{calc-inc-beta}) address@hidden command computes ! the incomplete beta function @expr{I(x,a,b)}. It is defined by ! @texline @tmath{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 address@hidden *************** *** 18367,18378 **** @tindex erf @tindex erfc The @kbd{f e} (@code{calc-erf}) address@hidden command computes the ! error function @c{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt} ! @cite{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}. The complementary error function @kbd{I f e} (@code{calc-erfc}) address@hidden is the corresponding integral from @samp{x} to infinity; the sum ! @c{\hbox{erf}(x) + \hbox{erfc}(x) = 1} ! @cite{erf(x) + erfc(x) = 1}. @kindex f j @kindex f y --- 18536,18548 ---- @tindex erf @tindex erfc The @kbd{f e} (@code{calc-erf}) address@hidden command computes the ! error function ! @texline @tmath{\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}) address@hidden is the corresponding integral from @samp{x} to infinity; the sum ! @texline @tmath{\hbox{erf}(x) + \hbox{erfc}(x) = 1}. ! @infoline @expr{erf(x) + erfc(x) = 1}. @kindex f j @kindex f y *************** *** 18384,18393 **** (@code{calc-bessel-Y}) address@hidden 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 ! @cite{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 address@hidden @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions @section Branch Cuts and Principal Values --- 18554,18563 ---- (@code{calc-bessel-Y}) address@hidden 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 *************** *** 18420,18428 **** 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 @cite{a<0} and @cite{b} is small but positive ! or zero, the result is close to the @cite{+i} axis. For @cite{b} small and ! negative, the result is close to the @cite{-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))}. --- 18590,18598 ---- 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))}. *************** *** 18431,18438 **** negative real axis. The following table describes these branch cuts in another way. ! If the real and imaginary parts of @cite{z} are as shown, then ! the real and imaginary parts of @cite{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. --- 18601,18608 ---- 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. *************** *** 18448,18455 **** 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 @i{-2} as you might expect, but to the complex ! number @cite{(1., 1.732)}. Both of these are valid cube roots ! of @i{-8} (as is @cite{(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))}. --- 18618,18625 ---- 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 @i{-2} as you might expect, but to the complex ! number @expr{(1., 1.732)}. Both of these are valid cube roots ! of @i{-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))}. *************** *** 18461,18471 **** 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 @cite{-i} and above @cite{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 @cite{-i} and ! above @cite{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 --- 18631,18641 ---- 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 *************** *** 18543,18580 **** The @kbd{k r} (@code{calc-random}) address@hidden command produces random numbers of various sorts. ! Given a positive numeric prefix argument @cite{M}, it produces a random ! integer @cite{N} in the range @c{0 \le N < M} ! @cite{0 <= N < M}. Each of the @cite{M} ! values appears with equal address@hidden 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 @cite{M} ! the result is a random integer less than @cite{M}. However, note that ! while numeric prefix arguments are limited to six digits or so, an @cite{M} ! taken from the stack can be arbitrarily large. If @cite{M} is negative, ! the result is a random integer in the range @c{M < N \le 0} ! @cite{M < N <= 0}. ! ! If the value on the stack is a floating-point number @cite{M}, the result ! is a random floating-point number @cite{N} in the range @c{0 \le N < M} ! @cite{0 <= N < M} ! or @c{M < N \le 0} ! @cite{M < N <= 0}, according to the sign of @cite{M}. ! If @cite{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 @cite{M} is an error form @c{m @code{+/-} \sigma} ! @samp{m +/- s} where @var{m} ! and @c{\sigma} ! @var{s} are both real numbers, the result uses a Gaussian ! distribution with mean @var{m} and standard deviation @c{\sigma} @var{s}. ! If @cite{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 --- 18713,18759 ---- The @kbd{k r} (@code{calc-random}) address@hidden 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 @tmath{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 @tmath{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 @tmath{0 \le N < M} ! @infoline @expr{0 <= N < M} ! or ! @texline @tmath{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 @tmath{m} @code{+/-} @tmath{\sigma} ! @infoline @samp{m +/- s} ! where @var{m} and ! @texline @tmath{\sigma} ! @infoline @var{s} ! are both real numbers, the result uses a Gaussian distribution with mean ! @var{m} and standard deviation ! @texline @tmath{\sigma}. @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 *************** *** 18586,18592 **** additionally return 2.00000, but the probability of this happening is extremely small.) ! If @cite{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 --- 18765,18771 ---- 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 *************** *** 18609,18617 **** @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 @cite{M}. With a numeric prefix argument @var{n}, it produces @var{n} more random numbers using ! that value of @cite{M}. @kindex k h @pindex calc-shuffle --- 18788,18796 ---- @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 *************** *** 18619,18630 **** 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 @cite{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 @cite{M} from second-to-top.) ! If @cite{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 --- 18798,18809 ---- 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 *************** *** 18632,18639 **** 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 @cite{M} is a vector, no given vector element will be drawn more ! than once, but if several elements of @cite{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 --- 18811,18818 ---- 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 *************** *** 18641,18652 **** @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 @cite{M}. Naturally, this is allowed only when @cite{M} specifies a small discrete set of possibilities. To do the equivalent of @kbd{k h} but with duplications allowed, ! given @cite{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 @cite{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}. --- 18820,18831 ---- @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}. *************** *** 18683,18692 **** If @code{RandSeed} contains an integer, Calc uses this integer to seed an additive congruential'' method (Knuth's algorithm 3.2.2A, ! computing @c{X_{n-55} - X_{n-24}} ! @cite{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 @cite{[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 --- 18862,18873 ---- If @code{RandSeed} contains an integer, Calc uses this integer to seed an additive congruential'' method (Knuth's algorithm 3.2.2A, ! computing ! @texline @tmath{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 *************** *** 18718,18733 **** To create a random floating-point number with precision @var{p}, Calc simply creates a random @var{p}-digit integer and multiplies by ! @c{10^{-p}} ! @cite{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 @c{10^{-9}} ! @cite{10^-9} or @c{10^{-10}} ! @cite{10^-10}, but those numbers ! will only have two or three random digits since they correspond to small ! integers times @c{10^{-12}} ! @cite{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 --- 18899,18919 ---- To create a random floating-point number with precision @var{p}, Calc simply creates a random @var{p}-digit integer and multiplies by ! @texline @tmath{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 @tmath{10^{-9}} ! @infoline @expr{10^-9} ! or ! @texline @tmath{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 @tmath{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 *************** *** 18761,18767 **** 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 address@hidden @kindex k l @pindex calc-lcm --- 18947,18953 ---- 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 *************** *** 18769,18783 **** The @kbd{k l} (@code{calc-lcm}) address@hidden 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 ! address@hidden @kindex k E @pindex calc-extended-gcd @tindex egcd The @kbd{k E} (@code{calc-extended-gcd}) address@hidden command computes ! the GCD of two integers @cite{x} and @cite{y} and returns a vector ! @cite{[g, a, b]} where @c{g = \gcd(x,y) = a x + b y} ! @cite{g = gcd(x,y) = a x + b y}. @kindex ! @pindex calc-factorial --- 18955,18970 ---- The @kbd{k l} (@code{calc-lcm}) address@hidden 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}) address@hidden command computes ! the GCD of two integers @expr{x} and @expr{y} and returns a vector ! @expr{[g, a, b]} where ! @texline @tmath{g = \gcd(x,y) = a x + b y}. ! @infoline @expr{g = gcd(x,y) = a x + b y}. @kindex ! @pindex calc-factorial *************** *** 18794,18800 **** 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 address@hidden @kindex k d @pindex calc-double-factorial --- 18981,18987 ---- 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 *************** *** 18805,18833 **** @tindex !! The @kbd{k d} (@code{calc-double-factorial}) address@hidden command computes the double factorial'' of an integer. For an even integer, ! this is the product of even integers from 2 to @cite{N}. For an odd ! integer, this is the product of odd integers from 3 to @cite{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 @cite{N!!} is also recognized for double address@hidden @kindex k c @pindex calc-choose @tindex choose The @kbd{k c} (@code{calc-choose}) address@hidden command computes the ! binomial coefficient @address@hidden, where @cite{M} is the number ! on the top of the stack and @cite{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 @c{N! \over M! (N-M)!\,} ! @cite{N! / M! (N-M)!}. @kindex H k c @pindex calc-perm @tindex perm @ifinfo The @kbd{H k c} (@code{calc-perm}) address@hidden command computes the ! number-of-permutations function @cite{N! / (N-M)!}. @end ifinfo @tex The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the --- 18992,19021 ---- @tindex !! The @kbd{k d} (@code{calc-double-factorial}) address@hidden 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}) address@hidden command computes the ! binomial coefficient @address@hidden, 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 @tmath{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}) address@hidden 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 *************** *** 18840,18850 **** @tindex bern The @kbd{k b} (@code{calc-bernoulli-number}) address@hidden command computes a given Bernoulli number. The value at the top of the stack ! is a nonnegative integer @cite{n} that specifies which Bernoulli number is desired. The @kbd{H k b} command computes a Bernoulli polynomial, ! taking @cite{n} from the second-to-top position and @cite{x} from the ! top of the stack. If @cite{x} is a variable or formula the result is ! a polynomial in @cite{x}; if @cite{x} is a number the result is a number. @kindex k e @kindex H k e --- 19028,19038 ---- @tindex bern The @kbd{k b} (@code{calc-bernoulli-number}) address@hidden 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 *************** *** 18861,18873 **** @tindex stir1 @tindex stir2 The @kbd{k s} (@code{calc-stirling-number}) address@hidden command ! computes a Stirling number of the first address@hidden n \brack m} ! @asis{}, given two integers ! @cite{n} and @cite{m} on the stack. The @kbd{H k s} address@hidden ! command computes a Stirling number of the second address@hidden n \brace m} ! @asis{}. These are ! the number of @cite{m}-cycle permutations of @cite{n} objects, and ! the number of ways to partition @cite{n} objects into @cite{m} non-empty sets, respectively. @kindex k p --- 19049,19063 ---- @tindex stir1 @tindex stir2 The @kbd{k s} (@code{calc-stirling-number}) address@hidden command ! computes a Stirling number of the first ! @texline address@hidden@tmath{n \brack m}, ! @infoline kind, ! given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s} ! address@hidden command computes a Stirling number of the second ! @texline address@hidden@tmath{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 *************** *** 18895,18901 **** 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 @cite{n} is (probably) prime and 0 if not. @kindex k f --- 19085,19091 ---- 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 *************** *** 18942,18958 **** @pindex calc-totient @tindex totient The @kbd{k t} (@code{calc-totient}) address@hidden command computes the ! Euler totient'' address@hidden \phi(n)} ! @asis{}, the number of integers less than @cite{n} which ! are relatively prime to @cite{n}. @kindex k m @pindex calc-moebius @tindex moebius The @kbd{k m} (@code{calc-moebius}) address@hidden command computes the ! @c{M\"obius \mu} ! @asis{Moebius mu''} function. If the input number is a product of @cite{k} ! distinct factors, this is @cite{(-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. --- 19132,19151 ---- @pindex calc-totient @tindex totient The @kbd{k t} (@code{calc-totient}) address@hidden command computes the ! Euler totient'' ! @texline address@hidden@tmath{\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}) address@hidden command computes the ! @texline M@"obius @tmath{\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. *************** *** 18962,18975 **** @noindent The functions in this section compute various probability distributions. For continuous distributions, this is the integral of the probability ! density function from @cite{x} to infinity. (These are the upper tail'' distribution functions; there are also corresponding lower ! tail'' functions which integrate from minus infinity to @cite{x}.) For discrete distributions, the upper tail function gives the sum ! from @cite{x} to infinity; the lower tail function gives the sum ! from minus infinity up to, but not including,@w{ address@hidden ! To integrate from @cite{x} to @cite{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)} --- 19155,19168 ---- @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{ address@hidden ! 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)} *************** *** 18999,19005 **** 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 address@hidden @key{DEL} 2.8 k N -}, using @address@hidden @key{DEL}} to ! recover the original arguments but substitute a new value for @cite{x}.) @kindex k C @pindex calc-utpc --- 19192,19198 ---- 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 address@hidden @key{DEL} 2.8 k N -}, using @address@hidden @key{DEL}} to ! recover the original arguments but substitute a new value for @expr{x}.) @kindex k C @pindex calc-utpc *************** *** 19013,19021 **** @end ignore @tindex ltpc The @samp{utpc(x,v)} function uses the chi-square distribution with ! @c{\nu} ! @cite{v} degrees of freedom. It is the probability that a model is ! correct if its chi-square statistic is @cite{x}. @kindex k F @pindex calc-utpf --- 19206,19215 ---- @end ignore @tindex ltpc The @samp{utpc(x,v)} function uses the chi-square distribution with ! @texline @tmath{\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 *************** *** 19029,19039 **** @end ignore @tindex ltpf The @samp{utpf(F,v1,v2)} function uses the F distribution, used in ! various statistical tests. The parameters @c{\nu_1} ! @cite{v1} and @c{\nu_2} ! @cite{v2} are the degrees of freedom in the numerator and denominator, ! respectively, used in computing the statistic @cite{F}. @kindex k N @pindex calc-utpn --- 19223,19236 ---- @end ignore @tindex ltpf The @samp{utpf(F,v1,v2)} function uses the F distribution, used in ! various statistical tests. The parameters ! @texline @tmath{\nu_1} ! @infoline @expr{v1} ! and ! @texline @tmath{\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 *************** *** 19047,19056 **** @end ignore @tindex ltpn The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution ! with mean @cite{m} and standard deviation @c{\sigma} ! @cite{s}. It is the ! probability that such a normal-distributed random variable would ! exceed @cite{x}. @kindex k P @pindex calc-utpp --- 19244,19254 ---- @end ignore @tindex ltpn The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution ! with mean @expr{m} and standard deviation ! @texline @tmath{\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 *************** *** 19064,19070 **** @end ignore @tindex ltpp The @samp{utpp(n,x)} function uses a Poisson distribution with ! mean @cite{x}. It is the probability that @cite{n} or more such Poisson random events will occur. @kindex k T --- 19262,19268 ---- @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 *************** *** 19079,19094 **** @end ignore @tindex ltpt The @samp{utpt(t,v)} function uses the Student's t'' distribution ! with @c{\nu} ! @cite{v} degrees of freedom. It is the probability that a ! t-distributed random variable will be greater than @cite{t}. ! (Note: This computes the distribution function @c{A(t|\nu)} ! @cite{A(t|v)} ! where @c{A(0|\nu) = 1} ! @cite{A(0|v) = 1} and @c{A(\infty|\nu) \to 0} ! @cite{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. --- 19277,19298 ---- @end ignore @tindex ltpt The @samp{utpt(t,v)} function uses the Student's t'' distribution ! with ! @texline @tmath{\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 @tmath{A(t|\nu)} ! @infoline @expr{A(t|v)} ! where ! @texline @tmath{A(0|\nu) = 1} ! @infoline @expr{A(0|v) = 1} ! and ! @texline @tmath{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. *************** *** 19155,19161 **** @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 ! @cite{(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 --- 19359,19365 ---- @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 *************** *** 19245,19252 **** 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 ! @c{2\times3} ! @asis{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}. --- 19449,19457 ---- 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 @tmath{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}. *************** *** 19346,19352 **** @noindent Vectors and matrices can be added, ! subtracted, multiplied, and divided; @pxref{Basic address@hidden @kindex | @pindex calc-concat --- 19551,19557 ---- @noindent Vectors and matrices can be added, ! subtracted, multiplied, and divided; @pxref{Basic Arithmetic}. @kindex | @pindex calc-concat *************** *** 19392,19402 **** 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 @c{3\times3} ! @asis{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 --- 19597,19608 ---- 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 @tmath{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 *************** *** 19407,19414 **** this command prompts for one. In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)}, ! except that @cite{a} is required to be a scalar (non-vector) quantity. ! If @cite{n} is omitted, @samp{idn(a)} represents @cite{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 --- 19613,19620 ---- 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 *************** *** 19531,19540 **** @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 @cite{x}, @cite{y}, or @cite{z} if ! @cite{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 @cite{i}, column @cite{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 --- 19737,19746 ---- @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 *************** *** 19561,19573 **** 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)address@hidden 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 @cite{i}, column @cite{j} ! of matrix @cite{m}. @kindex v s @pindex calc-subvector --- 19767,19779 ---- 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 *************** *** 19610,19624 **** The @kbd{v l} (@code{calc-vlength}) address@hidden 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 ! address@hidden @kindex H v l @tindex mdims With the Hyperbolic flag, @kbd{H v l} address@hidden 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 @c{2\times3} ! @asis{2x3} matrix. @kindex v f @pindex calc-vector-find --- 19816,19832 ---- The @kbd{v l} (@code{calc-vlength}) address@hidden 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} address@hidden 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 @tmath{2\times3} ! @infoline 2x3 ! matrix. @kindex v f @pindex calc-vector-find *************** *** 19647,19660 **** 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 @c{1\times4} ! @asis{1x4} matrix), @kbd{v a 1} produces ! @samp{[[1], [2], [3], [4]]} (a @c{4\times1} ! @asis{4x1} matrix), @kbd{v a 2} produces ! @samp{[[1, 2], [3, 4]]} (the original @c{2\times2} ! @asis{2x2} matrix), @address@hidden 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 --- 19855,19872 ---- 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 @tmath{1\times4} ! @infoline 1x4 ! matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a ! @texline @tmath{4\times1} ! @infoline 4x1 ! matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original ! @texline @tmath{2\times2} ! @infoline 2x2 ! matrix), @address@hidden 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 *************** *** 19810,19816 **** 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 address@hidden @kindex V J @pindex calc-conj-transpose --- 20022,20028 ---- 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 *************** *** 19832,19838 **** 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 address@hidden @kindex v n @pindex calc-rnorm --- 20044,20050 ---- 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 *************** *** 19851,19857 **** 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 @cite{k}-norms for @cite{k} other than one or infinity are not provided. @kindex V C --- 20063,20069 ---- 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 *************** *** 19877,19884 **** computed, the inverse and determinant of the matrix can be recomputed quickly in the future. ! If the argument to @kbd{&} is a plain number @cite{x}, this ! command simply computes @cite{1/x}. This is okay, because the @samp{/} operator also does a matrix inversion when dividing one by a matrix. --- 20089,20096 ---- 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. *************** *** 19936,19943 **** 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 @cite{A} ! is a subset of the set @cite{B}, use @samp{vdiff(A, B) = []}. @kindex V + @pindex calc-remove-duplicates --- 20148,20155 ---- 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 *************** *** 19969,19988 **** 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 address@hidden (A \cup B)} ! @asis{} and address@hidden (A \cap B)} ! @asis{}. @kindex V - @pindex calc-set-difference @tindex vdiff The @kbd{V -} (@code{calc-set-difference}) address@hidden command computes the difference between two sets. An object is in the difference ! @cite{A - B} if and only if it is in @cite{A} but not in @cite{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 @cite{A} is the set of ! all possible values, then @cite{A - B} is the complement'' of @cite{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). --- 20181,20203 ---- 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 address@hidden(@tmath{A \cup B}) ! @infoline union ! and ! @texline address@hidden(@tmath{A \cap B}). ! @infoline intersection. @kindex V - @pindex calc-set-difference @tindex vdiff The @kbd{V -} (@code{calc-set-difference}) address@hidden 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). *************** *** 20078,20085 **** 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 (@c{2^{100}} ! @cite{2^100}, a 31-digit integer, in this case). @node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions @section Statistical Operations on Vectors --- 20293,20301 ---- 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 (@tmath{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 *************** *** 20167,20173 **** (@code{calc-vector-prod}) address@hidden 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})address@hidden @kindex u X @kindex u N --- 20383,20389 ---- (@code{calc-vector-prod}) address@hidden 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 *************** *** 20189,20209 **** @cindex Mean of data values The @kbd{u M} (@code{calc-vector-mean}) address@hidden command computes the average (arithmetic mean) of the data values. ! If the inputs are error forms @c{x @code{+/-} \sigma} ! @samp{x +/- s}, this is the weighted ! mean of the @cite{x} values with weights @c{1 / \sigma^2} ! @cite{1 / s^2}. @tex \turnoffactive$$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over \displaystyle \sum { 1 \over \sigma_i^2 } } $$@end tex If the inputs are not error forms, this is simply the sum of the ! values divided by the count of the address@hidden Note that a plain number can be considered an error form with ! error @c{\sigma = 0} ! @cite{s = 0}. If the input to @kbd{u M} is a mixture of plain numbers and error forms, the result is the mean of the plain numbers, ignoring all values with non-zero errors. (By the above definitions it's clear that a plain number effectively --- 20405,20429 ---- @cindex Mean of data values The @kbd{u M} (@code{calc-vector-mean}) address@hidden command computes the average (arithmetic mean) of the data values. ! If the inputs are error forms ! @texline @tmath{x \pm \sigma}, ! @infoline @samp{x +/- s}, ! this is the weighted mean of the @expr{x} values with weights ! @texline @tmath{1 /\sigma^2}. ! @infoline @expr{1 / s^2}. @tex \turnoffactive$$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over \displaystyle \sum { 1 \over \sigma_i^2 } } $$@end tex If the inputs are not error forms, this is simply the sum of the ! values divided by the count of the values. Note that a plain number can be considered an error form with ! error ! @texline @tmath{\sigma = 0}. ! @infoline @expr{s = 0}. ! If the input to @kbd{u M} is a mixture of plain numbers and error forms, the result is the mean of the plain numbers, ignoring all values with non-zero errors. (By the above definitions it's clear that a plain number effectively *************** *** 20212,20218 **** This function also works for distributions (error forms or intervals). The mean of an error form address@hidden @t{+/-} @var{b}' is simply ! @cite{a}. The mean of an interval is the mean of the minimum and maximum values of the interval. @kindex I u M --- 20432,20438 ---- This function also works for distributions (error forms or intervals). The mean of an error form address@hidden @t{+/-} @var{b}' is simply ! @expr{a}. The mean of an interval is the mean of the minimum and maximum values of the interval. @kindex I u M *************** *** 20234,20240 **** divided by the square root of the number of values. (This works out to be equivalent to calculating the standard deviation and then assuming each value's error is equal to this standard ! deviation.)@refill @tex \turnoffactive$$ \sigma_\mu^2 = {\sigma^2 \over N} $$--- 20454,20460 ---- divided by the square root of the number of values. (This works out to be equivalent to calculating the standard deviation and then assuming each value's error is equal to this standard ! deviation.) @tex \turnoffactive$$ \sigma_\mu^2 = {\sigma^2 \over N} $$*************** *** 20309,20321 **** @cindex Standard deviation @cindex Sample statistics The @kbd{u S} (@code{calc-vector-sdev}) address@hidden command ! computes the standard address@hidden \sigma} ! @asis{} of the data values. If the ! values are error forms, the errors are used as weights just ! as for @kbd{u M}. This is the @emph{sample} standard deviation, ! whose value is the square root of the sum of the squares of the ! differences between the values and the mean of the @cite{N} values, ! divided by @cite{N-1}. @tex \turnoffactive$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$--- 20529,20542 ---- @cindex Standard deviation @cindex Sample statistics The @kbd{u S} (@code{calc-vector-sdev}) address@hidden command ! computes the standard ! @texline address@hidden@tmath{\sigma} ! @infoline deviation ! of the data values. If the values are error forms, the errors are used ! as weights just as for @kbd{u M}. This is the @emph{sample} standard ! deviation, whose value is the square root of the sum of the squares of ! the differences between the values and the mean of the @expr{N} values, ! divided by @expr{N-1}. @tex \turnoffactive$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$*************** *** 20324,20333 **** 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 @c{\sqrt{12}} ! @cite{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 --- 20545,20555 ---- 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 @tmath{\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 *************** *** 20336,20342 **** The @kbd{I u S} (@code{calc-vector-pop-sdev}) address@hidden command computes the @emph{population} standard deviation. It is defined by the same formula as above but dividing ! by @cite{N} instead of by @cite{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 --- 20558,20564 ---- The @kbd{I u S} (@code{calc-vector-pop-sdev}) address@hidden 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 *************** *** 20361,20368 **** The @kbd{H u S} (@code{calc-vector-variance}) address@hidden and @kbd{H I u S} (@code{calc-vector-pop-variance}) address@hidden commands compute the variance of the data values. The variance ! is the address@hidden \sigma^2} ! @asis{} 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.) --- 20583,20592 ---- The @kbd{H u S} (@code{calc-vector-variance}) address@hidden and @kbd{H I u S} (@code{calc-vector-pop-variance}) address@hidden commands compute the variance of the data values. The variance ! is the ! @texline address@hidden@tmath{\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.) *************** *** 20383,20392 **** 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 @c{N\times2} ! @asis{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 --- 20607,20617 ---- 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 @tmath{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 *************** *** 20397,20403 **** 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 @cite{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} --- 20622,20628 ---- 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} *************** *** 20419,20426 **** @tindex vpcov The @kbd{I u C} (@code{calc-vector-pop-covariance}) address@hidden command computes the population covariance, which is the same as the ! sample covariance computed by @kbd{u C} except dividing by @cite{N} ! instead of @cite{N-1}. @kindex H u C @pindex calc-vector-correlation --- 20644,20651 ---- @tindex vpcov The @kbd{I u C} (@code{calc-vector-pop-covariance}) address@hidden 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 *************** *** 20598,20604 **** 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 ! @address@hidden @ignore @starindex --- 20823,20829 ---- 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 *************** *** 20635,20658 **** 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 address@hidden 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 ! @cite{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to ! produce another @c{3\times2} ! @asis{3x2} matrix, @cite{[[1, 2, 3], [4, 5, 6]]}. @tindex mapr The command @kbd{V M _} address@hidden (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 @cite{[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, @cite{[[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 --- 20860,20885 ---- 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 @tmath{3\times2} ! @infoline 3x2 ! matrix, @expr{[[1, 2, 3], [4, 5, 6]]}. @tindex mapr The command @kbd{V M _} address@hidden (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 *************** *** 20665,20671 **** 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 ! @cite{[[-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 --- 20892,20898 ---- 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 *************** *** 20756,20768 **** @tindex reduced @tindex rreduced As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For ! example, given the matrix @cite{[[a, b, c], [d, e, f]]}, @kbd{V R +} will ! compute @cite{a + b + c + d + e + f}. You can type @kbd{V R _} or @kbd{V R :} to modify this behavior. The @kbd{V R _} address@hidden 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 @cite{[a + b + c, d + e + f]}. Similarly, @kbd{V R :} ! address@hidden reduces down; @kbd{V R : +} would produce @cite{[a + d, b + e, c + f]}. @tindex reducer --- 20983,20995 ---- @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 _} address@hidden 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 :} ! address@hidden reduces down; @kbd{V R : +} would produce @expr{[a + d, b + e, c + f]}. @tindex reducer *************** *** 20919,20925 **** The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >} (@code{calc-matrix-right-justify}), and @address@hidden =}} (@code{calc-matrix-center-justify}) control whether matrix elements ! are justified to the left, right, or center of their address@hidden @kindex V [ @pindex calc-vector-brackets --- 21146,21152 ---- The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >} (@code{calc-matrix-right-justify}), and @address@hidden =}} (@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 *************** *** 20936,20942 **** 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 address@hidden @kindex V ] @pindex calc-matrix-brackets --- 21163,21169 ---- 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 *************** *** 20989,20995 **** @kindex V , @pindex calc-vector-commas The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and ! off in vector and matrix address@hidden In vectors of length one, and in all vectors when commas have been turned off, Calc adds extra parentheses around formulas that might --- 21216,21222 ---- @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 *************** *** 21047,21053 **** for anything else'') prefix. @xref{Editing Stack Entries}, to see how to manipulate formulas ! using regular Emacs editing address@hidden When doing algebraic work, you may find several of the Calculator's modes to be helpful, including algebraic-simplification mode (@kbd{m A}) --- 21274,21280 ---- 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}) *************** *** 21055,21061 **** 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 address@hidden @menu * Selecting Subformulas:: --- 21282,21288 ---- 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:: *************** *** 21213,21219 **** 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 ! address@hidden 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. --- 21440,21446 ---- 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. *************** *** 21311,21317 **** 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 ! @address@hidden 1}} with @var{n} as a numeric prefix address@hidden @kindex j n @kindex j p --- 21538,21544 ---- 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 ! @address@hidden 1}} with @var{n} as a numeric prefix argument. @kindex j n @kindex j p *************** *** 21493,21499 **** 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})address@hidden @smallexample @group --- 21720,21726 ---- 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 *************** *** 21850,21856 **** 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 @cite{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. --- 22077,22083 ---- 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. *************** *** 21899,21905 **** 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 address@hidden 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 --- 22126,22132 ---- 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 *************** *** 21966,21991 **** @cindex Default simplifications This section describes the default simplifications,'' those which are normally applied to all results. For example, if you enter the variable ! @cite{x} on the stack twice and push @kbd{+}, Calc's default ! simplifications automatically change @cite{x + x} to @cite{2 x}. The @kbd{m O} command turns off the default simplifications, so that ! @cite{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, @cite{2 + 3} is evaluated to @cite{5}, and @address@hidden(9)} ! is evaluated to @cite{3}. Evaluation does not occur if the arguments ! to a function are somehow of the wrong type (@address@hidden([2,3,4])}), ! range (@address@hidden(90)}), or number (@address@hidden(3,5)}), or if the ! function name is not recognized (@address@hidden(5)}), or if symbolic'' ! mode (@pxref{Symbolic Mode}) prevents evaluation (@address@hidden(2)}). Calc simplifies (evaluates) the arguments to a function before it ! simplifies the function itself. Thus @address@hidden(5+4)} is ! simplified to @address@hidden(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{? :} --- 22193,22219 ---- @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 @address@hidden(9)} ! is evaluated to @expr{3}. Evaluation does not occur if the arguments ! to a function are somehow of the wrong type @address@hidden([2,3,4])}), ! range (@address@hidden(90)}), or number (@address@hidden(3,5)}), ! or if the function name is not recognized (@address@hidden(5)}), or if ! symbolic'' mode (@pxref{Symbolic Mode}) prevents evaluation ! (@address@hidden(2)}). Calc simplifies (evaluates) the arguments to a function before it ! simplifies the function itself. Thus @address@hidden(5+4)} is ! simplified to @address@hidden(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{? :} *************** *** 22024,22032 **** 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, @cite{((a + b) + c) + d}, and ! a right-associative form for products, @cite{a * (b * (c * d))}. ! Formulas like @cite{(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 --- 22252,22260 ---- 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 *************** *** 22034,22232 **** in the examples below. Sums and products are @emph{not} rearranged according to the ! commutative law (@cite{a + b} to @cite{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 @cite{a b + b a} can be simplified to @cite{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 @cite{a - b} are treated like sums @cite{a + (-b)} for purposes of simplification; one of the default simplifications ! is to rewrite @cite{a + (-b)} or @cite{(-b) + a}, where @cite{-b} ! represents a negative-looking'' term, into @cite{a - b} form. Negative-looking'' means negative numbers, negated formulas like ! @cite{-x}, and products or quotients in which either term is negative-looking. ! Other simplifications involving negation are @cite{-(-x)} to @cite{x}; ! @cite{-(a b)} or @cite{-(a/b)} where either @cite{a} or @cite{b} is negative-looking, simplified by negating that term, or else where ! @cite{a} or @cite{b} is any number, by negating that number; ! @cite{-(a + b)} to @cite{-a - b}, and @cite{-(b - a)} to @cite{a - b}. ! (This, and rewriting @cite{(-b) + a} to @cite{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: ! @cite{a x + b x} to @cite{(a + b) x}, where @cite{a} represents ! a number or an implicit 1 or @i{-1} (as in @cite{x} or @cite{-x}) ! and similarly for @cite{b}. Use the @kbd{a c}, @address@hidden 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 @cite{a + b + b + c} ! is simplified to @cite{a + 2 b + c}, but @cite{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, @cite{a + 0} and @cite{0 + a} are simplified to @cite{a}. ! A consequence of the above rules is that @cite{0 - a} is simplified ! to @cite{-a}. @tex \bigskip @end tex ! The products @cite{1 a} and @cite{a 1} are simplified to @cite{a}; ! @cite{(-1) a} and @cite{a (-1)} are simplified to @cite{-a}; ! @cite{0 a} and @cite{a 0} are simplified to @cite{0}, except that ! in matrix mode where @cite{a} is not provably scalar the result ! is the generic zero matrix @samp{idn(0)}, and that if @cite{a} is infinite the result is @samp{nan}. ! Also, @cite{(-a) b} and @cite{a (-b)} are simplified to @cite{-(a b)}, where this occurs for negated formulas but not for regular negative numbers. Products are commuted only to move numbers to the front: ! @cite{a b 2} is commuted to @cite{2 a b}. ! The product @cite{a (b + c)} is distributed over the sum only if ! @cite{a} and at least one of @cite{b} and @cite{c} are numbers: ! @cite{2 (x + 3)} goes to @cite{2 x + 6}. The formula ! @cite{(-a) (b - c)}, where @cite{-a} is a negative number, is ! rewritten to @cite{a (c - b)}. The distributive law of products and powers is used for adjacent ! terms of the product: @cite{x^a x^b} goes to @c{x^{a+b}} ! @cite{x^(a+b)} ! where @cite{a} is a number, or an implicit 1 (as in @cite{x}), ! or the implicit one-half of @address@hidden(x)}, and similarly for ! @cite{b}. The result is written using @samp{sqrt} or @samp{1/sqrt} ! if the sum of the powers is @cite{1/2} or @cite{-1/2}, respectively. If the sum of the powers is zero, the product is simplified to ! @cite{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: @c{x^{-2} y} ! @cite{x^(-2) y} goes to @cite{y / x^2} unless matrix mode is ! in effect and neither @cite{x} nor @cite{y} are scalar (in which case it is considered unsafe to rearrange the order of the terms). ! Finally, @cite{a (b/c)} is rewritten to @cite{(a b)/c}, and also ! @cite{(a/b) c} is changed to @cite{(a c)/b} unless in matrix mode. @tex \bigskip @end tex Simplifications for quotients are analogous to those for products. ! The quotient @cite{0 / x} is simplified to @cite{0}, with the same ! exceptions that were noted for @cite{0 x}. Likewise, @cite{x / 1} ! and @cite{x / (-1)} are simplified to @cite{x} and @cite{-x}, respectively. ! The quotient @cite{x / 0} is left unsimplified or changed to an infinite quantity, as directed by the current infinite mode. @xref{Infinite Mode}. ! The expression @c{a / b^{-c}} ! @cite{a / b^(-c)} is changed to @cite{a b^c}, ! where @cite{-c} is any negative-looking power. Also, @cite{1 / b^c} ! is changed to @c{b^{-c}} ! @cite{b^(-c)} for any power @cite{c}. ! ! Also, @cite{(-a) / b} and @cite{a / (-b)} go to @cite{-(a/b)}; ! @cite{(a/b) / c} goes to @cite{a / (b c)}; and @cite{a / (b/c)} ! goes to @cite{(a c) / b} unless matrix mode prevents this ! rearrangement. Similarly, @cite{a / (b:c)} is simplified to ! @cite{(c:b) a} for any fraction @cite{b:c}. ! The distributive law is applied to @cite{(a + b) / c} only if ! @cite{c} and at least one of @cite{a} and @cite{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, @cite{a x b / a y b} ! is cancelled to @cite{x b / y b} but not to @cite{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 @cite{(-a) / (-b)} to @cite{a / b}, @cite{(-a) / (b - c)} ! to @cite{a / (c - b)}, and @cite{(a - b) / (-c)} to @cite{(b - a) / c}. @tex \bigskip @end tex ! The formula @cite{x^0} is simplified to @cite{1}, or to @samp{idn(1)} ! in matrix mode. The formula @cite{0^x} is simplified to @cite{0} ! unless @cite{x} is a negative number or complex number, in which case the result is an infinity or an unsimplified formula according ! to the current infinite mode. Note that @cite{0^0} is an indeterminate form, as evidenced by the fact that the simplifications ! for @cite{x^0} and @cite{0^x} conflict when @cite{x=0}. ! Powers of products or quotients @cite{(a b)^c}, @cite{(a/b)^c} ! are distributed to @cite{a^c b^c}, @cite{a^c / b^c} only if @cite{c} ! is an integer, or if either @cite{a} or @cite{b} are nonnegative ! real numbers. Powers of powers @cite{(a^b)^c} are simplified to ! @c{a^{b c}} ! @cite{a^(b c)} only when @cite{c} is an integer and @cite{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, @c{((-3)^{1/2})^2} ! @cite{((-3)^1:2)^2} is safe to simplify, but ! @c{((-3)^2)^{1/2}} ! @cite{((-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, @address@hidden(x)^n} is simplified to ! @c{x^{n/2}} ! @cite{x^(n/2)} only for even integers @cite{n}. ! ! If @cite{a} is known to be real, @cite{b} is an even integer, and ! @cite{c} is a half- or quarter-integer, then @cite{(a^b)^c} is ! simplified to @address@hidden(a^{b c})} ! @address@hidden(a^(b c))}. ! ! Also, @cite{(-a)^b} is simplified to @cite{a^b} if @cite{b} is an ! even integer, or to @cite{-(a^b)} if @cite{b} is an odd integer, ! for any negative-looking expression @cite{-a}. ! ! Square roots @address@hidden(x)} generally act like one-half powers ! @c{x^{1:2}} ! @cite{x^1:2} for the purposes of the above-listed simplifications. ! ! Also, note that @c{1 / x^{1:2}} ! @cite{1 / x^1:2} is changed to @c{x^{-1:2}} ! @cite{x^(-1:2)}, ! but @cite{1 / @t{sqrt}(x)} is left alone. @tex \bigskip @end tex Generic identity matrices (@pxref{Matrix Mode}) are simplified by the ! following rules: @address@hidden(a) + b} to @cite{a + b} if @cite{b} ! is provably scalar, or expanded out if @cite{b} is a matrix; ! @address@hidden(a) + @t{idn}(b)} to @address@hidden(a + b)}; ! @address@hidden(a)} to @address@hidden(-a)}; @cite{a @t{idn}(b)} to ! @address@hidden(a b)} if @cite{a} is provably scalar, or to @cite{a b} ! if @cite{a} is provably non-scalar; @address@hidden(a) @t{idn}(b)} ! to @address@hidden(a b)}; analogous simplifications for quotients ! involving @code{idn}; and @address@hidden(a)^n} to @address@hidden(a^n)} ! where @cite{n} is an integer. @tex \bigskip --- 22262,22474 ---- 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 @i{-1} (as in @expr{x} or @expr{-x}) ! and similarly for @expr{b}. Use the @kbd{a c}, @address@hidden 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 @tmath{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 @address@hidden(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 @tmath{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 @tmath{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 @tmath{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 or complex number, in which case the result is an infinity or an unsimplified formula according ! to the current infinite mode. Note that @expr{0^0} is an indeterminate form, as evidenced by the fact that the simplifications ! for @expr{x^0} and @expr{0^x} conflict when @expr{x=0}. ! 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 @tmath{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 @tmath{((-3)^{1/2})^2} ! @infoline @expr{((-3)^1:2)^2} ! is safe to simplify, but ! @texline @tmath{((-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, @address@hidden(x)^n} is simplified to ! @texline @tmath{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 @address@hidden(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 @address@hidden(x)} generally act like one-half powers ! @texline @tmath{x^{1:2}} ! @infoline @expr{x^1:2} ! for the purposes of the above-listed simplifications. ! ! Also, note that ! @texline @tmath{1 / x^{1:2}} ! @infoline @expr{1 / x^1:2} ! is changed to ! @texline @tmath{x^{-1:2}}, ! @infoline @expr{x^(-1:2)}, ! but @expr{1 / @t{sqrt}(x)} is left alone. @tex \bigskip @end tex Generic identity matrices (@pxref{Matrix Mode}) are simplified by the ! following rules: @address@hidden(a) + b} to @expr{a + b} if @expr{b} ! is provably scalar, or expanded out if @expr{b} is a matrix; ! @address@hidden(a) + @t{idn}(b)} to @address@hidden(a + b)}; ! @address@hidden(a)} to @address@hidden(-a)}; @expr{a @t{idn}(b)} to ! @address@hidden(a b)} if @expr{a} is provably scalar, or to @expr{a b} ! if @expr{a} is provably non-scalar; @address@hidden(a) @t{idn}(b)} to ! @address@hidden(a b)}; analogous simplifications for quotients involving ! @code{idn}; and @address@hidden(a)^n} to @address@hidden(a^n)} where ! @expr{n} is an integer. @tex \bigskip *************** *** 22234,22259 **** The @code{floor} function and other integer truncation functions vanish if the argument is provably integer-valued, so that ! @address@hidden(@t{round}(x))} simplifies to @address@hidden(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 @address@hidden(-x)} changes to @address@hidden(x)}. ! The expression @address@hidden(@t{abs}(x))} changes to @address@hidden(x)}; ! in fact, @address@hidden(x)} changes to @cite{x} or @cite{-x} if @cite{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 ! @address@hidden(@t{i})} and @address@hidden(address@hidden)} just for convenience. ! The expression @address@hidden(@t{conj}(x))} simplifies to @cite{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, ! @address@hidden(a + b i)} is changed to @address@hidden(a) - @t{conj}(b) i}, ! or to @cite{a - b i} if @cite{a} and @cite{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 --- 22476,22503 ---- The @code{floor} function and other integer truncation functions vanish if the argument is provably integer-valued, so that ! @address@hidden(@t{round}(x))} simplifies to @address@hidden(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 @address@hidden(-x)} changes to @address@hidden(x)}. ! The expression @address@hidden(@t{abs}(x))} changes to ! @address@hidden(x)}; in fact, @address@hidden(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 ! @address@hidden(@t{i})} and @address@hidden(address@hidden)} just for convenience. ! The expression @address@hidden(@t{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, ! @address@hidden(a + b i)} is changed to ! @address@hidden(a) - @t{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 *************** *** 22261,22278 **** described in the next section does provide some simplifications for these functions, though. ! One important simplification that does occur is that @address@hidden(@t{e})} ! is simplified to 1, and @address@hidden(@t{e}^x)} is simplified to @cite{x} ! for any @cite{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, @t{(@var{a} <= @var{b})} changes to @address@hidden > @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 ! @cite{!!x} to @cite{x}, this would not be valid in general because ! @cite{!!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 --- 22505,22522 ---- described in the next section does provide some simplifications for these functions, though. ! One important simplification that does occur is that ! @address@hidden(@t{e})} is simplified to 1, and @address@hidden(@t{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, @t{(@var{a} <= @var{b})} changes to @address@hidden > @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 *************** *** 22309,22321 **** @end tex Sums are simplified in two ways. Constant terms are commuted to the ! end of the sum, so that @cite{a + 2 + b} changes to @cite{a + b + 2}. The only exception is that a constant will not be commuted away ! from the first position of a difference, i.e., @cite{2 - x} is not ! commuted to @cite{-x + 2}. Also, terms of sums are combined by the distributive law, as in ! @cite{x + y + 2 x} to @cite{y + 3 x}. This always occurs for adjacent terms, but @kbd{a s} compares all pairs of terms including non-adjacent ones. --- 22553,22565 ---- @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. *************** *** 22324,22333 **** @end tex Products are sorted into a canonical order using the commutative ! law. For example, @cite{b c a} is commuted to @cite{a b c}. This allows easier comparison of products; for example, the default ! simplifications will not change @cite{x y + y x} to @cite{2 x y}, ! but @kbd{a s} will; it first rewrites the sum to @cite{x y + x y}, and then the default simplifications are able to recognize a sum of identical terms. --- 22568,22577 ---- @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. *************** *** 22346,22360 **** Even though sums are not sorted, the commutative law is still taken into account when terms of a product are being compared. ! Thus @cite{(x + y) (y + x)} will be simplified to @cite{(x + y)^2}. ! A subtle point is that @cite{(x - y) (y - x)} will @emph{not} ! be simplified to @cite{-(x - y)^2}; Calc does not notice that one term can be written as a constant times the other, even if that constant is @i{-1}. ! A fraction times any expression, @cite{(a:b) x}, is changed to ! a quotient involving integers: @cite{a x / b}. This is not ! done for floating-point numbers like @cite{0.5}, however. This is one reason why you may find it convenient to turn Fraction mode on while doing algebra; @pxref{Fraction Mode}. --- 22590,22604 ---- 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 @i{-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}. *************** *** 22364,22388 **** 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, @cite{a x^2 b / c x^3 d} will ! cancel @cite{x^2} from both sides to get @cite{a b / c x d}. ! (The terms in the denominator will then be rearranged to @cite{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, @cite{(4 x + 6) / 8 x} simplifies to @cite{(2 x + 3) / 4 x}. Non-constant common factors are not found even by @kbd{a s}. To ! cancel the factor @cite{a} in @cite{(a x + a) / a^2} you could first ! use @kbd{j M} on the product @cite{a x} to Merge the numerator to ! @cite{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 @cite{i^2 = -1}. If you store a new value other ! than the complex number @cite{(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 --- 22608,22632 ---- 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 both sides 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 *************** *** 22392,22417 **** 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 @address@hidden(8)} is rewritten as ! @c{2\,\t{sqrt}(2)} ! @cite{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: @cite{1 / @t{sqrt}(3)} changes to @address@hidden(3) / 3}. The same effect occurs for the square root of a fraction: ! @address@hidden(2:3)} changes to @address@hidden(6) / 3}. @tex \bigskip @end tex The @code{%} (modulo) operator is simplified in several ways ! when the modulus @cite{M} is a positive real number. First, if ! the argument is of the form @cite{x + n} for some real number ! @cite{n}, then @cite{n} is itself reduced modulo @cite{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 --- 22636,22661 ---- 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 @address@hidden(8)} is rewritten as ! @texline @tmath{2\,\t{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 / @t{sqrt}(3)} changes to @address@hidden(3) / 3}. The same effect occurs for the square root of a fraction: ! @address@hidden(2:3)} changes to @address@hidden(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 *************** *** 22435,22450 **** @end tex Trigonometric functions are simplified in several ways. First, ! @address@hidden(@t{arcsin}(x))} is simplified to @cite{x}, and similarly for @code{cos} and @code{tan}. If the argument to ! @code{sin} is negative-looking, it is simplified to @address@hidden(x)}, ! and similarly for @code{cos} and @code{tan}. Finally, certain ! special values of the argument are recognized; @pxref{Trigonometric and Hyperbolic Functions}. Trigonometric functions of inverses of different trigonometric ! functions can also be simplified, as in @address@hidden(@t{arccos}(x))} ! to @address@hidden(1 - x^2)}. Hyperbolic functions of their inverses and of negative-looking arguments are also handled, as are exponentials of inverse --- 22679,22694 ---- @end tex Trigonometric functions are simplified in several ways. First, ! @address@hidden(@t{arcsin}(x))} is simplified to @expr{x}, and similarly for @code{cos} and @code{tan}. If the argument to ! @code{sin} is negative-looking, it is simplified to ! @address@hidden(x),}, and similarly for @code{cos} and @code{tan}. ! Finally, certain special values of the argument are recognized; @pxref{Trigonometric and Hyperbolic Functions}. Trigonometric functions of inverses of different trigonometric ! functions can also be simplified, as in @address@hidden(@t{arccos}(x))} ! to @address@hidden(1 - x^2)}. Hyperbolic functions of their inverses and of negative-looking arguments are also handled, as are exponentials of inverse *************** *** 22453,22478 **** 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 ! @address@hidden(@t{sin}(x))} can @emph{not} safely change to ! @cite{x}, since this only correct within an integer multiple ! of @c{2 \pi} ! @cite{2 pi} radians or 360 degrees. However, ! @address@hidden(@t{sinh}(x))} is simplified to @cite{x} if ! @cite{x} is known to be real. Several simplifications that apply to logarithms and exponentials ! are that @address@hidden(@t{ln}(x))}, @address@hidden(x)}} ! @address@hidden(x)}, and ! @c{10^{{\rm log10}(x)}} ! @address@hidden(x)} all reduce to @cite{x}. ! Also, @address@hidden(@t{exp}(x))}, etc., can reduce to @cite{x} if ! @cite{x} is provably real. The form @address@hidden(x)^y} is simplified ! to @address@hidden(x y)}. If @cite{x} is a suitable multiple of @c{\pi i} ! @cite{pi i} ! (as described above for the trigonometric functions), then @address@hidden(x)} ! or @cite{e^x} will be expanded. Finally, @address@hidden(x)} is simplified ! to a form involving @code{pi} and @code{i} where @cite{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} --- 22697,22727 ---- 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 ! @address@hidden(@t{sin}(x))} can @emph{not} safely change to ! @expr{x}, since this only correct within an integer multiple of ! @texline @tmath{2 \pi} ! @infoline @expr{2 pi} ! radians or 360 degrees. However, @address@hidden(@t{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 @address@hidden(@t{ln}(x))}, ! @texline @address@hidden(x)}}, ! @infoline @address@hidden(x)}, ! and ! @texline @tmath{10^{{\rm log10}(x)}} ! @infoline @address@hidden(x)} ! all reduce to @expr{x}. Also, @address@hidden(@t{exp}(x))}, etc., can ! reduce to @expr{x} if @expr{x} is provably real. The form ! @address@hidden(x)^y} is simplified to @address@hidden(x y)}. If @expr{x} ! is a suitable multiple of ! @texline @tmath{\pi i} ! @infoline @expr{pi i} ! (as described above for the trigonometric functions), then ! @address@hidden(x)} or @expr{e^x} will be expanded. Finally, ! @address@hidden(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} *************** *** 22485,22491 **** 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 @cite{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 --- 22734,22740 ---- 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 *************** *** 22493,22503 **** Simplification also replaces an equation or inequality with 1 or 0 (true'' or false'') if it can through the use of ! declarations. If @cite{x} is declared to be an integer greater ! than 5, then @cite{x < 3}, @cite{x = 3}, and @cite{x = 7.5} are ! all simplified to 0, but @cite{x > 3} is simplified to 1. ! By a similar analysis, @cite{abs(x) >= 0} is simplified to 1, ! as is @cite{x^2 >= 0} if @cite{x} is known to be real. @node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas @subsection Unsafe'' Simplifications --- 22742,22752 ---- 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 *************** *** 22519,22526 **** 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 @cite{a} only,'' Calc (like most other symbolic ! integration programs) will simply produce an unqualified address@hidden 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 --- 22768,22775 ---- 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 *************** *** 22542,22586 **** Inverse trigonometric or hyperbolic functions, called with their corresponding non-inverse functions as arguments, are simplified ! by @kbd{a e}. For example, @address@hidden(@t{sin}(x))} changes ! to @cite{x}. Also, @address@hidden(@t{cos}(x))} and ! @address@hidden(@t{sin}(x))} both change to @address@hidden/2 - x}. These simplifications are unsafe because they are valid only for ! values of @cite{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 @cite{(x^a)^b} are simplified to @c{x^{a b}} ! @cite{x^(a b)} ! for all @cite{a} and @cite{b}. These results will be valid only ! in a restricted range of @cite{x}; for example, in @c{(x^2)^{1:2}} ! @cite{(x^2)^1:2} ! the powers cancel to get @cite{x}, which is valid for positive values ! of @cite{x} but not for negative or complex values. ! ! Similarly, @address@hidden(x^a)} and @address@hidden(x)^a} are both ! simplified (possibly unsafely) to @c{x^{a/2}} ! @cite{x^(a/2)}. ! Forms like @address@hidden(1 - @t{sin}(x)^2)} are simplified to, e.g., ! @address@hidden(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, ! @address@hidden(a^2 b^3 + a^3 b^2)} simplifies to @cite{a b @t{sqrt}(a+b)}. ! The simplifications of @address@hidden(@t{exp}(x))}, @address@hidden(@t{e}^x)}, ! and @address@hidden(10^x)} to @cite{x} are also unsafe because ! of problems with principal values (although these simplifications ! are safe if @cite{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: @cite{a x / b x} ! to @cite{a / b}. Such factors are never cancelled from inequalities: Even @kbd{a e} is not bold enough to reduce ! @cite{a x < b x} to @cite{a < b} (or @cite{a > b}, depending ! on whether you believe @cite{x} is positive or negative). The @kbd{a M /} command can be used to divide a factor out of both sides of an inequality. --- 22791,22839 ---- Inverse trigonometric or hyperbolic functions, called with their corresponding non-inverse functions as arguments, are simplified ! by @kbd{a e}. For example, @address@hidden(@t{sin}(x))} changes ! to @expr{x}. Also, @address@hidden(@t{cos}(x))} and ! @address@hidden(@t{sin}(x))} both change to @address@hidden/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 @tmath{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 @tmath{(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, @address@hidden(x^a)} and @address@hidden(x)^a} are both ! simplified (possibly unsafely) to ! @texline @tmath{x^{a/2}}. ! @infoline @expr{x^(a/2)}. ! Forms like @address@hidden(1 - sin(x)^2)} are simplified to, e.g., ! @address@hidden(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, ! @address@hidden(a^2 b^3 + a^3 b^2)} simplifies to ! @expr{a b @t{sqrt}(a+b)}. ! The simplifications of @address@hidden(@t{exp}(x))}, ! @address@hidden(@t{e}^x)}, and @address@hidden(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. *************** *** 22600,22611 **** Scalar mode is automatically put into effect when simplifying units. @xref{Matrix Mode}. ! Sums @cite{a + b} involving units are simplified by extracting the ! units of @cite{a} as if by the @kbd{u x} command (call the result ! @cite{u_a}), then simplifying the expression @cite{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 @cite{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 --- 22853,22864 ---- 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 *************** *** 22617,22655 **** However, compatible but different units like @code{ft} and @code{in} are not combined in this way. ! Quotients @cite{a / b} are simplified in three additional ways. First, ! if @cite{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 @cite{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 ! @cite{(a b)^c} to @cite{a^c b^c}, @cite{(a/b)^c} to @cite{a^c / b^c}, ! and @cite{(a^b)^c} to @c{a^{b c}} ! @cite{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 @cite{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 @cite{3:2}. Thus, @code{acre^1.5} is ! replaced by approximately @c{(4046 m^2)^{1.5}} ! @cite{(4046 m^2)^1.5}, which is then ! changed to @c{4046^{1.5} \, (m^2)^{1.5}} ! @cite{4046^1.5 (m^2)^1.5}, then to @cite{257440 m^3}. The functions @code{float}, @code{frac}, @code{clean}, @code{abs}, as well as @code{floor} and the other integer truncation functions, --- 22870,22913 ---- 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 @tmath{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 @tmath{(4046 m^2)^{1.5}} ! @infoline @expr{(4046 m^2)^1.5}, ! which is then changed to ! @texline @tmath{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, *************** *** 22667,22676 **** @section Polynomials A @dfn{polynomial} is a sum of terms which are coefficients times ! various powers of a base'' variable. For example, @cite{2 x^2 + 3 x - 4} ! is a polynomial in @cite{x}. Some formulas can be considered ! polynomials in several different variables: @cite{1 + 2 x + 3 y + 4 x y^2} ! is a polynomial in both @cite{x} and @cite{y}. Polynomial coefficients are often numbers, but they may in general be any formulas not involving the base variable. --- 22925,22934 ---- @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. *************** *** 22679,22687 **** @tindex factor The @kbd{a f} (@code{calc-factor}) address@hidden command factors a polynomial into a product of terms. For example, the polynomial ! @cite{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another ! example, @cite{a c + b d + b c + a d} is factored into the product ! @cite{(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 --- 22937,22945 ---- @tindex factor The @kbd{a f} (@code{calc-factor}) address@hidden 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 *************** *** 22689,22695 **** 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 ! (@cite{x}, @cite{x+1}, and @cite{x+1} again). Finally, formulas which do not fit the above criteria are handled by the algebraic rewrite mechanism. --- 22947,22953 ---- 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. *************** *** 22721,22727 **** @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, ! @cite{(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 --- 22979,22985 ---- @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 *************** *** 22742,22754 **** 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, ! @cite{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @cite{(x + 7) x^2 (x - 3)^2} ! in @kbd{a f}, or to @cite{[ [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 @cite{x} with ! respect to the specific variable @cite{v}. The default is to factor with ! respect to all the variables that appear in @cite{x}. @kindex a c @pindex calc-collect --- 23000,23012 ---- 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 *************** *** 22756,22767 **** The @kbd{a c} (@code{calc-collect}) address@hidden command rearranges a formula as a polynomial in a given variable, ordered in decreasing powers of that ! variable. For example, given @cite{1 + 2 x + 3 y + 4 x y^2} on ! the stack, @kbd{a c x} would produce @cite{(2 + 4 y^2) x + (1 + 3 y)}, ! and @kbd{a c y} would produce @cite{(4 x) y^2 + 3 y + (1 + 2 x)}. The polynomial will be expanded out using the distributive law as ! necessary: Collecting @cite{x} in @cite{(x - 1)^3} produces ! @cite{x^3 - 3 x^2 + 3 x - 1}. Terms not involving @cite{x} will not be expanded. The variable'' you specify at the prompt can actually be any --- 23014,23025 ---- The @kbd{a c} (@code{calc-collect}) address@hidden 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 *************** *** 22790,22799 **** do not do.) Calc's automatic simplifications will sometimes reverse a partial ! expansion. For example, the first step in expanding @cite{(x+1)^3} is ! to write @cite{(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 @cite{(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. --- 23048,23057 ---- 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. *************** *** 22814,22834 **** @tindex nrat The @kbd{a n} (@code{calc-normalize-rat}) address@hidden command attempts to arrange a formula into a quotient of two polynomials. ! For example, given @cite{1 + (a + b/c) / d}, the result would be ! @cite{(b + a c + c d) / c d}. The quotient is reduced, so that ! @kbd{a n} will simplify @cite{(x^2 + 2x + 1) / (x^2 - 1)} by dividing ! out the common factor @cite{x + 1}, yielding @cite{(x + 1) / (x - 1)}. @kindex a \ @pindex calc-poly-div @tindex pdiv The @kbd{a \} (@code{calc-poly-div}) address@hidden command divides ! two polynomials @cite{u} and @cite{v}, yielding a new polynomial ! @cite{q}. If several variables occur in the inputs, the inputs are considered multivariate polynomials. (Calc divides by the variable ! with the largest power in @cite{u} first, or, in the case of equal powers, chooses the variables in alphabetical order.) For example, ! dividing @cite{x^2 + 3 x + 2} by @cite{x + 2} yields @cite{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. --- 23072,23092 ---- @tindex nrat The @kbd{a n} (@code{calc-normalize-rat}) address@hidden 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}) address@hidden 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. *************** *** 22842,22850 **** @pindex calc-poly-rem @tindex prem The @kbd{a %} (@code{calc-poly-rem}) address@hidden command divides ! two polynomials and keeps the remainder @cite{r}. The quotient ! @cite{q} is discarded. For any formulas @cite{a} and @cite{b}, the ! results of @kbd{a \} and @kbd{a %} satisfy @cite{a = q b + r}. (This is analogous to plain @kbd{\} and @kbd{%}, which compute the integer quotient and remainder from dividing two numbers.) --- 23100,23108 ---- @pindex calc-poly-rem @tindex prem The @kbd{a %} (@code{calc-poly-rem}) address@hidden 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.) *************** *** 22855,22864 **** @tindex pdivide The @kbd{a /} (@code{calc-poly-div-rem}) address@hidden command divides two polynomials and reports both the quotient and the ! remainder as a vector @cite{[q, r]}. The @kbd{H a /} address@hidden command divides two polynomials and constructs the formula ! @cite{q + r/b} on the stack. (Naturally if the remainder is zero, ! this will immediately simplify to @cite{q}.) @kindex a g @pindex calc-poly-gcd --- 23113,23122 ---- @tindex pdivide The @kbd{a /} (@code{calc-poly-div-rem}) address@hidden command divides two polynomials and reports both the quotient and the ! remainder as a vector @expr{[q, r]}. The @kbd{H a /} address@hidden 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 *************** *** 22936,22943 **** 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 @cite{f} with respect to @cite{x}, evaluated at the point @c{x=x_0} ! @cite{x=x0}. If the formula being differentiated contains functions which Calc does not know, the derivatives of those functions are produced by adding --- 23194,23202 ---- 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 @tmath{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 *************** *** 22960,22966 **** 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)address@hidden @node Integration, Customizing the Integrator, Differentiation, Calculus @subsection Integration --- 23219,23225 ---- 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 *************** *** 22975,22985 **** 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; @c{x/(1+x^{-2})} ! @cite{x/(1+x^-2)} is not strictly a quotient of polynomials, but it is equivalent to ! @cite{x^3/(x^2+1)}, which is.) Also, square roots of terms involving ! @cite{x} and @cite{x^2} may appear in rational functions being integrated. Finally, rational functions involving trigonometric or hyperbolic functions can be integrated. --- 23234,23245 ---- 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 @tmath{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. *************** *** 23000,23016 **** 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 @c{1/(x+\sqrt{x^2+1})} ! @cite{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 @c{1/(1+\tan x)} ! @cite{1/(1+tan(x))} differs from the solution given ! in the @emph{CRC Math Tables} by a constant factor of @c{\pi i / 2} ! @cite{pi i / 2}, due to a different choice of constant of integration. The Calculator remembers all the integrals it has done. If conditions --- 23260,23280 ---- 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 @tmath{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 @tmath{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 @tmath{\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 *************** *** 23068,23075 **** @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 @c{{\rm Ei}(x)} ! @cite{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 --- 23332,23341 ---- @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 @tmath{{\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 *************** *** 23215,23221 **** 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 address@hidden 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 --- 23481,23487 ---- 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 *************** *** 23232,23259 **** @cindex Solving equations The @kbd{a S} (@code{calc-solve-for}) address@hidden command rearranges an equation to solve for a specific variable. An equation is an ! expression of the form @cite{L = R}. For example, the command @kbd{a S x} ! will rearrange @cite{y = 3x + 6} to the form, @cite{x = y/3 - 2}. If the input is not an equation, it is treated like an equation of the ! form @cite{X = 0}. ! This command also works for inequalities, as in @cite{y < 3x + 6}. Some inequalities cannot be solved where the analogous equation could ! be; for example, solving @c{a < b \, c} ! @cite{a < b c} for @cite{b} is impossible ! without knowing the sign of @cite{c}. In this case, @kbd{a S} will ! produce the result @c{b \mathbin{\hbox{\code{!=}}} a/c} ! @cite{b != a/c} (using the not-equal-to operator) ! to signify that the direction of the inequality is now unknown. The ! inequality @c{a \le b \, c} ! @cite{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 @cite{x = y/3 - 2} ! to @cite{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates ! another formula with @cite{x} set equal to @cite{y/3 - 2}. @menu * Multiple Solutions:: --- 23498,23529 ---- @cindex Solving equations The @kbd{a S} (@code{calc-solve-for}) address@hidden 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 @tmath{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 @tmath{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 @tmath{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:: *************** *** 23293,23299 **** 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 @cite{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}. --- 23563,23569 ---- 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}. *************** *** 23337,23343 **** With the Inverse flag, @kbd{I a S} address@hidden 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 @cite{2x + 6} to @cite{x/2 - 3}. You can use both Inverse and Hyperbolic address@hidden to obtain a fully general inverse, as described above. --- 23607,23613 ---- With the Inverse flag, @kbd{I a S} address@hidden 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 address@hidden to obtain a fully general inverse, as described above. *************** *** 23361,23372 **** 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 @cite{x^4} can always be solved exactly; polynomials of higher degree sometimes ! can be: @cite{x^6 + x^3 + 1} is converted to @cite{(x^3)^2 + (x^3) + 1}, ! which can be solved for @cite{x^3} using the quadratic equation, and then ! for @cite{x} by taking cube roots. But in many cases, like ! @cite{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 --- 23631,23642 ---- 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 *************** *** 23392,23398 **** 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 ! @cite{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 --- 23662,23668 ---- 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 *************** *** 23453,23475 **** 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 ! @cite{[0, 2, 0, 1]}. If the input is not a polynomial in @cite{x}, the call to @code{poly} is left in symbolic form. If the input does ! not involve the variable @cite{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 @cite{[0, 1]}. The last element of the returned vector is guaranteed to be nonzero; ! note that @samp{poly(0, x)} returns the empty vector @cite{[]}. ! Note also that @cite{x} may actually be any formula; for example, ! @samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @cite{[3, -1, 1]}. @cindex Coefficients of polynomial @cindex Degree of polynomial ! To get the @cite{x^k} coefficient of polynomial @cite{p}, use ! @samp{poly(p, x)_(k+1)}. To get the degree of polynomial @cite{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 @cite{x^2} coefficient of this polynomial, 6. @ignore @starindex --- 23723,23745 ---- 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 *************** *** 23494,23506 **** 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 @cite{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 @cite{x^2} is easier to solve than ! a quartic polynomial in @cite{x}. A few more examples of the kinds of polynomials @code{gpoly} can discover: --- 23764,23776 ---- 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: *************** *** 23517,23523 **** 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 @cite{n}, then only @var{c} vectors of length @cite{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 --- 23787,23793 ---- 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 *************** *** 23549,23555 **** 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 @cite{p = 0}. @ignore @starindex --- 23819,23825 ---- 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 *************** *** 23618,23624 **** The @kbd{a R} (@code{calc-find-root}) address@hidden 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 @cite{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 --- 23888,23894 ---- The @kbd{a R} (@code{calc-find-root}) address@hidden 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 *************** *** 23704,23725 **** with the minimum value itself. Note that this command looks for a @emph{local} minimum. Many functions ! have more than one minimum; some, like @c{x \sin x} ! @cite{x sin(x)}, have infinitely ! many. In fact, there is no easy way to define the global'' minimum ! of @c{x \sin x} ! @cite{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 @cite{17 x} ! over @cite{[2..3]} will return @cite{[2, 38]}, but minimizing over ! @cite{(2..3]} would report no minimum found. In general, you should use closed intervals to find literally the minimum value in that ! range of @cite{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 --- 23974,23998 ---- 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 @tmath{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 @tmath{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 *************** *** 23781,23789 **** @noindent The @kbd{a F} command fits a set of data to a @dfn{model formula}, ! such as @cite{y = m x + b} where @cite{m} and @cite{b} are parameters to be determined. For a typical set of measured data there will be ! no single @cite{m} and @cite{b} that exactly fit the data; in this case, Calc chooses values of the parameters that provide the closest possible fit. --- 24054,24062 ---- @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. *************** *** 23806,23851 **** @cindex Linear regression @cindex Least-squares fits The @kbd{a F} (@code{calc-curve-fit}) address@hidden command attempts ! to fit a set of data (@cite{x} and @cite{y} vectors of numbers) to a ! straight line, polynomial, or other function of @cite{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 @cite{(x,y)} ! data points that we wish to fit to the model @cite{y = m x + b} ! by adjusting the parameters @cite{m} and @cite{b} to make the @cite{y} values calculated from the formula be as close as possible to the actual ! @cite{y} values in the data set. (In a polynomial fit, the model is ! instead, say, @cite{y = a x^3 + b x^2 + c x + d}. In a multilinear fit, ! we have data points of the form @cite{(x_1,x_2,x_3,y)} and our model is ! @cite{y = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.) ! ! In the model formula, variables like @cite{x} and @cite{x_2} are called ! the @dfn{independent variables}, and @cite{y} is the @dfn{dependent ! variable}. Variables like @cite{m}, @cite{a}, and @cite{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 @c{2\times N} ! @asis{2xN} matrix where ! the first row is a list of @cite{x} values and the second row has the ! corresponding @cite{y} values. For the multilinear fit shown above, ! the matrix would have four rows (@cite{x_1}, @cite{x_2}, @cite{x_3}, and ! @cite{y}, respectively). ! ! If you happen to have an @c{N\times2} ! @asis{Nx2} matrix instead of a @c{2\times N} ! @asis{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 @cite{x} and @cite{y} for independent variables and ! low letters like @cite{a} and @cite{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 --- 24079,24128 ---- @cindex Linear regression @cindex Least-squares fits The @kbd{a F} (@code{calc-curve-fit}) address@hidden 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 @tmath{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 @tmath{N\times2} ! @infoline Nx2 ! matrix instead of a ! @texline @tmath{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 *************** *** 23874,23890 **** @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 @cite{3 + 2 x} on the stack. Calc has created the model expression @kbd{a + b x}, ! then found the optimal values of @cite{a} and @cite{b} to fit the data. (In this case, it was able to find an exact fit.) Calc then ! substituted those values for @cite{a} and @cite{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: ! @cite{[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}. --- 24151,24167 ---- @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}. *************** *** 23930,23946 **** @end tex @noindent ! which is clearly zero if @cite{a + b x} exactly fits all data points, ! and increases as various @cite{a + b x_i} values fail to match the ! corresponding @cite{y_i} values. There are several reasons why the ! summand is squared, one of them being to ensure that @c{\chi^2 \ge 0} ! @cite{chi^2 >= 0}. ! Least-squares fitting simply chooses the values of @cite{a} and @cite{b} ! for which the error @c{\chi^2} ! @cite{chi^2} is as small as possible. Other kinds of models do the same thing but with a different model ! formula in place of @cite{a + b x_i}. @tex \bigskip --- 24207,24226 ---- @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 @tmath{\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 @tmath{\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 *************** *** 23953,23961 **** 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 @cite{x} values, and a ! vector of @cite{y} values. If there is only one independent variable, ! the @cite{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 @address@hidden 2}} prefix. @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting --- 24233,24241 ---- 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 @address@hidden 2}} prefix. @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting *************** *** 23974,23988 **** 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 @cite{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 @cite{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 @cite{x^2} contribution, as Calc bends the line slightly to improve the fit. @example --- 24254,24268 ---- 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 *************** *** 24000,24006 **** @end example The actual coefficients we get with a precision of 12, like ! @cite{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 --- 24280,24286 ---- @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 *************** *** 24023,24030 **** @end tex Another generalization of the linear model is to assume the ! @cite{y} values are a sum of linear contributions from several ! @cite{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.) --- 24303,24310 ---- @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.) *************** *** 24039,24047 **** @end example @noindent ! the command @kbd{a F 1 @key{RET}} will call the first row @cite{x} and the ! second row @cite{y}, and will fit the values in the third row to the ! model @cite{a + b x + c y}. @example 8. + 3. x + 0.5 y --- 24319,24327 ---- @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 *************** *** 24056,24075 **** 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 @cite{a x}; in the multilinear ! case, the model might be @cite{a x + b y + c z}; and in the polynomial ! case, the model could be @cite{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 @cite{2.3 + a x} or @cite{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., @cite{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 --- 24336,24355 ---- 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 *************** *** 24096,24104 **** 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 ! address@hidden@w{ @t{+/-} address@hidden ! @var{sigma_i}', then the @c{\chi^2} ! @cite{chi^2} statistic is now, @ifinfo --- 24376,24386 ---- 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 address@hidden@w{ @t{+/-} address@hidden', ! @infoline address@hidden@w{ @t{+/-} address@hidden', ! then the ! @texline @tmath{\chi^2} ! @infoline @expr{chi^2} statistic is now, @ifinfo *************** *** 24119,24142 **** 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 @c{\sigma_i} ! @cite{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 @c{\sigma_i} ! @cite{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 (@c{\chi^2} ! @cite{chi^2} ! is simply scaled uniformly by @c{1 / \sigma^2} ! @cite{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 --- 24401,24429 ---- 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 @tmath{\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 @tmath{\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 (@tmath{\chi^2} ! @infoline (@expr{chi^2} ! is simply scaled uniformly by ! @texline @tmath{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 *************** *** 24161,24214 **** 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 @cite{d}, this vector ! will have length @cite{M = d+1} with the constant term first. @item ! The covariance matrix @cite{C} computed from the fit. This is an @address@hidden symmetric matrix; the diagonal elements ! @c{C_{jj}} ! @cite{C_j_j} are the variances @c{\sigma_j^2} ! @cite{sigma_j^2} of the parameters. ! The other elements are covariances @c{\sigma_{ij}^2} ! @cite{sigma_i_j^2} that describe the ! correlation between pairs of parameters. (A related set of ! numbers, the @dfn{linear correlation coefficients} @c{r_{ij}} ! @cite{r_i_j}, ! are defined as @c{\sigma_{ij}^2 / \sigma_i \, \sigma_j} ! @cite{sigma_i_j^2 / sigma_i sigma_j}.) @item ! A vector of @cite{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 @c{\chi^2} ! @cite{chi^2} for the fit, calculated by the formulas ! shown above. This gives a measure of the quality of the fit; ! statisticians consider @c{\chi^2 \approx N - M} ! @cite{chi^2 = N - M} to indicate a moderately good fit ! (where again @cite{N} is the number of data points and @cite{M} ! is the number of parameters). @item ! A measure of goodness of fit expressed as a probability @cite{Q}. This is computed from the @code{utpc} probability distribution ! function using @c{\chi^2} ! @cite{chi^2} with @cite{N - M} degrees of freedom. A value of 0.5 implies a good fit; some texts recommend that often ! @cite{Q = 0.1} or even 0.001 can signify an acceptable fit. In ! particular, @c{\chi^2} ! @cite{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 @cite{Q}. ! The @cite{Q} value is computed only if the input included error estimates. Otherwise, Calc will report the symbol @code{nan} ! for @cite{Q}. The reason is that in this case the @c{\chi^2} ! @cite{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. --- 24448,24513 ---- 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 @address@hidden symmetric matrix; the diagonal elements ! @texline @tmath{C_{jj}} ! @infoline @expr{C_j_j} ! are the variances ! @texline @tmath{\sigma_j^2} ! @infoline @expr{sigma_j^2} ! of the parameters. The other elements are covariances ! @texline @tmath{\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 @tmath{r_{ij}}, ! @infoline @expr{r_i_j}, ! are defined as ! @texline @tmath{\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 @tmath{\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 @tmath{\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 @tmath{\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 @tmath{\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 @tmath{\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. *************** *** 24248,24255 **** @item q Quadratic. @i{a + b (x-c)^2 + d (x-e)^2}. @item g ! Gaussian. @c{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)} ! @i{(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 --- 24547,24555 ---- @item q Quadratic. @i{a + b (x-c)^2 + d (x-e)^2}. @item g ! Gaussian. ! @texline @tmath{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}. ! @infoline @i{(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 *************** *** 24260,24271 **** 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 @cite{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, @cite{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 --- 24560,24571 ---- 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 *************** *** 24275,24282 **** 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, ! @cite{a} is what the HP-48 identifies as the intercept,'' and ! @cite{b} is what it calls the slope.'' @tex \bigskip --- 24575,24582 ---- 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 *************** *** 24287,24299 **** 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 @cite{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 ! (@cite{m} and @cite{b}) and two independent variables (@cite{x} ! and @cite{y}), with no dependent'' variables. Model equations ! do not need to take this @cite{y =} form. For example, the ! implicit line equation @cite{a x + b y = 1} works fine as a model. When you enter a model, Calc makes an alphabetical list of all --- 24587,24599 ---- 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 *************** *** 24303,24314 **** 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 @cite{a,mu,sigma,t,x}, and the data matrix has three rows (meaning two independent variables), ! Calc will use @cite{a,mu,sigma} as the default parameters, and the ! data rows will be named @cite{t} and @cite{x}, respectively. If you ! enter an equation instead of a plain formula, Calc will use @cite{a,mu} ! as the parameters, and @cite{sigma,t,x} as the three independent variables. You can, of course, override these choices by entering something --- 24603,24614 ---- 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 *************** *** 24332,24342 **** which are parameters. Models taken from the stack can also be expressed as vectors of ! two or three elements, @address@hidden, @var{vars}]} or ! @address@hidden, @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.)@refill 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. --- 24632,24642 ---- which are parameters. Models taken from the stack can also be expressed as vectors of ! two or three elements, @address@hidden, @var{vars}]} or ! @address@hidden, @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. *************** *** 24364,24377 **** returns results in the range from @i{-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 @c{3\times360} ! @i{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 @cite{t}, but the lefthand side will bounce back and forth between @i{-90} and 90. ! No values of @cite{a} and @cite{b} can make the two sides match, even approximately. There is no good solution to this problem at present. You could --- 24664,24679 ---- returns results in the range from @i{-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 @tmath{3\times360} ! @infoline @i{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 @i{-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 *************** *** 24388,24401 **** @noindent Calc's internal least-squares fitter can only handle multilinear models. More precisely, it can handle any model of the form ! @cite{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @cite{a,b,c} ! are the parameters and @cite{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 @cite{a + b x + c x^2}, then @cite{f(x) = 1}, @cite{g(x) = x}, ! and @cite{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 --- 24690,24703 ---- @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 *************** *** 24405,24419 **** @end smallexample @noindent ! where @cite{Y,A,B,C,F,G,H} are arbitrary functions. It computes ! @cite{Y}, @cite{F}, @cite{G}, and @cite{H} for all the data points, ! does a standard linear fit to find the values of @cite{A}, @cite{B}, ! and @cite{C}, then uses the equation solver to solve for @cite{a,b,c} ! in terms of @cite{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 @cite{y = a x^b} with two independent variables and two parameters can be rewritten as follows: @example --- 24707,24721 ---- @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 *************** *** 24424,24438 **** @end example @noindent ! which matches the desired form with @c{Y = \ln(y)} ! @cite{Y = ln(y)}, @c{A = \ln(a)} ! @cite{A = ln(a)}, ! @cite{F = 1}, @cite{B = b}, and @c{G = \ln(x)} ! @cite{G = ln(x)}. Calc thus computes ! the logarithms of your @cite{y} and @cite{x} values, does a linear fit ! for @cite{A} and @cite{B}, then solves to get @c{a = \exp(A)} ! @cite{a = exp(A)} and ! @cite{b = B}. Another interesting example is the quadratic'' model, which can be handled by expanding according to the distributive law. --- 24726,24744 ---- @end example @noindent ! which matches the desired form with ! @texline @tmath{Y = \ln(y)}, ! @infoline @expr{Y = ln(y)}, ! @texline @tmath{A = \ln(a)}, ! @infoline @expr{A = ln(a)}, ! @expr{F = 1}, @expr{B = b}, and ! @texline @tmath{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 @tmath{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. *************** *** 24443,24469 **** @end example @noindent ! which matches with @cite{Y = y}, @cite{A = a + b c^2}, @cite{F = 1}, ! @cite{B = -2 b c}, @cite{G = x} (the @i{-2} factor could just as easily ! have been put into @cite{G} instead of @cite{B}), @cite{C = b}, and ! @cite{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 @cite{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., ! @cite{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 ! address@hidden model can be linearized by setting @cite{d} ! (the background) to a constant, or by setting @cite{b} (the standard ! deviation) and @cite{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 --- 24749,24775 ---- @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 @i{-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 ! address@hidden 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 *************** *** 24475,24482 **** 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 @c{\chi^2} ! @cite{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} --- 24781,24789 ---- 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 @tmath{\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} *************** *** 24485,24493 **** 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 @c{\chi^2} ! @cite{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) --- 24792,24801 ---- 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 @tmath{\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) *************** *** 24496,24502 **** @noindent where @code{gaus} represents the Gaussian model with background, @code{data} represents the data matrix, and @code{guess} represents ! the initial guess for @cite{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). --- 24804,24810 ---- @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). *************** *** 24507,24523 **** The @kbd{I a F} address@hidden command is somewhat trickier when nonlinear models are used. The second item in the result is the ! vector of raw'' parameters @cite{A}, @cite{B}, @cite{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 @cite{A = a}, @cite{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 @cite{A}, @samp{fitdummy(2)} for @cite{B}, and so on. When Calc needs to modify the model to return the result, it replaces --- 24815,24831 ---- The @kbd{I a F} address@hidden 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 *************** *** 24537,24566 **** nontrivial filter functions. Things are also complicated when the input contains error forms. ! Suppose there are three independent and dependent variables, @cite{x}, ! @cite{y}, and @cite{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 @cite{x} ! and @cite{y} to be plain numbers, and makes @cite{z} into an error ! form with this combined error. The @cite{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 @c{\sigma_i} ! @cite{sigma_i} for ! the data point. If for some reason @cite{Y(x,y,z)} does not return ! an error form, the combined error from @cite{z} is used directly ! for @c{\sigma_i} ! @cite{sigma_i}. Finally, @cite{z} is also stripped of its error ! for use in computing @cite{F(x,y,z)}, @cite{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 @cite{Y(x,y,z)} ! depends only on the dependent variable @cite{z}, and in fact is ! often simply equal to @cite{z}. For common cases like polynomials and multilinear models, the combined error is simply used as the ! @c{\sigma} ! @cite{sigma} for the data point with no further ado.) @tex \bigskip --- 24845,24877 ---- 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 @tmath{\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 @tmath{\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 @tmath{\sigma} ! @infoline @expr{sigma} ! for the data point with no further ado.) @tex \bigskip *************** *** 24617,24623 **** @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 @cite{a x^b} is converted to @cite{y = a x^b}, then to @smallexample @group --- 24928,24934 ---- @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 *************** *** 24637,24647 **** @end example @noindent ! where @var{Y} is a formula that describes the function @cite{Y(x,y,z)}, ! @var{FGH} is the vector of formulas @cite{[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 @cite{A}, @samp{fitdummy(2)} ! for @cite{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. --- 24948,24958 ---- @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. *************** *** 24664,24670 **** 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 @cite{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. --- 24975,24981 ---- 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. *************** *** 24672,24679 **** 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 @cite{a=b} ! to @cite{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 --- 24983,24990 ---- 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 *************** *** 24719,24725 **** @tindex hasfitvars Two functions which are useful in connection with @code{FitRules} are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check ! whether @cite{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 --- 25030,25036 ---- @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 *************** *** 24767,24814 **** @pindex calc-poly-interp @tindex polint The @kbd{a p} (@code{calc-poly-interp}) address@hidden command does ! a polynomial interpolation at a particular @cite{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 @cite{x} value. Calc effectively does an exact polynomial fit as if by @kbd{a F i}, ! then substitutes the @cite{x} value into the result in order to get an ! approximate @cite{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 @cite{y} ! value approximation, and an error measure @cite{dy} that reflects Calc's estimation of the probable error of the approximation at that value of ! @cite{x}. If the input @cite{x} is equal to any of the @cite{x} values ! in the data matrix, the output @cite{y} will be the corresponding @cite{y} ! value from the matrix, and the output @cite{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 @cite{x} is a vector of numbers, @kbd{a p} will return a matrix of ! interpolated results for each of those @cite{x} values. (The matrix will ! have two columns, the @cite{y} values and the @cite{dy} values.) ! If @cite{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 @cite{x} value is replaced by the result. @kindex H a p @tindex ratint The @kbd{H a p} address@hidden 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 ! @cite{N} data points, the numerator and denominator polynomials will ! each have degree @cite{N/2} (if @cite{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 @cite{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]}. --- 25078,25125 ---- @pindex calc-poly-interp @tindex polint The @kbd{a p} (@code{calc-poly-interp}) address@hidden 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} address@hidden 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]}. *************** *** 24842,24850 **** 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 @cite{(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'' @cite{(0, 1)}! If you really want to use @code{i} as an index variable, use @address@hidden u i @key{RET}}} first to unstore'' this variable. (@xref{Storing Variables}.) --- 25153,25161 ---- 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 @address@hidden u i @key{RET}}} first to unstore'' this variable. (@xref{Storing Variables}.) *************** *** 24878,24884 **** and @samp{inf}, respectively. Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)} ! returns @cite{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 --- 25189,25195 ---- 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 *************** *** 24916,24946 **** 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 @cite{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 @cite{k^2}, where @cite{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 @cite{f(k)} for all @cite{k} from 1 to @cite{n}, excluding ! one value @cite{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 @cite{1/(k-k_0)} except at @cite{k = k_0}, where ! this would be a division by zero. But at @cite{k = k_0}, this ! formula works out to the indeterminate form @cite{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 @c{k \ne k_0} ! @cite{k != k_0}, ! then @cite{1/(k-k_0)}, else zero.'' Now the formula @cite{1/(k-k_0)} ! will not even be evaluated by Calc when @cite{k = k_0}. @cindex Alternating sums @kindex a - --- 25227,25258 ---- 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 @tmath{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 - *************** *** 24991,24997 **** which @code{dnonzero} returns 0 or cannot decide is assumed false.'' Note that this means that @address@hidden [ Z ]}} will execute the then'' portion if its condition is provably true, but it will execute the ! else'' portion for any condition like @cite{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 --- 25303,25309 ---- which @code{dnonzero} returns 0 or cannot decide is assumed false.'' Note that this means that @address@hidden [ 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 *************** *** 25004,25013 **** @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 @cite{a} and @cite{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 @cite{a} and @cite{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. --- 25316,25325 ---- @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. *************** *** 25035,25043 **** @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 @cite{a} and @cite{b} are not equal. This also works with more than two arguments; @samp{a != b != c != d} ! tests that all four of @cite{a}, @cite{b}, @cite{c}, and @cite{d} are distinct numbers. @kindex a < --- 25347,25355 ---- @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 < *************** *** 25087,25093 **** @end ignore @tindex >= The @kbd{a <} (@code{calc-less-than}) address@hidden(a,b)} or @samp{a < b}] ! operation is true if @cite{a} is less than @cite{b}. Similar functions are @kbd{a >} (@code{calc-greater-than}) address@hidden(a,b)} or @samp{a > b}], @kbd{a [} (@code{calc-less-equal}) address@hidden(a,b)} or @samp{a <= b}], and @kbd{a ]} (@code{calc-greater-equal}) address@hidden(a,b)} or @samp{a >= b}]. --- 25399,25405 ---- @end ignore @tindex >= The @kbd{a <} (@code{calc-less-than}) address@hidden(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}) address@hidden(a,b)} or @samp{a > b}], @kbd{a [} (@code{calc-less-equal}) address@hidden(a,b)} or @samp{a <= b}], and @kbd{a ]} (@code{calc-greater-equal}) address@hidden(a,b)} or @samp{a >= b}]. *************** *** 25121,25128 **** @tindex && The @kbd{a &} (@code{calc-logical-and}) address@hidden(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 @cite{a} or ! @cite{b}, chosen arbitrarily. If either argument is zero, the result is zero. Otherwise, the formula is left in symbolic form. @kindex a | --- 25433,25440 ---- @tindex && The @kbd{a &} (@code{calc-logical-and}) address@hidden(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 | *************** *** 25132,25138 **** The @kbd{a |} (@code{calc-logical-or}) address@hidden(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 @cite{a} and @cite{b} are zero, the result is zero. @kindex a ! --- 25444,25450 ---- The @kbd{a |} (@code{calc-logical-or}) address@hidden(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 ! *************** *** 25140,25147 **** @tindex lnot @tindex ! The @kbd{a !} (@code{calc-logical-not}) address@hidden(a)} or @samp{!@: a}] ! function is true if @cite{a} is false (zero), or false if @cite{a} is ! true (nonzero). It is left in symbolic form if @cite{a} is not a number. @kindex a : --- 25452,25459 ---- @tindex lnot @tindex ! The @kbd{a !} (@code{calc-logical-not}) address@hidden(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 : *************** *** 25157,25165 **** @tindex : @cindex Arguments, not evaluated The @kbd{a :} (@code{calc-logical-if}) address@hidden(a,b,c)} or @samp{a ? b :@: c}] ! function is equal to either @cite{b} or @cite{c} if @cite{a} is a nonzero ! number or zero, respectively. If @cite{a} is not a number, the test is ! left in symbolic form and neither @cite{b} nor @cite{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 --- 25469,25477 ---- @tindex : @cindex Arguments, not evaluated The @kbd{a :} (@code{calc-logical-if}) address@hidden(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 *************** *** 25170,25193 **** as three separate symbols. Type something like @samp{a ? 3 : 4} or @samp{a?(3):4} instead. ! As a special case, if @cite{a} evaluates to a vector, then both @cite{b} ! and @cite{c} are evaluated; the result is a vector of the same length ! as @cite{a} whose elements are chosen from corresponding elements of ! @cite{b} and @cite{c} according to whether each element of @cite{a} ! is zero or nonzero. Each of @cite{b} and @cite{c} must be either a ! vector of the same length as @cite{a}, or a non-vector which is matched ! with all elements of @cite{a}. @kindex a @{ @pindex calc-in-set @tindex in The @kbd{a @{} (@code{calc-in-set}) address@hidden(a,b)}] function is true if ! the number @cite{a} is in the set of numbers represented by @cite{b}. ! If @cite{b} is an interval form, @cite{a} must be one of the values ! encompassed by the interval. If @cite{b} is a vector, @cite{a} must be equal to one of the elements of the vector. (If any vector elements are ! intervals, @cite{a} must be in any of the intervals.) If @cite{b} is a ! plain number, @cite{a} must be numerically equal to @cite{b}. @xref{Set Operations}, for a group of commands that manipulate sets of this sort. --- 25482,25505 ---- 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}) address@hidden(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. *************** *** 25196,25202 **** @end ignore @tindex typeof The @samp{typeof(a)} function produces an integer or variable which ! characterizes @cite{a}. If @cite{a} is a number, vector, or variable, the result will be one of the following numbers: @example --- 25508,25514 ---- @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 *************** *** 25217,25223 **** 102 Matrix @end example ! Otherwise, @cite{a} is a formula, and the result is a variable which represents the name of the top-level function call. @ignore --- 25529,25535 ---- 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 *************** *** 25232,25246 **** @starindex @end ignore @tindex constant ! The @samp{integer(a)} function returns true if @cite{a} is an integer. The @samp{real(a)} function ! is true if @cite{a} is a real number, either integer, fraction, or ! float. The @samp{constant(a)} function returns true if @cite{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 @address@hidden @xref{Declarations}, for a set of similar functions that recognize formulas as well as actual numbers. For example, @samp{dint(floor(x))} --- 25544,25558 ---- @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))} *************** *** 25253,25273 **** @end ignore @tindex refers The @samp{refers(a,b)} function is true if the variable (or sub-expression) ! @cite{b} appears in @cite{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 @cite{a} is a plain ! variable (different from @cite{b}). @ignore @starindex @end ignore @tindex negative ! The @samp{negative(a)} function returns true if @cite{a} looks'' negative, ! because it is a negative number, because it is of the form @cite{-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 @cite{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). --- 25565,25585 ---- @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). *************** *** 25276,25283 **** @starindex @end ignore @tindex variable ! The @samp{variable(a)} function is true if @cite{a} is a variable, ! or false if not. If @cite{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. --- 25588,25595 ---- @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. *************** *** 25285,25291 **** @starindex @end ignore @tindex nonvar ! The @samp{nonvar(a)} function is true if @cite{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 --- 25597,25603 ---- @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 *************** *** 25310,25324 **** @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 ! @cite{a + b x} for some constants @cite{a} and @cite{b}, and some ! variable or subformula @cite{x}. The function @samp{islin(f,x)} checks ! if formula @cite{f} is linear in @cite{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 ! @cite{[a, b, x]}. For the above examples, this vector would be ! @cite{[0, 1, x]}, @cite{[0, -1, x]}, @cite{[3, 0, x]}, and ! @cite{[-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))} --- 25622,25636 ---- @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))} *************** *** 25326,25344 **** The @code{linnt} and @code{islinnt} functions perform a similar check, but require a non-trivial'' linear form, which means that the ! @cite{b} coefficient must be non-zero. For example, @samp{lin(2,x)} ! returns @cite{[2, 0, x]} and @samp{lin(y,x)} returns @cite{[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 @cite{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 @cite{a=0}, @cite{b=1} case is also considered ! trivial, and only constant values for @cite{a} and @cite{b} are ! recognized. Thus, @samp{lin(2 x y)} returns @cite{[0, 2, x y]}, ! @samp{lin(2 - x y)} returns @cite{[2, -1, x y]}, and @samp{lin(x y)} ! returns @cite{[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. --- 25638,25656 ---- 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. *************** *** 25347,25354 **** @starindex @end ignore @tindex istrue ! The @samp{istrue(a)} function returns 1 if @cite{a} is a nonzero ! number or provably nonzero formula, or 0 if @cite{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; --- 25659,25666 ---- @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; *************** *** 25408,25414 **** 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 address@hidden 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 --- 25720,25726 ---- 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 *************** *** 25451,25457 **** (@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 address@hidden Rewrite rules are compiled into a special internal form for faster matching. If you enter a rule set directly it must be recompiled --- 25763,25769 ---- (@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 *************** *** 25468,25477 **** @subsection Basic Rewrite Rules @noindent ! To match a particular formula @cite{x} with a particular rewrite rule ! @address@hidden := @var{new}}, Calc compares the structure of @cite{x} with the structure of @var{old}. Variables that appear in @var{old} are ! treated as @dfn{meta-variables}; the corresponding positions in @cite{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 --- 25780,25789 ---- @subsection Basic Rewrite Rules @noindent ! To match a particular formula @expr{x} with a particular rewrite rule ! @address@hidden := @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 *************** *** 25481,25487 **** 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 @cite{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.) --- 25793,25799 ---- 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.) *************** *** 25525,25538 **** 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 address@hidden ! For example, the formula @samp{n > 0} simplifies to 1 or 0 if @cite{n} is replaced by a positive or nonpositive number, respectively (or if ! @cite{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 @cite{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. --- 25837,25850 ---- 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. *************** *** 25562,25573 **** matched. Certain conditions are handled as special cases by the rewrite rule ! system and are tested very efficiently: Where @cite{x} is any meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)}, ! @samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @cite{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 @cite{a} and @cite{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. --- 25874,25885 ---- 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. *************** *** 25776,25782 **** 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 @cite{b = 1/z} by @code{lin}, but it will not match the above pattern because that pattern calls for a multiplication, not a division. --- 26088,26094 ---- 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. *************** *** 25796,25802 **** @end example Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2} ! because one @cite{a} would have matched'' 1 while the other matched 6. Calc automatically converts a rule like --- 26108,26114 ---- @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 *************** *** 25944,25963 **** @end ignore @tindex import One kind of marker, @samp{import(x)}, takes the place of a whole ! rule. Here @cite{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 @cite{x} with all occurrences of @c{v_1} ! @cite{v1}, as either ! a variable name or a function name, replaced with @c{x_1} ! @cite{x1} and ! so on. (If @c{v_1} ! @cite{v1} is used as a function name, then @c{x_1} ! @cite{x1} must be either a function name itself or a @address@hidden< >}} nameless function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0, import(linearF, f, g)]} applies the linearity rules to the function --- 26256,26280 ---- @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 @tmath{v_1}, ! @infoline @expr{v1}, ! as either a variable name or a function name, replaced with ! @texline @tmath{x_1} ! @infoline @expr{x1} ! and so on. (If ! @texline @tmath{v_1} ! @infoline @expr{v1} ! is used as a function name, then ! @texline @tmath{x_1} ! @infoline @expr{x1} must be either a function name itself or a @address@hidden< >}} nameless function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0, import(linearF, f, g)]} applies the linearity rules to the function *************** *** 25972,25978 **** @starindex @end ignore @tindex quote ! This pattern matches exactly @cite{x}; variable names in @cite{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)}. --- 26289,26295 ---- @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)}. *************** *** 25986,25995 **** @starindex @end ignore @tindex plain ! Here @cite{x} must be a function call @samp{f(x1,x2,@dots{})}. This ! pattern matches a call to function @cite{f} with the specified argument patterns. No special knowledge of the properties of the ! function @cite{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: --- 26303,26312 ---- @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: *************** *** 26000,26023 **** @starindex @end ignore @tindex opt ! Here @cite{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 @cite{x} and the other to @cite{y}, and it ! matches anything else by binding the whole expression to @cite{x} and ! zero to @cite{y}. The other operators above work address@hidden 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 @cite{b} are supplied in this example for the omitted arguments. Note that ! the literal variable @cite{b} will be the default in the latter ! case, @emph{not} the value that matched the meta-variable @cite{b}. In other words, the default @var{def} is effectively quoted. @item condition(x,c) --- 26317,26340 ---- @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) *************** *** 26026,26033 **** @end ignore @tindex condition @tindex :: ! This matches the pattern @cite{x}, with the attached condition ! @cite{c}. It is the same as @samp{x :: c}. @item pand(x,y) @ignore --- 26343,26350 ---- @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 *************** *** 26035,26042 **** @end ignore @tindex pand @tindex &&& ! This matches anything that matches both pattern @cite{x} and ! pattern @cite{y}. It is the same as @samp{x &&& y}. @pxref{Composing Patterns in Rewrite Rules}. @item por(x,y) --- 26352,26359 ---- @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) *************** *** 26045,26052 **** @end ignore @tindex por @tindex ||| ! This matches anything that matches either pattern @cite{x} or ! pattern @cite{y}. It is the same as @address@hidden ||| y}}. @item pnot(x) @ignore --- 26362,26369 ---- @end ignore @tindex por @tindex ||| ! This matches anything that matches either pattern @expr{x} or ! pattern @expr{y}. It is the same as @address@hidden ||| y}}. @item pnot(x) @ignore *************** *** 26054,26060 **** @end ignore @tindex pnot @tindex !!! ! This matches anything that does not match pattern @cite{x}. It is the same as @samp{!!! x}. @item cons(h,t) --- 26371,26377 ---- @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) *************** *** 26063,26070 **** @end ignore @tindex cons (rewrites) This matches any vector of one or more elements. The first ! element is matched to @cite{h}; a vector of the remaining ! elements is matched to @cite{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]}. --- 26380,26387 ---- @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]}. *************** *** 26075,26082 **** @end ignore @tindex rcons (rewrites) This is like @code{cons}, except that the @emph{last} element ! is matched to @cite{h}, with the remaining elements matched ! to @cite{t}. @item apply(f,args) @ignore --- 26392,26399 ---- @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 *************** *** 26084,26090 **** @end ignore @tindex apply (rewrites) This matches any function call. The name of the function, in ! the form of a variable, is matched to @cite{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, --- 26401,26407 ---- @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, *************** *** 26152,26158 **** @item plain(x) Special properties of and simplifications for the function call ! @cite{x} are not used. One interesting case where @code{plain} is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a shorthand notation for the @code{quote} function. This rule will not work as shown; instead of replacing @samp{q(foo)} with --- 26469,26475 ---- @item plain(x) Special properties of and simplifications for the function call ! @expr{x} are not used. One interesting case where @code{plain} is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a shorthand notation for the @code{quote} function. This rule will not work as shown; instead of replacing @samp{q(foo)} with *************** *** 26160,26166 **** rule would be @samp{q(x) := plain(quote(x))}. @item cons(h,t) ! Where @cite{t} is a vector, this is converted into an expanded vector during rewrite processing. Note that @code{cons} is a regular Calc function which normally does this anyway; the only way @code{cons} is treated specially by rewrites is that @code{cons} on the righthand --- 26477,26483 ---- rule would be @samp{q(x) := plain(quote(x))}. @item cons(h,t) ! Where @expr{t} is a vector, this is converted into an expanded vector during rewrite processing. Note that @code{cons} is a regular Calc function which normally does this anyway; the only way @code{cons} is treated specially by rewrites is that @code{cons} on the righthand *************** *** 26168,26178 **** have been turned off. @item rcons(t,h) ! Analogous to @code{cons} except putting @cite{h} at the @emph{end} of ! the vector @cite{t}. @item apply(f,args) ! Where @cite{f} is a variable and @var{args} is a vector, this is converted to a function call. Once again, note that @code{apply} is also a regular Calc function. --- 26485,26495 ---- have been turned off. @item rcons(t,h) ! Analogous to @code{cons} except putting @expr{h} at the @emph{end} of ! the vector @expr{t}. @item apply(f,args) ! Where @expr{f} is a variable and @var{args} is a vector, this is converted to a function call. Once again, note that @code{apply} is also a regular Calc function. *************** *** 26181,26187 **** @starindex @end ignore @tindex eval ! The formula @cite{x} is handled in the usual way, then the default simplifications are applied to it even if they have been turned off normally. This allows you to treat any function similarly to the way @code{cons} and @code{apply} are always --- 26498,26504 ---- @starindex @end ignore @tindex eval ! The formula @expr{x} is handled in the usual way, then the default simplifications are applied to it even if they have been turned off normally. This allows you to treat any function similarly to the way @code{cons} and @code{apply} are always *************** *** 26194,26200 **** @starindex @end ignore @tindex evalsimp ! The formula @cite{x} has meta-variables substituted in the usual way, then algebraically simplified as if by the @kbd{a s} command. @item evalextsimp(x) --- 26511,26517 ---- @starindex @end ignore @tindex evalsimp ! The formula @expr{x} has meta-variables substituted in the usual way, then algebraically simplified as if by the @kbd{a s} command. @item evalextsimp(x) *************** *** 26202,26208 **** @starindex @end ignore @tindex evalextsimp ! The formula @cite{x} has meta-variables substituted in the normal way, then extendedly'' simplified as if by the @kbd{a e} command. @item select(x) --- 26519,26525 ---- @starindex @end ignore @tindex evalextsimp ! The formula @expr{x} has meta-variables substituted in the normal way, then extendedly'' simplified as if by the @kbd{a e} command. @item select(x) *************** *** 26217,26238 **** @starindex @end ignore @tindex let ! The expression @cite{x} is evaluated with meta-variables substituted. The @kbd{a s} command's simplifications are @emph{not} applied by ! default, but @cite{x} can include calls to @code{evalsimp} or @code{evalextsimp} as described above to invoke higher levels of simplification. The ! result of @cite{x} is then bound to the meta-variable @cite{v}. As usual, if this meta-variable has already been matched to something else the two values must be equal; if the meta-variable is new then it is bound to the result of the expression. This variable can then appear in later conditions, and on the righthand side of the rule. ! In fact, @cite{v} may be any pattern in which case the result of ! evaluating @cite{x} is matched to that pattern, binding any meta-variables that appear in that pattern. Note that @code{let} can only appear by itself as a condition, or as one term of an @samp{&&} which is a whole condition: It cannot be inside ! an @samp{||} term or otherwise address@hidden The alternate, equivalent form @samp{let(v, x)} is also recognized. Note that the use of @samp{:=} by @code{let}, while still being --- 26534,26555 ---- @starindex @end ignore @tindex let ! The expression @expr{x} is evaluated with meta-variables substituted. The @kbd{a s} command's simplifications are @emph{not} applied by ! default, but @expr{x} can include calls to @code{evalsimp} or @code{evalextsimp} as described above to invoke higher levels of simplification. The ! result of @expr{x} is then bound to the meta-variable @expr{v}. As usual, if this meta-variable has already been matched to something else the two values must be equal; if the meta-variable is new then it is bound to the result of the expression. This variable can then appear in later conditions, and on the righthand side of the rule. ! In fact, @expr{v} may be any pattern in which case the result of ! evaluating @expr{x} is matched to that pattern, binding any meta-variables that appear in that pattern. Note that @code{let} can only appear by itself as a condition, or as one term of an @samp{&&} which is a whole condition: It cannot be inside ! an @samp{||} term or otherwise buried. The alternate, equivalent form @samp{let(v, x)} is also recognized. Note that the use of @samp{:=} by @code{let}, while still being *************** *** 26249,26255 **** Note that, because the meta-variable @samp{ia} is otherwise unbound in this rule, the @code{let} condition itself always succeeds'' because no matter what @samp{1/a} evaluates to, it can successfully ! be bound to @address@hidden Here's another example, for integrating cosines of linear terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}. --- 26566,26572 ---- Note that, because the meta-variable @samp{ia} is otherwise unbound in this rule, the @code{let} condition itself always succeeds'' because no matter what @samp{1/a} evaluates to, it can successfully ! be bound to @code{ia}. Here's another example, for integrating cosines of linear terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}. *************** *** 26260,26266 **** the coefficients @code{a} and @code{b} for use elsewhere in the rule. (It would have been possible to use @samp{sin(a x + b)/b} for the righthand side instead, but using @samp{sin(y)/b} avoids gratuitous ! rearrangement of the argument of the sine.)@refill @ignore @starindex --- 26577,26583 ---- the coefficients @code{a} and @code{b} for use elsewhere in the rule. (It would have been possible to use @samp{sin(a x + b)/b} for the righthand side instead, but using @samp{sin(y)/b} avoids gratuitous ! rearrangement of the argument of the sine.) @ignore @starindex *************** *** 26321,26327 **** @starindex @end ignore @tindex remember ! Remember the match as described above, but only if condition @cite{c} is true. For example, @samp{remember(n % 4 = 0)} in the above factorial rule remembers only every fourth result. Note that @samp{remember(1)} is equivalent to @samp{remember}, and @samp{remember(0)} has no effect. --- 26638,26644 ---- @starindex @end ignore @tindex remember ! Remember the match as described above, but only if condition @expr{c} is true. For example, @samp{remember(n % 4 = 0)} in the above factorial rule remembers only every fourth result. Note that @samp{remember(1)} is equivalent to @samp{remember}, and @samp{remember(0)} has no effect. *************** *** 26883,26889 **** Another subtle difference between @code{EvalRules} and regular rewrites concerns rules that rewrite a formula into an identical formula. For ! example, @samp{f(n) := f(floor(n))} fails to match'' when @cite{n} is already an integer. But in @code{EvalRules} this case is detected only if the righthand side literally becomes the original formula before any further simplification. This means that @samp{f(n) := f(floor(n))} will --- 27200,27206 ---- Another subtle difference between @code{EvalRules} and regular rewrites concerns rules that rewrite a formula into an identical formula. For ! example, @samp{f(n) := f(floor(n))} fails to match'' when @expr{n} is already an integer. But in @code{EvalRules} this case is detected only if the righthand side literally becomes the original formula before any further simplification. This means that @samp{f(n) := f(floor(n))} will *************** *** 26927,26940 **** functions directly rather than going through the default simplifications. When it does this, @code{EvalRules} will not be able to override those functions. For example, when you take the absolute value of the complex ! number @cite{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling the multiplication, addition, and square root functions directly rather than applying the default simplifications to this formula. So an @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6} would not apply. (However, if you put Calc into symbolic mode so that @samp{sqrt(13)} will be left in symbolic form by the built-in square root function, your rule will be able to apply. But if the complex ! number were @cite{(3,4)}, so that @samp{sqrt(25)} must be calculated, then symbolic mode will not help because @samp{sqrt(25)} can be evaluated exactly to 5.) --- 27244,27257 ---- functions directly rather than going through the default simplifications. When it does this, @code{EvalRules} will not be able to override those functions. For example, when you take the absolute value of the complex ! number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling the multiplication, addition, and square root functions directly rather than applying the default simplifications to this formula. So an @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6} would not apply. (However, if you put Calc into symbolic mode so that @samp{sqrt(13)} will be left in symbolic form by the built-in square root function, your rule will be able to apply. But if the complex ! number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated, then symbolic mode will not help because @samp{sqrt(25)} can be evaluated exactly to 5.) *************** *** 27028,27034 **** finding suitable cases. Another solution would be to use the rule @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification if necessary. This rule will be the most effective way to do the job, ! but at the expense of making some changes that you might not address@hidden Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}. To make this work with the @address@hidden r}} command so that it can be --- 27345,27351 ---- finding suitable cases. Another solution would be to use the rule @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification if necessary. This rule will be the most effective way to do the job, ! but at the expense of making some changes that you might not desire. Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}. To make this work with the @address@hidden r}} command so that it can be *************** *** 27036,27045 **** you might wish to write the rule as @samp{select(exp(x+y)) := select(exp(x) exp(y))}. The @samp{select} markers will be ignored by the regular @kbd{a r} command ! (@pxref{Selections with Rewrite Rules})address@hidden A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}. ! This will simplify the formula whenever @cite{b} and/or @cite{c} can be made simpler by squaring. For example, applying this rule to @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming Symbolic Mode has been enabled to keep the square root from being --- 27353,27362 ---- you might wish to write the rule as @samp{select(exp(x+y)) := select(exp(x) exp(y))}. The @samp{select} markers will be ignored by the regular @kbd{a r} command ! (@pxref{Selections with Rewrite Rules}). A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}. ! This will simplify the formula whenever @expr{b} and/or @expr{c} can be made simpler by squaring. For example, applying this rule to @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming Symbolic Mode has been enabled to keep the square root from being *************** *** 27059,27075 **** @code{tri} to the value on the top of the stack. @xref{Programming}. @cindex Quaternions ! The following rule set, contributed by @c{Fran\c cois} ! @asis{Francois} Pinard, implements ! @dfn{quaternions}, a generalization of the concept of complex numbers. ! Quaternions have four components, and are here represented by function ! calls @samp{quat(@var{w}, address@hidden, @var{y}, @var{z}])} with real ! part'' @var{w} and the three imaginary'' parts collected into a ! vector. Various arithmetical operations on quaternions are supported. ! To use these rules, either add them to @code{EvalRules}, or create a ! command based on @kbd{a r} for simplifying quaternion formulas. ! A convenient way to enter quaternions would be a command defined by ! a keyboard macro containing: @kbd{' quat($$$$, [$$, $$, ]) @key{RET}}. @smallexample [ quat(w, x, y, z) := quat(w, [x, y, z]), --- 27376,27394 ---- @code{tri} to the value on the top of the stack. @xref{Programming}. @cindex Quaternions ! The following rule set, contributed by ! @texline Fran\c cois ! @infoline Francois ! Pinard, implements @dfn{quaternions}, a generalization of the concept of ! complex numbers. Quaternions have four components, and are here ! represented by function calls @samp{quat(@var{w}, address@hidden, @var{y}, ! @var{z}])} with real part'' @var{w} and the three imaginary'' parts ! collected into a vector. Various arithmetical operations on quaternions ! are supported. To use these rules, either add them to @code{EvalRules}, ! or create a command based on @kbd{a r} for simplifying quaternion ! formulas. A convenient way to enter quaternions would be a command ! defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, ]) ! @key{RET}}. @smallexample [ quat(w, x, y, z) := quat(w, [x, y, z]), *************** *** 27094,27101 **** @end smallexample Quaternions, like matrices, have non-commutative multiplication. ! In other words, @cite{q1 * q2 = q2 * q1} is not necessarily true if ! @cite{q1} and @cite{q2} are @code{quat} forms. The @samp{quat*quat} rule above uses @code{plain} to prevent Calc from rearranging the product. It may also be wise to add the line @samp{[quat(), matrix]} to the @code{Decls} matrix, to ensure that Calc's other algebraic --- 27413,27420 ---- @end smallexample Quaternions, like matrices, have non-commutative multiplication. ! In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if ! @expr{q1} and @expr{q2} are @code{quat} forms. The @samp{quat*quat} rule above uses @code{plain} to prevent Calc from rearranging the product. It may also be wise to add the line @samp{[quat(), matrix]} to the @code{Decls} matrix, to ensure that Calc's other algebraic *************** *** 27142,27154 **** or @samp{u} (for micro'') followed by a name in the unit table. A substantial table of built-in units is provided with Calc; @pxref{Predefined Units}. You can also define your own unit names; ! @pxref{User-Defined address@hidden Note that if the value part of a units expression is exactly @samp{1}, it will be removed by the Calculator's automatic algebra routines: The formula @samp{1 mm} is simplified'' to @samp{mm}. This is only a display anomaly, however; @samp{mm} will work just fine as a ! representation of one address@hidden You may find that Algebraic Mode (@pxref{Algebraic Entry}) makes working with units expressions easier. Otherwise, you will have to remember --- 27461,27473 ---- or @samp{u} (for micro'') followed by a name in the unit table. A substantial table of built-in units is provided with Calc; @pxref{Predefined Units}. You can also define your own unit names; ! @pxref{User-Defined Units}. Note that if the value part of a units expression is exactly @samp{1}, it will be removed by the Calculator's automatic algebra routines: The formula @samp{1 mm} is simplified'' to @samp{mm}. This is only a display anomaly, however; @samp{mm} will work just fine as a ! representation of one millimeter. You may find that Algebraic Mode (@pxref{Algebraic Entry}) makes working with units expressions easier. Otherwise, you will have to remember *************** *** 27169,27175 **** simplify to @samp{5.023 m}. When different but compatible units are added, the righthand term's units are converted to match those of the lefthand term. @xref{Simplification Modes}, for a way to have this done ! automatically at all address@hidden Units simplification also handles quotients of two units with the same dimensionality, as in @address@hidden in s/L cm}} to @samp{5.08 s/L}; fractional --- 27488,27494 ---- simplify to @samp{5.023 m}. When different but compatible units are added, the righthand term's units are converted to match those of the lefthand term. @xref{Simplification Modes}, for a way to have this done ! automatically at all times. Units simplification also handles quotients of two units with the same dimensionality, as in @address@hidden in s/L cm}} to @samp{5.08 s/L}; fractional *************** *** 27180,27186 **** applied to units expressions, in which case the operation in question is applied only to the numeric part of the expression. Finally, trigonometric functions of quantities with units ! of angle are evaluated, regardless of the current angular address@hidden @kindex u c @pindex calc-convert-units --- 27499,27505 ---- applied to units expressions, in which case the operation in question is applied only to the numeric part of the expression. Finally, trigonometric functions of quantities with units ! of angle are evaluated, regardless of the current angular mode. @kindex u c @pindex calc-convert-units *************** *** 27261,27267 **** absolute temperatures. The value on the stack must be a simple units expression with units of temperature only. This command would convert @samp{10 degC} to @samp{50 degF}, the equivalent temperature on the ! Fahrenheit address@hidden @kindex u r @pindex calc-remove-units --- 27580,27586 ---- absolute temperatures. The value on the stack must be a simple units expression with units of temperature only. This command would convert @samp{10 degC} to @samp{50 degF}, the equivalent temperature on the ! Fahrenheit scale. @kindex u r @pindex calc-remove-units *************** *** 27272,27278 **** (@code{calc-extract-units}) command extracts only the units portion of a formula. These commands essentially replace every term of the formula that does or doesn't (respectively) look like a unit name by the ! constant 1, then resimplify the address@hidden @kindex u a @pindex calc-autorange-units --- 27591,27597 ---- (@code{calc-extract-units}) command extracts only the units portion of a formula. These commands essentially replace every term of the formula that does or doesn't (respectively) look like a unit name by the ! constant 1, then resimplify the formula. @kindex u a @pindex calc-autorange-units *************** *** 27397,27408 **** The unit @code{e} stands for the elementary (electron) unit of charge; because algebra command could mistake this for the special constant ! @cite{e}, Calc provides the alternate unit name @code{ech} which is preferable to @code{e}. The name @code{g} stands for one gram of mass; there is also @code{gf}, one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.) ! Meanwhile, one address@hidden'' of acceleration is denoted @code{ga}. The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is a metric ton of @samp{1000 kg}. --- 27716,27727 ---- The unit @code{e} stands for the elementary (electron) unit of charge; because algebra command could mistake this for the special constant ! @expr{e}, Calc provides the alternate unit name @code{ech} which is preferable to @code{e}. The name @code{g} stands for one gram of mass; there is also @code{gf}, one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.) ! Meanwhile, one address@hidden'' of acceleration is denoted @code{ga}. The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is a metric ton of @samp{1000 kg}. *************** *** 27634,27645 **** and @kbd{s ]} which decrease or increase a variable by one. All the arithmetic stores accept the Inverse prefix to reverse the ! order of the operands. If @cite{v} represents the contents of the ! variable, and @cite{a} is the value drawn from the stack, then regular ! @address@hidden -}} assigns @c{v \coloneq v - a} ! @cite{v := v - a}, but @kbd{I s -} assigns ! @c{v \coloneq a - v} ! @cite{v := a - v}. While @kbd{I s *} might seem pointless, it is useful if matrix multiplication is involved. Actually, all the arithmetic stores use formulas designed to behave usefully both forwards and backwards: --- 27953,27967 ---- and @kbd{s ]} which decrease or increase a variable by one. All the arithmetic stores accept the Inverse prefix to reverse the ! order of the operands. If @expr{v} represents the contents of the ! variable, and @expr{a} is the value drawn from the stack, then regular ! @address@hidden -}} assigns ! @texline @tmath{v \coloneq v - a}, ! @infoline @expr{v := v - a}, ! but @kbd{I s -} assigns ! @texline @tmath{v \coloneq a - v}. ! @infoline @expr{v := a - v}. ! While @kbd{I s *} might seem pointless, it is useful if matrix multiplication is involved. Actually, all the arithmetic stores use formulas designed to behave usefully both forwards and backwards: *************** *** 27666,27672 **** The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -}, etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous ! arithmetic stores that don't remove the value @cite{a} from the stack. All arithmetic stores report the new value of the variable in the Trail for your information. They signal an error if the variable --- 27988,27994 ---- The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -}, etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous ! arithmetic stores that don't remove the value @expr{a} from the stack. All arithmetic stores report the new value of the variable in the Trail for your information. They signal an error if the variable *************** *** 27692,27699 **** If the mapping function takes two or more arguments, the additional arguments are taken from the stack; the old value of the variable ! is provided as the first argument. Thus @kbd{s m -} with @cite{a} ! on the stack computes @cite{v - a}, just like @kbd{s -}. With the Inverse prefix, the variable's original value becomes the @emph{last} argument instead of the first. Thus @kbd{I s m -} is also equivalent to @kbd{I s -}. --- 28014,28021 ---- If the mapping function takes two or more arguments, the additional arguments are taken from the stack; the old value of the variable ! is provided as the first argument. Thus @kbd{s m -} with @expr{a} ! on the stack computes @expr{v - a}, just like @kbd{s -}. With the Inverse prefix, the variable's original value becomes the @emph{last} argument instead of the first. Thus @kbd{I s m -} is also equivalent to @kbd{I s -}. *************** *** 27716,27722 **** contain no value at all. If they appear in an algebraic formula they will be left alone even if you press @kbd{=} (@code{calc-evaluate}). The @kbd{s u} (@code{calc-unstore}) command returns a variable to the ! void address@hidden The only variables with predefined values are the special constants'' @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free --- 28038,28044 ---- contain no value at all. If they appear in an algebraic formula they will be left alone even if you press @kbd{=} (@code{calc-evaluate}). The @kbd{s u} (@code{calc-unstore}) command returns a variable to the ! void state. The only variables with predefined values are the special constants'' @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free *************** *** 27728,27735 **** normally void). Note that @code{var-pi} doesn't actually have 3.14159265359 stored ! in it, but rather a special magic value that evaluates to @c{\pi} ! @cite{pi} at the current precision. Likewise @code{var-e}, @code{var-i}, and @code{var-phi} evaluate according to the current precision or polar mode. If you recall a value from @code{pi} and store it back, this magic --- 28050,28056 ---- normally void). Note that @code{var-pi} doesn't actually have 3.14159265359 stored ! in it, but rather a special magic value that evaluates to @cpi{} at the current precision. Likewise @code{var-e}, @code{var-i}, and @code{var-phi} evaluate according to the current precision or polar mode. If you recall a value from @code{pi} and store it back, this magic *************** *** 27913,27919 **** @code{FitRules}, @code{DistribRules}, and other built-in rewrite rules; and @address@hidden variables generated by the graphics commands. (You can still save these variables by ! explicitly naming them in an @kbd{s p} command.)@refill @kindex s i @pindex calc-insert-variables --- 28234,28240 ---- @code{FitRules}, @code{DistribRules}, and other built-in rewrite rules; and @address@hidden variables generated by the graphics commands. (You can still save these variables by ! explicitly naming them in an @kbd{s p} command.) @kindex s i @pindex calc-insert-variables *************** *** 27936,27944 **** @cindex Variables, temporary assignment @cindex Temporary assignment to variables If you have an expression like @samp{a+b^2} on the stack and you wish to ! compute its value where @cite{b=3}, you can simply store 3 in @cite{b} and then press @kbd{=} to reevaluate the formula. This has the side-effect ! of leaving the stored value of 3 in @cite{b} for future operations. The @kbd{s l} (@code{calc-let}) command evaluates a formula under a @emph{temporary} assignment of a variable. It stores the value on the --- 28257,28265 ---- @cindex Variables, temporary assignment @cindex Temporary assignment to variables If you have an expression like @samp{a+b^2} on the stack and you wish to ! compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and then press @kbd{=} to reevaluate the formula. This has the side-effect ! of leaving the stored value of 3 in @expr{b} for future operations. The @kbd{s l} (@code{calc-let}) command evaluates a formula under a @emph{temporary} assignment of a variable. It stores the value on the *************** *** 27961,27967 **** The @kbd{a b} (@code{calc-substitute}) command is another way to substitute a variable with a value in a formula. It does an actual substitution rather than temporarily assigning the variable and evaluating. For ! example, letting @cite{n=2} in @samp{f(n pi)} with @kbd{a b} will produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)} since the evaluation step will also evaluate @code{pi}. --- 28282,28288 ---- The @kbd{a b} (@code{calc-substitute}) command is another way to substitute a variable with a value in a formula. It does an actual substitution rather than temporarily assigning the variable and evaluating. For ! example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)} since the evaluation step will also evaluate @code{pi}. *************** *** 28145,28152 **** the interval (whether the interval is open or closed is ignored). The x'' entry may also be a number, in which case Calc uses the ! sequence of x'' values @cite{x}, @cite{x+1}, @cite{x+2}, etc. ! (Generally the number 0 or 1 would be used for @cite{x} in this case.) The y'' entry may be any formula instead of a vector. Calc effectively uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula; --- 28466,28473 ---- the interval (whether the interval is open or closed is ignored). The x'' entry may also be a number, in which case Calc uses the ! sequence of x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc. ! (Generally the number 0 or 1 would be used for @expr{x} in this case.) The y'' entry may be any formula instead of a vector. Calc effectively uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula; *************** *** 28169,28175 **** In this case the x'' vector or interval you specified is not directly visible in the graph. For example, if x'' is the interval @samp{[0..360]} and y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph ! will be a address@hidden Also, x'' and y'' may each be variable names, in which case Calc looks for suitable vectors, intervals, or formulas stored in those --- 28490,28496 ---- In this case the x'' vector or interval you specified is not directly visible in the graph. For example, if x'' is the interval @samp{[0..360]} and y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph ! will be a circle. Also, x'' and y'' may each be variable names, in which case Calc looks for suitable vectors, intervals, or formulas stored in those *************** *** 28224,28232 **** the same length); either or both may instead be interval forms. The z'' value must be a matrix with the same number of rows as elements in x'', and the same number of columns as elements in y''. The ! result is a surface plot where @c{z_{ij}} ! @cite{z_ij} is the height of the point ! at coordinate @cite{(x_i, y_j)} on the surface. The 3D graph will be displayed from a certain default viewpoint; you can change this viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*} buffer as described later. See the GNUPLOT 3.0 documentation for a --- 28545,28555 ---- the same length); either or both may instead be interval forms. The z'' value must be a matrix with the same number of rows as elements in x'', and the same number of columns as elements in y''. The ! result is a surface plot where ! @texline @tmath{z_{ij}} ! @infoline @expr{z_ij} ! is the height of the point ! at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will be displayed from a certain default viewpoint; you can change this viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*} buffer as described later. See the GNUPLOT 3.0 documentation for a *************** *** 28317,28342 **** A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way stack entries are interpreted as curves. With a positive prefix ! argument @cite{n}, the top @cite{n} stack entries are y'' values ! for @cite{n} different curves which share a common x'' value in ! the @cite{n+1}st stack entry. (Thus @kbd{g a} with no prefix argument is equivalent to @kbd{C-u 1 g a}.) A prefix of zero or plain @kbd{C-u} means to take two stack entries, x'' and y'' as usual, but to interpret y'' as a vector of y'' values for several curves that share a common x''. ! A negative prefix argument tells Calc to read @cite{n} vectors from ! the stack; each vector @cite{[x, y]} describes an independent curve. This is the only form of @kbd{g a} that creates several curves at once that don't have common x'' values. (Of course, the range of x'' values covered by all the curves ought to be roughly the same if they are to look nice on the same graph.) ! For example, to plot @c{\sin n x} ! @cite{sin(n x)} for integers @cite{n} from 1 to 5, you could use @kbd{v x} to create a vector of integers ! (@cite{n}), then @kbd{V M '} or @kbd{V M } to map @samp{sin(n x)} across this vector. The resulting vector of formulas is suitable for use as the y'' argument to a @kbd{C-u g a} or @kbd{C-u g f} command. --- 28640,28667 ---- A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way stack entries are interpreted as curves. With a positive prefix ! argument @expr{n}, the top @expr{n} stack entries are y'' values ! for @expr{n} different curves which share a common x'' value in ! the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix argument is equivalent to @kbd{C-u 1 g a}.) A prefix of zero or plain @kbd{C-u} means to take two stack entries, x'' and y'' as usual, but to interpret y'' as a vector of y'' values for several curves that share a common x''. ! A negative prefix argument tells Calc to read @expr{n} vectors from ! the stack; each vector @expr{[x, y]} describes an independent curve. This is the only form of @kbd{g a} that creates several curves at once that don't have common x'' values. (Of course, the range of x'' values covered by all the curves ought to be roughly the same if they are to look nice on the same graph.) ! For example, to plot ! @texline @tmath{\sin n x} ! @infoline @expr{sin(n x)} ! for integers @expr{n} from 1 to 5, you could use @kbd{v x} to create a vector of integers ! (@expr{n}), then @kbd{V M '} or @kbd{V M } to map @samp{sin(n x)} across this vector. The resulting vector of formulas is suitable for use as the y'' argument to a @kbd{C-u g a} or @kbd{C-u g f} command. *************** *** 28346,28356 **** The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve to the graph. It is not legal to intermix 2D and 3D curves in a single graph. This command takes three arguments, x'', y'', ! and z'', from the stack. With a positive prefix @cite{n}, it ! takes @cite{n+2} arguments (common x'' and y'', plus @cite{n} separate z''s). With a zero prefix, it takes three stack entries but the z'' entry is a vector of curve values. With a negative ! prefix @cite{-n}, it takes @cite{n} vectors of the form @cite{[x, y, z]}. The @kbd{g A} command works by adding a @code{splot} (surface-plot) command to the @samp{*Gnuplot Commands*} buffer. --- 28671,28681 ---- The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve to the graph. It is not legal to intermix 2D and 3D curves in a single graph. This command takes three arguments, x'', y'', ! and z'', from the stack. With a positive prefix @expr{n}, it ! takes @expr{n+2} arguments (common x'' and y'', plus @expr{n} separate z''s). With a zero prefix, it takes three stack entries but the z'' entry is a vector of curve values. With a negative ! prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}. The @kbd{g A} command works by adding a @code{splot} (surface-plot) command to the @samp{*Gnuplot Commands*} buffer. *************** *** 28469,28481 **** graphs created by @kbd{g a} that don't specify the resolution explicitly. With a negative prefix argument, this command changes or displays the default value (initially 5) used for 3D graphs created by @kbd{g A}. ! Note that a 3D setting of 5 means that a total of @cite{5^2 = 25} points will be computed for the surface. Data values in the graph of a function are normally computed to a precision of five digits, regardless of the current precision at the time. This is usually more than adequate, but there are cases where ! it will not be. For example, plotting @cite{1 + x} with @cite{x} in the interval @samp{[0 ..@: 1e-6]} will round all the data points down to 1.0! Putting the command @samp{set precision @var{n}} in the @samp{*Gnuplot Commands*} buffer will cause the data to be computed --- 28794,28806 ---- graphs created by @kbd{g a} that don't specify the resolution explicitly. With a negative prefix argument, this command changes or displays the default value (initially 5) used for 3D graphs created by @kbd{g A}. ! Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points will be computed for the surface. Data values in the graph of a function are normally computed to a precision of five digits, regardless of the current precision at the time. This is usually more than adequate, but there are cases where ! it will not be. For example, plotting @expr{1 + x} with @expr{x} in the interval @samp{[0 ..@: 1e-6]} will round all the data points down to 1.0! Putting the command @samp{set precision @var{n}} in the @samp{*Gnuplot Commands*} buffer will cause the data to be computed *************** *** 28621,28628 **** picture of the graph composed of characters like @code{-} and @code{|} to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays. The graph is made the same size as the Emacs screen, which on most ! dumb terminals will be @c{80\times24} ! @asis{80x24} characters. The graph is displayed in an Emacs recursive edit''; type @kbd{q} or @kbd{M-# M-#} to exit the recursive edit and return to Calc. Note that the @code{dumb} device is present only in GNUPLOT 3.0 and later versions. --- 28946,28955 ---- picture of the graph composed of characters like @code{-} and @code{|} to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays. The graph is made the same size as the Emacs screen, which on most ! dumb terminals will be ! @texline @tmath{80\times24} ! @infoline 80x24 ! characters. The graph is displayed in an Emacs recursive edit''; type @kbd{q} or @kbd{M-# M-#} to exit the recursive edit and return to Calc. Note that the @code{dumb} device is present only in GNUPLOT 3.0 and later versions. *************** *** 28824,28831 **** it appears on the screen, including line numbers if they are enabled. A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number ! of lines killed. A positive argument kills the current line and @cite{n-1} ! lines below it. A negative argument kills the @cite{-n} lines above the current line. Again this mirrors the behavior of the standard Emacs @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k} with no argument copies only the number itself into the kill ring, whereas --- 29151,29158 ---- it appears on the screen, including line numbers if they are enabled. A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number ! of lines killed. A positive argument kills the current line and @expr{n-1} ! lines below it. A negative argument kills the @expr{-n} lines above the current line. Again this mirrors the behavior of the standard Emacs @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k} with no argument copies only the number itself into the kill ring, whereas *************** *** 28868,28876 **** A numeric prefix argument grabs the specified number of lines around point, ignoring the mark. A positive prefix grabs from point to the ! @cite{n}th following newline (so that @kbd{M-1 M-# g} grabs from point to the end of the current line); a negative prefix grabs from point ! back to the @cite{n+1}st preceding newline. In these cases the text that is grabbed is exactly the same as the text that @kbd{C-k} would delete given that prefix argument. --- 29195,29203 ---- A numeric prefix argument grabs the specified number of lines around point, ignoring the mark. A positive prefix grabs from point to the ! @expr{n}th following newline (so that @kbd{M-1 M-# g} grabs from point to the end of the current line); a negative prefix grabs from point ! back to the @expr{n+1}st preceding newline. In these cases the text that is grabbed is exactly the same as the text that @kbd{C-k} would delete given that prefix argument. *************** *** 28926,28937 **** will be split up into columns of width @var{n}; each column is parsed separately as a matrix element. If a line contained @address@hidden +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8 ! would correctly split the line into two error address@hidden @xref{Matrix Functions}, to see how to pull the matrix apart into its ! constituent rows and columns. (If it is a @c{1\times1} ! @asis{1x1} matrix, just hit @kbd{v u} ! (@code{calc-unpack}) twice.) @kindex M-# : @kindex M-# _ --- 29253,29265 ---- will be split up into columns of width @var{n}; each column is parsed separately as a matrix element. If a line contained @address@hidden +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8 ! would correctly split the line into two error forms. @xref{Matrix Functions}, to see how to pull the matrix apart into its ! constituent rows and columns. (If it is a ! @texline @tmath{1\times1} ! @infoline 1x1 ! matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.) @kindex M-# : @kindex M-# _ *************** *** 28984,28990 **** With a prefix argument, @kbd{y} inserts several numbers, one per line. A positive argument inserts the specified number of values from the top ! of the stack. A negative argument inserts the @cite{n}th value from the top of the stack. An argument of zero inserts the entire stack. Note that @kbd{y} with an argument of 1 is slightly different from @kbd{y} with no argument; the former always copies full lines, whereas the --- 29312,29318 ---- With a prefix argument, @kbd{y} inserts several numbers, one per line. A positive argument inserts the specified number of values from the top ! of the stack. A negative argument inserts the @expr{n}th value from the top of the stack. An argument of zero inserts the entire stack. Note that @kbd{y} with an argument of 1 is slightly different from @kbd{y} with no argument; the former always copies full lines, whereas the *************** *** 28999,29005 **** display style (@pxref{Vector and Matrix Formats}) or change the current display language (@pxref{Language Modes}) before doing this. Also, note that this command replaces a linear region of text (as grabbed by ! @kbd{M-# g}), not a rectangle (as grabbed by @kbd{M-# r})address@hidden If the editing buffer is in overwrite (as opposed to insert) mode, and the @kbd{C-u} prefix was not used, then the yanked number will --- 29327,29333 ---- display style (@pxref{Vector and Matrix Formats}) or change the current display language (@pxref{Language Modes}) before doing this. Also, note that this command replaces a linear region of text (as grabbed by ! @kbd{M-# g}), not a rectangle (as grabbed by @kbd{M-# r}). If the editing buffer is in overwrite (as opposed to insert) mode, and the @kbd{C-u} prefix was not used, then the yanked number will *************** *** 29187,29193 **** @item INV * is the same as @key{y^x}. @item INV / ! is the same as @key{INV y^x} (the @cite{x}th root of @cite{y}). @item HYP/INV 1 are the same as @key{SIN} / @kbd{INV SIN}. @item HYP/INV 2 --- 29515,29521 ---- @item INV * is the same as @key{y^x}. @item INV / ! is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}). @item HYP/INV 1 are the same as @key{SIN} / @kbd{INV SIN}. @item HYP/INV 2 *************** *** 29253,29259 **** prefix keys. @key{IMAG} multiplies the number on the stack by the imaginary ! number @cite{i = (0, 1)}. @key{RE} extracts the real part a complex number. @kbd{INV RE} extracts the imaginary part. --- 29581,29587 ---- prefix keys. @key{IMAG} multiplies the number on the stack by the imaginary ! number @expr{i = (0, 1)}. @key{RE} extracts the real part a complex number. @kbd{INV RE} extracts the imaginary part. *************** *** 29266,29273 **** @key{INV GCD} computes the LCM (least common multiple) function. ! @key{INV FACT} is the gamma function. @c{\Gamma(x) = (x-1)!} ! @cite{gamma(x) = (x-1)!}. @key{PERM} is the number-of-permutations function, which is on the @kbd{H k c} key in normal Calc. --- 29594,29602 ---- @key{INV GCD} computes the LCM (least common multiple) function. ! @key{INV FACT} is the gamma function. ! @texline @tmath{\Gamma(x) = (x-1)!}. ! @infoline @expr{gamma(x) = (x-1)!}. @key{PERM} is the number-of-permutations function, which is on the @kbd{H k c} key in normal Calc. *************** *** 29382,29394 **** For example, you could simulate @key{MAP^} using @key{MAP} with the formula @samp{x^y}. ! The @kbd{"x"} key pushes the variable name @cite{x} onto the ! stack. To build the formula @cite{x^2 + 6}, you would use the key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be suitable for use with the @key{MAP} key described above. With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the ! @kbd{"x"} key pushes the variable names @cite{y}, @cite{z}, and ! @cite{t}, respectively. @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode @section Modes Menu --- 29711,29723 ---- For example, you could simulate @key{MAP^} using @key{MAP} with the formula @samp{x^y}. ! The @kbd{"x"} key pushes the variable name @expr{x} onto the ! stack. To build the formula @expr{x^2 + 6}, you would use the key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be suitable for use with the @key{MAP} key described above. With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the ! @kbd{"x"} key pushes the variable names @expr{y}, @expr{z}, and ! @expr{t}, respectively. @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode @section Modes Menu *************** *** 29553,29559 **** @end example @noindent ! The formula @cite{n>2} will be pushed onto the Calc stack, and the top of stack will be copied back into the editing buffer. This means that spaces will appear around the @samp{>} symbol to match Calc's usual display style: --- 29882,29888 ---- @end example @noindent ! The formula @expr{n>2} will be pushed onto the Calc stack, and the top of stack will be copied back into the editing buffer. This means that spaces will appear around the @samp{>} symbol to match Calc's usual display style: *************** *** 29681,29688 **** Embedded buffers remember active formulas for as long as they exist in Emacs memory. Suppose you have an embedded formula ! which is @c{\pi} ! @cite{pi} to the normal 12 decimal places, and then type @address@hidden 5 d n}} to display only five decimal places. If you then type @kbd{d n}, all 12 places reappear because the full number is still there on the Calc stack. More surprisingly, --- 30010,30016 ---- Embedded buffers remember active formulas for as long as they exist in Emacs memory. Suppose you have an embedded formula ! which is @cpi{} to the normal 12 decimal places, and then type @address@hidden 5 d n}} to display only five decimal places. If you then type @kbd{d n}, all 12 places reappear because the full number is still there on the Calc stack. More surprisingly, *************** *** 29820,29828 **** @end example @noindent ! records @cite{5} as the stored value of @code{foo} for the purposes of Embedded mode operations in the current buffer. It ! does @emph{not} actually store @cite{5} as the global'' value of @code{foo}, however. Regular Calc operations, and Embedded formulas in other buffers, will not see this assignment. --- 30148,30156 ---- @end example @noindent ! records @expr{5} as the stored value of @code{foo} for the purposes of Embedded mode operations in the current buffer. It ! does @emph{not} actually store @expr{5} as the global'' value of @code{foo}, however. Regular Calc operations, and Embedded formulas in other buffers, will not see this assignment. *************** *** 29891,29897 **** mode. Then, we have to find a way to get Embedded mode to notice the change. The @kbd{M-# u} or @kbd{M-# =} (@code{calc-embedded-update-formula}) command is a convenient way ! to do address@hidden @example foo := 6 --- 30219,30225 ---- mode. Then, we have to find a way to get Embedded mode to notice the change. The @kbd{M-# u} or @kbd{M-# =} (@code{calc-embedded-update-formula}) command is a convenient way ! to do this. @example foo := 6 *************** *** 30458,30464 **** Press address@hidden (or the standard Emacs key sequence @kbd{C-x e}) to execute your keyboard macro by replaying the recorded keystrokes. @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further ! address@hidden When you use @kbd{X} to invoke a keyboard macro, the entire macro is treated as a single command by the undo and trail features. The stack --- 30786,30792 ---- Press address@hidden (or the standard Emacs key sequence @kbd{C-x e}) to execute your keyboard macro by replaying the recorded keystrokes. @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further ! information. When you use @kbd{X} to invoke a keyboard macro, the entire macro is treated as a single command by the undo and trail features. The stack *************** *** 30504,30510 **** just the @key{RET} key as in this example) will be something like @samp{calc-User-n}. The keyboard macro will now be available as both @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more ! descriptive command name if you address@hidden Macros defined by @kbd{Z K} act like single commands; they are executed in the same way as by the @kbd{X} key. If you wish to define the macro --- 30832,30838 ---- just the @key{RET} key as in this example) will be something like @samp{calc-User-n}. The keyboard macro will now be available as both @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more ! descriptive command name if you wish. Macros defined by @kbd{Z K} act like single commands; they are executed in the same way as by the @kbd{X} key. If you wish to define the macro *************** *** 30522,30528 **** the macro definition into full Emacs command names, like @code{calc-pop} and @code{calc-add}. Type @kbd{M-# M-#} to finish editing and update the definition stored on the key, or, to cancel the edit, type ! @kbd{M-# address@hidden If you give a negative numeric prefix argument to @kbd{Z E}, the keyboard macro is edited in spelled-out keystroke form. For example, the editing --- 30850,30856 ---- the macro definition into full Emacs command names, like @code{calc-pop} and @code{calc-add}. Type @kbd{M-# M-#} to finish editing and update the definition stored on the key, or, to cancel the edit, type ! @kbd{M-# x}. If you give a negative numeric prefix argument to @kbd{Z E}, the keyboard macro is edited in spelled-out keystroke form. For example, the editing *************** *** 30536,30548 **** same as is used in all of this manual's examples, except that the manual takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}}, we take it for granted that it is clear we really mean @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}, ! which is what @code{read-kbd-macro} wants to address@hidden If @file{macedit} is not available, @kbd{Z E} edits the keyboard macro in raw'' form; the editing buffer simply contains characters like @samp{1^M2+} (here @samp{^M} represents the carriage-return character). Editing in this mode, you will have to use @kbd{C-q} to enter new ! control characters into the address@hidden @kindex M-# m @pindex read-kbd-macro --- 30864,30876 ---- same as is used in all of this manual's examples, except that the manual takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}}, we take it for granted that it is clear we really mean @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}, ! which is what @code{read-kbd-macro} wants to see. If @file{macedit} is not available, @kbd{Z E} edits the keyboard macro in raw'' form; the editing buffer simply contains characters like @samp{1^M2+} (here @samp{^M} represents the carriage-return character). Editing in this mode, you will have to use @kbd{C-q} to enter new ! control characters into the buffer. @kindex M-# m @pindex read-kbd-macro *************** *** 30646,30652 **** computes two to a nonnegative integer power. First, we push 1 on the stack and then swap the integer argument back to the top. The @kbd{Z <} pops that argument leaving the 1 back on top of the stack. Then, we ! repeat a multiply-by-two step however many address@hidden Once again, the keyboard macro is executed as it is being entered. In this case it is especially important to set up reasonable initial --- 30974,30980 ---- computes two to a nonnegative integer power. First, we push 1 on the stack and then swap the integer argument back to the top. The @kbd{Z <} pops that argument leaving the 1 back on top of the stack. Then, we ! repeat a multiply-by-two step however many times. Once again, the keyboard macro is executed as it is being entered. In this case it is especially important to set up reasonable initial *************** *** 30666,30672 **** innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no effect. Thus @address@hidden Z /} is similar to @samp{if (@var{cond}) break;} ! in the C address@hidden @kindex Z ( @kindex Z ) --- 30994,31000 ---- innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no effect. Thus @address@hidden Z /} is similar to @samp{if (@var{cond}) break;} ! in the C language. @kindex Z ( @kindex Z ) *************** *** 30680,30686 **** a temporary internal counter and initializes it with the value @var{init}. The @kbd{Z (} command then repeatedly pushes the counter value onto the stack and executes @var{body} and @var{step}, adding @var{step} to the ! counter each time until the loop address@hidden @cindex Summations (by keyboard macros) By default, the loop finishes when the counter becomes greater than (or --- 31008,31014 ---- a temporary internal counter and initializes it with the value @var{init}. The @kbd{Z (} command then repeatedly pushes the counter value onto the stack and executes @var{body} and @var{step}, adding @var{step} to the ! counter each time until the loop finishes. @cindex Summations (by keyboard macros) By default, the loop finishes when the counter becomes greater than (or *************** *** 30880,30886 **** new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}. Then the new function can be invoked by pushing two numbers on the stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic ! formula @samp{yow(x,y)address@hidden The fourth prompt is for the function's argument list. This is used to associate values on the stack with the variables that appear in the formula. --- 31208,31214 ---- new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}. Then the new function can be invoked by pushing two numbers on the stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic ! formula @samp{yow(x,y)}. The fourth prompt is for the function's argument list. This is used to associate values on the stack with the variables that appear in the formula. *************** *** 30891,30899 **** @samp{b} (respectively) in the formula, then simplify the formula and push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m} would replace the 10 and 100 on the stack with the number 210, which is ! @cite{a + 2 b} with @cite{a=10} and @cite{b=100}. Likewise, the formula ! @samp{yow(10, 100)} will be evaluated by substituting @cite{a=10} and ! @cite{b=100} in the definition. You can rearrange the order of the names before pressing @key{RET} to control which stack positions go to which variables in the formula. If --- 31219,31227 ---- @samp{b} (respectively) in the formula, then simplify the formula and push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m} would replace the 10 and 100 on the stack with the number 210, which is ! @expr{a + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula ! @samp{yow(10, 100)} will be evaluated by substituting @expr{a=10} and ! @expr{b=100} in the definition. You can rearrange the order of the names before pressing @key{RET} to control which stack positions go to which variables in the formula. If *************** *** 30911,30922 **** The final prompt is a y-or-n question concerning what to do if symbolic arguments are given to your function. If you answer @kbd{y}, then executing @kbd{z m} (using the original argument list @samp{(a b)}) with ! arguments @cite{10} and @cite{x} will leave the function in symbolic form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n}, then the formula will always be expanded, even for non-constant arguments: @samp{10 + 2 x}. If you never plan to feed algebraic formulas to your new function, it doesn't matter how you answer this ! address@hidden If you answered @kbd{y} to this question you can still cause a function call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}). --- 31239,31250 ---- The final prompt is a y-or-n question concerning what to do if symbolic arguments are given to your function. If you answer @kbd{y}, then executing @kbd{z m} (using the original argument list @samp{(a b)}) with ! arguments @expr{10} and @expr{x} will leave the function in symbolic form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n}, then the formula will always be expanded, even for non-constant arguments: @samp{10 + 2 x}. If you never plan to feed algebraic formulas to your new function, it doesn't matter how you answer this ! question. If you answered @kbd{y} to this question you can still cause a function call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}). *************** *** 30931,30937 **** key onto the stack. Actually, it pushes a nameless function that specifies both the argument list and the defining formula. You will get an error message if the key is undefined, or if the key was not defined ! by a @kbd{Z F} address@hidden The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has been defined by a formula uses a variant of the @code{calc-edit} command --- 31259,31265 ---- key onto the stack. Actually, it pushes a nameless function that specifies both the argument list and the defining formula. You will get an error message if the key is undefined, or if the key was not defined ! by a @kbd{Z F} command. The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has been defined by a formula uses a variant of the @code{calc-edit} command *************** *** 30951,30957 **** used as a function definition. For example, the formula @samp{deriv(a^2,v)} which might be used to define a new function @samp{dsqr(a,v)} will be simplified'' to 0 immediately upon entry since @code{deriv} considers ! @cite{a} to be constant with respect to @cite{v}. Turning off default simplifications cures this problem: The definition will be stored in symbolic form without ever activating the @code{deriv} function. Press @kbd{m D} to turn the default simplifications back on afterwards. --- 31279,31285 ---- used as a function definition. For example, the formula @samp{deriv(a^2,v)} which might be used to define a new function @samp{dsqr(a,v)} will be simplified'' to 0 immediately upon entry since @code{deriv} considers ! @expr{a} to be constant with respect to @expr{v}. Turning off default simplifications cures this problem: The definition will be stored in symbolic form without ever activating the @code{deriv} function. Press @kbd{m D} to turn the default simplifications back on afterwards. *************** *** 31049,31055 **** @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=}, @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor}, @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for ! @code{math-nearly-equal}, which is useful in implementing Taylor address@hidden For other functions @var{func}, if a function by the name @address@hidden exists it is used, otherwise if a function by the --- 31377,31383 ---- @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=}, @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor}, @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for ! @code{math-nearly-equal}, which is useful in implementing Taylor series. For other functions @var{func}, if a function by the name @address@hidden exists it is used, otherwise if a function by the *************** *** 31057,31069 **** is defined as a function it is used, otherwise @address@hidden is used on the assumption that this is a to-be-defined math function. Also, if the function name is quoted as in @samp{('integerp a)} the function name is ! always used exactly as written (but not quoted)address@hidden Variable names have @samp{var-} prepended to them unless they appear in the function's argument list or in an enclosing @code{let}, @code{let*}, @code{for}, or @code{foreach} form, or their names already contain a @samp{-} character. Thus a reference to ! @samp{foo} is the same as a reference to @address@hidden A few other Lisp extensions are available in @code{defmath} definitions: --- 31385,31397 ---- is defined as a function it is used, otherwise @address@hidden is used on the assumption that this is a to-be-defined math function. Also, if the function name is quoted as in @samp{('integerp a)} the function name is ! always used exactly as written (but not quoted). Variable names have @samp{var-} prepended to them unless they appear in the function's argument list or in an enclosing @code{let}, @code{let*}, @code{for}, or @code{foreach} form, or their names already contain a @samp{-} character. Thus a reference to ! @samp{foo} is the same as a reference to @samp{var-foo}. A few other Lisp extensions are available in @code{defmath} definitions: *************** *** 31081,31098 **** a synonym of @code{setq}.) Specifically, the first argument of @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form, in which case the effect is to store into the specified ! element of a list. Thus, @samp{(setq (elt m i j) x)} stores @cite{x} into one element of a matrix. @item A @code{for} looping construct is available. For example, @samp{(for ((i 0 10)) body)} executes @code{body} once for each ! binding of @cite{i} from zero to 10. This is like a @code{let} ! form in that @cite{i} is temporarily bound to the loop count without disturbing its value outside the @code{for} construct. Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)}, ! are also available. For each value of @cite{i} from zero to 10, ! @cite{j} counts from 0 to @cite{i-1} in steps of two. Note that @code{for} has the same general outline as @code{let*}, except that each element of the header is a list of three or four things, not just two. --- 31409,31426 ---- a synonym of @code{setq}.) Specifically, the first argument of @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form, in which case the effect is to store into the specified ! element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x} into one element of a matrix. @item A @code{for} looping construct is available. For example, @samp{(for ((i 0 10)) body)} executes @code{body} once for each ! binding of @expr{i} from zero to 10. This is like a @code{let} ! form in that @expr{i} is temporarily bound to the loop count without disturbing its value outside the @code{for} construct. Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)}, ! are also available. For each value of @expr{i} from zero to 10, ! @expr{j} counts from 0 to @expr{i-1} in steps of two. Note that @code{for} has the same general outline as @code{let*}, except that each element of the header is a list of three or four things, not just two. *************** *** 31100,31107 **** @item The @code{foreach} construct loops over elements of a list. For example, @samp{(foreach ((x (cdr v))) body)} executes ! @code{body} with @cite{x} bound to each element of Calc vector ! @cite{v} in turn. The purpose of @code{cdr} here is to skip over the initial @code{vec} symbol in the vector. @item --- 31428,31435 ---- @item The @code{foreach} construct loops over elements of a list. For example, @samp{(foreach ((x (cdr v))) body)} executes ! @code{body} with @expr{x} bound to each element of Calc vector ! @expr{v} in turn. The purpose of @code{cdr} here is to skip over the initial @code{vec} symbol in the vector. @item *************** *** 31112,31118 **** @item The @code{return} function prematurely returns from the enclosing ! function. For example, @samp{(return (+ x y))} returns @cite{x+y} as the value of a function. You can use @code{return} anywhere inside the body of the function. @end itemize --- 31440,31446 ---- @item The @code{return} function prematurely returns from the enclosing ! function. For example, @samp{(return (+ x y))} returns @expr{x+y} as the value of a function. You can use @code{return} anywhere inside the body of the function. @end itemize *************** *** 31464,31470 **** In this example, the command @code{calc-foo} will evaluate the expression @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if ! executed with a numeric prefix argument of @cite{n}. The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"} code as used with @code{defun}). It uses the numeric prefix argument as the --- 31792,31798 ---- In this example, the command @code{calc-foo} will evaluate the expression @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if ! executed with a numeric prefix argument of @expr{n}. The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"} code as used with @code{defun}). It uses the numeric prefix argument as the *************** *** 31501,31507 **** The argument must not be an incomplete vector, interval, or complex number. (This is rarely needed since the Calculator itself will never call your function with an incomplete argument. But there is nothing stopping your ! own Lisp code from calling your function with an incomplete argument.)@refill @item integer @findex integer --- 31829,31835 ---- The argument must not be an incomplete vector, interval, or complex number. (This is rarely needed since the Calculator itself will never call your function with an incomplete argument. But there is nothing stopping your ! own Lisp code from calling your function with an incomplete argument.) @item integer @findex integer *************** *** 31665,31672 **** @end ignore @tindex mysin A somewhat limited sine function could be defined as follows, using the ! well-known Taylor series expansion for @c{\sin x} ! @samp{sin(x)}: @smallexample (defmath mysin ((float (anglep x))) --- 31993,32001 ---- @end ignore @tindex mysin A somewhat limited sine function could be defined as follows, using the ! well-known Taylor series expansion for ! @texline @tmath{\sin x}: ! @infoline @samp{sin(x)}: @smallexample (defmath mysin ((float (anglep x))) *************** *** 31688,31695 **** @end smallexample The actual @code{sin} function in Calc works by first reducing the problem ! to a sine or cosine of a nonnegative number less than @c{\pi \over 4} ! @cite{pi/4}. This ensures that the Taylor series will converge quickly. Also, the calculation is carried out with two extra digits of precision to guard against cumulative round-off in @samp{sum}. Finally, complex arguments are allowed and handled --- 32017,32023 ---- @end smallexample The actual @code{sin} function in Calc works by first reducing the problem ! to a sine or cosine of a nonnegative number less than @cpiover{4}. This ensures that the Taylor series will converge quickly. Also, the calculation is carried out with two extra digits of precision to guard against cumulative round-off in @samp{sum}. Finally, complex arguments are allowed and handled *************** *** 31724,31733 **** series as before, and @code{mycos-raw} is a function analogous to @code{mysin-raw} for cosines. ! The strategy is to ensure that @cite{x} is nonnegative before calling @code{mysin-raw}. This function then recursively reduces its argument ! to a suitable range, namely, plus-or-minus @c{\pi \over 4} ! @cite{pi/4}. Note that each test, and particularly the first comparison against 7, is designed so that small roundoff errors cannot produce an infinite loop. (Suppose we compared with @samp{(two-pi)} instead; if due to roundoff problems --- 32052,32060 ---- series as before, and @code{mycos-raw} is a function analogous to @code{mysin-raw} for cosines. ! The strategy is to ensure that @expr{x} is nonnegative before calling @code{mysin-raw}. This function then recursively reduces its argument ! to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each test, and particularly the first comparison against 7, is designed so that small roundoff errors cannot produce an infinite loop. (Suppose we compared with @samp{(two-pi)} instead; if due to roundoff problems *************** *** 31877,31884 **** when the user has left Calc in symbolic mode or no-simplify mode. As another example, @samp{(equal (calc-eval '("<$$") nil a b) "1")} ! checks if the number in string @cite{a} is less than the one in ! string @cite{b}. Without using a list, the integer 1 might come out in a variety of formats which would be hard to test for conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But see Predicates'' mode, below.) --- 32204,32211 ---- when the user has left Calc in symbolic mode or no-simplify mode. As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")} ! checks if the number in string @expr{a} is less than the one in ! string @expr{b}. Without using a list, the integer 1 might come out in a variety of formats which would be hard to test for conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But see Predicates'' mode, below.) *************** *** 32218,32230 **** are that the number 0.0 is always stored as @samp{(float 0 0)}, and, except for the 0.0 case, the rightmost base-10 digit of @var{mant} is always nonzero. (If the rightmost digit is zero, the number is ! rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)@refill Rectangular complex numbers are stored in the form @samp{(cplx @var{re} @var{im})}, where @var{re} and @var{im} are each real numbers, either integers, fractions, or floats. The value is @address@hidden + @var{im}i}. The @var{im} part is nonzero; complex numbers with zero imaginary ! components are converted to real numbers address@hidden Polar complex numbers are stored in the form @samp{(polar @var{r} @var{theta})}, where @var{r} is a positive real value and @var{theta} --- 32545,32557 ---- are that the number 0.0 is always stored as @samp{(float 0 0)}, and, except for the 0.0 case, the rightmost base-10 digit of @var{mant} is always nonzero. (If the rightmost digit is zero, the number is ! rearranged by dividing @var{mant} by ten and incrementing @var{exp}.) Rectangular complex numbers are stored in the form @samp{(cplx @var{re} @var{im})}, where @var{re} and @var{im} are each real numbers, either integers, fractions, or floats. The value is @address@hidden + @var{im}i}. The @var{im} part is nonzero; complex numbers with zero imaginary ! components are converted to real numbers automatically. Polar complex numbers are stored in the form @samp{(polar @var{r} @var{theta})}, where @var{r} is a positive real value and @var{theta} *************** *** 32233,32245 **** or @samp{(-pi ..@: pi)} radians, according to the current angular mode. If the angle is 0 the value is converted to a real number automatically. (If the angle is 180 degrees, the value is usually also converted to a ! negative real number.)@refill Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m} @var{s})}, where @var{h} is an integer or an integer-valued float (i.e., a float with @address@hidden >= 0}), @var{m} is an integer or integer-valued float in the range @address@hidden ..@: 60)}}, and @var{s} is any real number ! in the range @samp{[0 ..@: 60)address@hidden Date forms are stored as @samp{(date @var{n})}, where @var{n} is a real number that counts days since midnight on the morning of --- 32560,32572 ---- or @samp{(-pi ..@: pi)} radians, according to the current angular mode. If the angle is 0 the value is converted to a real number automatically. (If the angle is 180 degrees, the value is usually also converted to a ! negative real number.) Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m} @var{s})}, where @var{h} is an integer or an integer-valued float (i.e., a float with @address@hidden >= 0}), @var{m} is an integer or integer-valued float in the range @address@hidden ..@: 60)}}, and @var{s} is any real number ! in the range @samp{[0 ..@: 60)}. Date forms are stored as @samp{(date @var{n})}, where @var{n} is a real number that counts days since midnight on the morning of *************** *** 32293,32299 **** which represent units are not stored in any special way; they are units only because their names appear in the units table. If the value cell contains a string, it is parsed to get the variable's value when ! the variable is address@hidden A Lisp list with any other symbol as the first element is a function call. The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^}, --- 32620,32626 ---- which represent units are not stored in any special way; they are units only because their names appear in the units table. If the value cell contains a string, it is parsed to get the variable's value when ! the variable is used. A Lisp list with any other symbol as the first element is a function call. The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^}, *************** *** 32313,32319 **** wish to return multiple values. (The latter case is allowed only for functions which are the outer-level call in an expression whose value is about to be pushed on the stack; this feature is considered obsolete ! and is not used by any built-in Calc functions.)@refill @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals @subsubsection Interactive Functions --- 32640,32646 ---- wish to return multiple values. (The latter case is allowed only for functions which are the outer-level call in an expression whose value is about to be pushed on the stack; this feature is considered obsolete ! and is not used by any built-in Calc functions.) @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals @subsubsection Interactive Functions *************** *** 32349,32355 **** previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})}, which means that to undo requires calling the function @samp{(@var{undo} @var{args} @dots{})} and, if the undo is later redone, calling ! @samp{(@var{redo} @var{args} @dots{})address@hidden @end defun @defun calc-record-why msg args --- 32676,32682 ---- previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})}, which means that to undo requires calling the function @samp{(@var{undo} @var{args} @dots{})} and, if the undo is later redone, calling ! @samp{(@var{redo} @var{args} @dots{})}. @end defun @defun calc-record-why msg args *************** *** 32364,32370 **** (such as @code{integerp} or @code{numvecp}) which the arguments did not satisfy; it is expanded to a suitable string such as Expected an integer.'' The @code{reject-arg} function calls @code{calc-record-why} ! automatically; @address@hidden @end defun @defun calc-is-inverse --- 32691,32697 ---- (such as @code{integerp} or @code{numvecp}) which the arguments did not satisfy; it is expanded to a suitable string such as Expected an integer.'' The @code{reject-arg} function calls @code{calc-record-why} ! automatically; @pxref{Predicates}. @end defun @defun calc-is-inverse *************** *** 32391,32397 **** end up at level @var{n}, the next-to-last at level @var{n}+1, etc. The elements of @var{vals} are assumed to be valid Calc objects, and are not evaluated, rounded, or renormalized in any way. If @var{vals} ! is an empty list, nothing address@hidden The stack elements are pushed without any sub-formula selections. You can give an optional third argument to this function, which must --- 32718,32724 ---- end up at level @var{n}, the next-to-last at level @var{n}+1, etc. The elements of @var{vals} are assumed to be valid Calc objects, and are not evaluated, rounded, or renormalized in any way. If @var{vals} ! is an empty list, nothing happens. The stack elements are pushed without any sub-formula selections. You can give an optional third argument to this function, which must *************** *** 32410,32416 **** element will be next-to-last, etc. If @var{n} or @var{m} are out of range, the command is aborted with a suitable error message. If @var{n} is zero, the function returns an empty list. The stack elements are not ! evaluated, rounded, or address@hidden If any stack elements contain selections, and selections have not been disabled by the @kbd{j e} (@code{calc-enable-selections}) command, --- 32737,32743 ---- element will be next-to-last, etc. If @var{n} or @var{m} are out of range, the command is aborted with a suitable error message. If @var{n} is zero, the function returns an empty list. The stack elements are not ! evaluated, rounded, or renormalized. If any stack elements contain selections, and selections have not been disabled by the @kbd{j e} (@code{calc-enable-selections}) command, *************** *** 32506,32512 **** argument to apply the operator over many stack entries. If the prefix argument @var{arg} is @code{nil}, this uses @code{calc-enter-result} as outlined above. Otherwise, it maps the function over several stack ! elements; @pxref{Prefix Arguments}. For example,@refill @smallexample (defun calc-zeta (arg) --- 32833,32839 ---- argument to apply the operator over many stack entries. If the prefix argument @var{arg} is @code{nil}, this uses @code{calc-enter-result} as outlined above. Otherwise, it maps the function over several stack ! elements; @pxref{Prefix Arguments}. For example, @smallexample (defun calc-zeta (arg) *************** *** 32527,32533 **** the binary function @var{func} is reduced across the top @var{arg} stack elements; when the argument is negative, the function is mapped between the next-to-top @address@hidden stack elements and the ! top address@hidden @end defun @defun calc-stack-size --- 32854,32860 ---- the binary function @var{func} is reduced across the top @var{arg} stack elements; when the argument is negative, the function is mapped between the next-to-top @address@hidden stack elements and the ! top element. @end defun @defun calc-stack-size *************** *** 32541,32547 **** will be the @samp{.} line. If @var{n} is from 1 to the current stack size, this will be the beginning of the first line of that stack entry's display. If line numbers are enabled, this will move to the first character of the ! line number, not the stack entry address@hidden @end defun @defun calc-substack-height n --- 32868,32874 ---- will be the @samp{.} line. If @var{n} is from 1 to the current stack size, this will be the beginning of the first line of that stack entry's display. If line numbers are enabled, this will move to the first character of the ! line number, not the stack entry itself. @end defun @defun calc-substack-height n *************** *** 32551,32557 **** one line long (i.e., no matrices are displayed), the return value will be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big mode, the return value includes the blank lines that separate stack ! entries.)@refill @end defun @defun calc-refresh --- 32878,32884 ---- one line long (i.e., no matrices are displayed), the return value will be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big mode, the return value includes the blank lines that separate stack ! entries.) @end defun @defun calc-refresh *************** *** 32560,32566 **** display radix, which might change the appearance of existing stack entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing is suppressed, but a flag is set so that the entire stack will be refreshed ! rather than just the top few elements when the macro finishes.)@refill @end defun @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals --- 32887,32893 ---- display radix, which might change the appearance of existing stack entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing is suppressed, but a flag is set so that the entire stack will be refreshed ! rather than just the top few elements when the macro finishes.) @end defun @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals *************** *** 32739,32745 **** by checking whether @address@hidden - @var{y}} is @code{negp}. In @code{defmath}, the expression @samp{(< x y)} will automatically be converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=}, ! and @code{>=} are similarly converted in terms of @address@hidden @end defun @defun beforep x y --- 33066,33072 ---- by checking whether @address@hidden - @var{y}} is @code{negp}. In @code{defmath}, the expression @samp{(< x y)} will automatically be converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=}, ! and @code{>=} are similarly converted in terms of @code{lessp}. @end defun @defun beforep x y *************** *** 32771,32777 **** Returns true if @var{x} and @var{n} are numerically equal, where @var{n} is a fixnum which is not a multiple of 10. This will automatically be used by @code{defmath} in place of the more general @code{math-equal} ! whenever address@hidden @end defun @defun nearly-equal x y --- 33098,33104 ---- Returns true if @var{x} and @var{n} are numerically equal, where @var{n} is a fixnum which is not a multiple of 10. This will automatically be used by @code{defmath} in place of the more general @code{math-equal} ! whenever possible. @end defun @defun nearly-equal x y *************** *** 32808,32814 **** Abort the current function evaluation due to unacceptable argument values. This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a Lisp error which @code{normalize} will trap. The net effect is that the ! function call which led here will be left in symbolic address@hidden @end defun @defun inexact-value --- 33135,33141 ---- Abort the current function evaluation due to unacceptable argument values. This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a Lisp error which @code{normalize} will trap. The net effect is that the ! function call which led here will be left in symbolic form. @end defun @defun inexact-value *************** *** 32819,32825 **** @code{sin} function will call @code{inexact-value}, which will cause your function to be left unsimplified. You may instead wish to call @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic Mode will ! return the formula @samp{sin(5)} to your address@hidden @end defun @defun overflow --- 33146,33152 ---- @code{sin} function will call @code{inexact-value}, which will cause your function to be left unsimplified. You may instead wish to call @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic Mode will ! return the formula @samp{sin(5)} to your function. @end defun @defun overflow *************** *** 32840,32852 **** the documentation refers to the @code{calc-sqrt} address@hidden command, this means @code{calc-sqrt} is an interactive stack-based square-root command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt}) ! is the actual Lisp function for taking square address@hidden The functions @code{math-add}, @code{math-sub}, @code{math-mul}, @code{math-div}, @code{math-mod}, and @code{math-neg} are not included in this list, since @code{defmath} allows you to write native Lisp @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-}, ! respectively, address@hidden @defun normalize val (Full form: @code{math-normalize}.) --- 33167,33179 ---- the documentation refers to the @code{calc-sqrt} address@hidden command, this means @code{calc-sqrt} is an interactive stack-based square-root command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt}) ! is the actual Lisp function for taking square roots. The functions @code{math-add}, @code{math-sub}, @code{math-mul}, @code{math-div}, @code{math-mod}, and @code{math-neg} are not included in this list, since @code{defmath} allows you to write native Lisp @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-}, ! respectively, instead. @defun normalize val (Full form: @code{math-normalize}.) *************** *** 32857,32868 **** small. All the various data types are similarly converted to their standard forms. Variables are left alone, but function calls are actually evaluated in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will ! return address@hidden If a function call fails, because the function is void or has the wrong number of parameters, or because it returns @code{nil} or calls @code{reject-arg} or @code{inexact-result}, @code{normalize} returns ! the formula still in symbolic address@hidden If the current Simplification Mode is none'' or numeric arguments only,'' @code{normalize} will act appropriately. However, the more --- 33184,33195 ---- small. All the various data types are similarly converted to their standard forms. Variables are left alone, but function calls are actually evaluated in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will ! return 6. If a function call fails, because the function is void or has the wrong number of parameters, or because it returns @code{nil} or calls @code{reject-arg} or @code{inexact-result}, @code{normalize} returns ! the formula still in symbolic form. If the current Simplification Mode is none'' or numeric arguments only,'' @code{normalize} will act appropriately. However, the more *************** *** 32871,32883 **** which calls @code{normalize} and possibly some other routines, such as @code{simplify} or @code{simplify-units}. Programs generally will never call @code{calc-normalize} except when popping or pushing values ! on the address@hidden @end defun @defun evaluate-expr expr Replace all variables in @var{expr} that have values with their values, then use @code{normalize} to simplify the result. This is what happens ! when you press the @kbd{=} key address@hidden @end defun @defmac with-extra-prec n body --- 33198,33210 ---- which calls @code{normalize} and possibly some other routines, such as @code{simplify} or @code{simplify-units}. Programs generally will never call @code{calc-normalize} except when popping or pushing values ! on the stack. @end defun @defun evaluate-expr expr Replace all variables in @var{expr} that have values with their values, then use @code{normalize} to simplify the result. This is what happens ! when you press the @kbd{=} key interactively. @end defun @defmac with-extra-prec n body *************** *** 32948,32954 **** Compare the numbers @var{x} and @var{y}, and return @i{-1} if @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})}, 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is ! undefined or cannot be address@hidden @end defun @defun numdigs n --- 33275,33281 ---- Compare the numbers @var{x} and @var{y}, and return @i{-1} if @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})}, 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is ! undefined or cannot be determined. @end defun @defun numdigs n *************** *** 33000,33012 **** @defun idivmod x y Divide integer @var{x} by integer @var{y}; return a cons cell whose @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr} ! is @samp{(imod @var{x} @var{y})address@hidden @end defun @defun pow x y Compute @var{x} to the power @var{y}. In @code{defmath} code, this can also be written @samp{(^ @var{x} @var{y})} or ! @address@hidden(expt @var{x} @var{y})address@hidden @end defun @defun abs-approx x --- 33327,33339 ---- @defun idivmod x y Divide integer @var{x} by integer @var{y}; return a cons cell whose @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr} ! is @samp{(imod @var{x} @var{y})}. @end defun @defun pow x y Compute @var{x} to the power @var{y}. In @code{defmath} code, this can also be written @samp{(^ @var{x} @var{y})} or ! @address@hidden(expt @var{x} @var{y})}}. @end defun @defun abs-approx x *************** *** 33030,33036 **** @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi}, @code{e}, @code{sqrt-e}, @code{ln-2}, and @code{ln-10}. Each function returns a floating-point value in the current precision, and each uses ! caching so that all calls after the first are essentially address@hidden @end defun @defmac math-defcache @var{func} @var{initial} @var{form} --- 33357,33363 ---- @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi}, @code{e}, @code{sqrt-e}, @code{ln-2}, and @code{ln-10}. Each function returns a floating-point value in the current precision, and each uses ! caching so that all calls after the first are essentially free. @end defun @defmac math-defcache @var{func} @var{initial} @var{form} *************** *** 33045,33051 **** two least significant digits is stored in the cache. For example, calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34 digits, rounds it down to 32 digits for future use, then rounds it ! again to 30 digits for use in the present address@hidden @end defmac @findex half-circle --- 33372,33378 ---- two least significant digits is stored in the cache. For example, calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34 digits, rounds it down to 32 digits for future use, then rounds it ! again to 30 digits for use in the present request. @end defmac @findex half-circle *************** *** 33072,33078 **** @defun div-mod a b m Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if ! there is no solution, or if any of the arguments are not address@hidden @end defun @defun pow-mod a b m --- 33399,33405 ---- @defun div-mod a b m Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if ! there is no solution, or if any of the arguments are not integers. @end defun @defun pow-mod a b m *************** *** 33147,33153 **** @var{iters} parameter is the number of Fermat iterations to use, in the case that this is necessary. If @code{prime-test} returns maybe,'' you can call it again with the same @var{n} to get a greater certainty; ! @code{prime-test} remembers where it left address@hidden @end defun @defun to-simple-fraction f --- 33474,33480 ---- @var{iters} parameter is the number of Fermat iterations to use, in the case that this is necessary. If @code{prime-test} returns maybe,'' you can call it again with the same @var{n} to get a greater certainty; ! @code{prime-test} remembers where it left off. @end defun @defun to-simple-fraction f *************** *** 33198,33204 **** produce lists of more than two dimensions. Note that the object @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size, and is treated by this and other Calc routines as a plain vector of two ! address@hidden @end defun @defun dimension-error --- 33525,33531 ---- produce lists of more than two dimensions. Note that the object @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size, and is treated by this and other Calc routines as a plain vector of two ! elements. @end defun @defun dimension-error *************** *** 33246,33252 **** For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v} with each element increased by one. Note that using @samp{'+} would not work here, since @code{defmath} does not expand function names everywhere, ! just where they are in the function position of a Lisp address@hidden @end defun @defun reduce-vec f v --- 33573,33579 ---- For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v} with each element increased by one. Note that using @samp{'+} would not work here, since @code{defmath} does not expand function names everywhere, ! just where they are in the function position of a Lisp expression. @end defun @defun reduce-vec f v *************** *** 33296,33302 **** element of the result matrix will be @code{eq} to the corresponding element of @var{m}, but none of the @code{cons} cells that make up the structure of the matrix will be @code{eq}. If @var{m} is a plain ! vector, this is the same as @address@hidden @end defun @defun swap-rows m r1 r2 --- 33623,33629 ---- element of the result matrix will be @code{eq} to the corresponding element of @var{m}, but none of the @code{cons} cells that make up the structure of the matrix will be @code{eq}. If @var{m} is a plain ! vector, this is the same as @code{copy-sequence}. @end defun @defun swap-rows m r1 r2 *************** *** 33305,33311 **** function changes @var{m} itself rather than building up a new result matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)} is true, with the side effect of exchanging the first two rows of ! @address@hidden @end defun @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals --- 33632,33638 ---- function changes @var{m} itself rather than building up a new result matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)} is true, with the side effect of exchanging the first two rows of ! @var{m}. @end defun @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals *************** *** 33443,33449 **** implemented by temporarily binding the variable @code{math-living-dangerously} to @code{t} (using a @code{let} form) and calling @code{simplify}. Dangerous simplification rules are written to check this variable ! before taking any address@hidden @end defun @defun simplify-units expr --- 33770,33776 ---- implemented by temporarily binding the variable @code{math-living-dangerously} to @code{t} (using a @code{let} form) and calling @code{simplify}. Dangerous simplification rules are written to check this variable ! before taking any action. @end defun @defun simplify-units expr *************** *** 33465,33471 **** if it returns a result @code{equal} to the original @code{expr}, it is ignored and Calc goes on to try the next simplification rule that applies. If the function body returns something different, that new formula is ! substituted for @var{expr} in the original address@hidden At each point in the formula, rules are tried in the order of the original calls to @code{math-defsimplify}; the search stops after the --- 33792,33798 ---- if it returns a result @code{equal} to the original @code{expr}, it is ignored and Calc goes on to try the next simplification rule that applies. If the function body returns something different, that new formula is ! substituted for @var{expr} in the original formula. At each point in the formula, rules are tried in the order of the original calls to @code{math-defsimplify}; the search stops after the *************** *** 33498,33504 **** This is really a pair of rules written with one @code{math-defsimplify} for convenience; the first replaces @samp{arcsinh(-x)} with @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x}, ! replaces @samp{arcsinh(sinh(x))} with @address@hidden @end defmac @defun common-constant-factor expr --- 33825,33831 ---- This is really a pair of rules written with one @code{math-defsimplify} for convenience; the first replaces @samp{arcsinh(-x)} with @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x}, ! replaces @samp{arcsinh(sinh(x))} with @samp{x}. @end defmac @defun common-constant-factor expr *************** *** 33533,33539 **** rational numbers. This is the fraction composed of the GCD of the numerators of @var{a} and @var{b}, over the GCD of the denominators. It is used by @code{common-constant-factor}. Note that the standard ! @code{gcd} function uses the LCM to combine the address@hidden @end defun @defun map-tree func expr many --- 33860,33866 ---- rational numbers. This is the fraction composed of the GCD of the numerators of @var{a} and @var{b}, over the GCD of the denominators. It is used by @code{common-constant-factor}. Note that the standard ! @code{gcd} function uses the LCM to combine the denominators. @end defun @defun map-tree func expr many *************** *** 33548,33554 **** @var{func} functions may @emph{not} make destructive changes to @var{expr}. If a third argument @var{many} is provided, it is an integer which says how many times @var{func} may be applied; the ! default, as described above, is infinitely many address@hidden @end defun @defun compile-rewrites rules --- 33875,33881 ---- @var{func} functions may @emph{not} make destructive changes to @var{expr}. If a third argument @var{many} is provided, it is an integer which says how many times @var{func} may be applied; the ! default, as described above, is infinitely many times. @end defun @defun compile-rewrites rules *************** *** 33665,33678 **** cosines of more complicated arguments. An integration rule should return @code{nil} if it can't do the integral; if several rules are defined for the same function, they are tried in order until one returns a address@hidden ! address@hidden @end defmac @defmac math-defintegral-2 funcs body Define a rule for integrating a function or functions of two arguments. This is exactly analogous to @code{math-defintegral}, except that @var{body} is written as the body of a function with two arguments, @var{u} and ! @address@hidden @end defmac @defun solve-for lhs rhs var full --- 33992,34005 ---- cosines of more complicated arguments. An integration rule should return @code{nil} if it can't do the integral; if several rules are defined for the same function, they are tried in order until one returns a address@hidden ! result. @end defmac @defmac math-defintegral-2 funcs body Define a rule for integrating a function or functions of two arguments. This is exactly analogous to @code{math-defintegral}, except that @var{body} is written as the body of a function with two arguments, @var{u} and ! @var{v}. @end defmac @defun solve-for lhs rhs var full *************** *** 33685,33691 **** which return a rearranged equation or a functional inverse, respectively. If @var{full} is address@hidden, a full solution including dummy signs and dummy integers will be produced. User-defined inverses are provided ! as properties in a manner similar to derivatives:@refill @smallexample (put 'calcFunc-ln 'math-inverse --- 34012,34018 ---- which return a rearranged equation or a functional inverse, respectively. If @var{full} is address@hidden, a full solution including dummy signs and dummy integers will be produced. User-defined inverses are provided ! as properties in a manner similar to derivatives: @smallexample (put 'calcFunc-ln 'math-inverse *************** *** 33721,33732 **** @code{expr-contains} uses @code{equal} to test for matches, whereas @code{calc-find-sub-formula} uses @code{eq}. In the formula @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not ! @code{eq} to each address@hidden @end defun @defun expr-contains-count expr var Returns the number of occurrences of @var{var} as a subexpression ! of @var{expr}, or @code{nil} if there are no address@hidden @end defun @defun expr-depends expr var --- 34048,34059 ---- @code{expr-contains} uses @code{equal} to test for matches, whereas @code{calc-find-sub-formula} uses @code{eq}. In the formula @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not ! @code{eq} to each other. @end defun @defun expr-contains-count expr var Returns the number of occurrences of @var{var} as a subexpression ! of @var{expr}, or @code{nil} if there are no occurrences. @end defun @defun expr-depends expr var *************** *** 33744,33750 **** Returns a copy of @var{expr}, with all occurrences of @var{old} replaced by @var{new}. This treats @code{lambda} forms specially with respect to the dummy argument variables, so that the effect is always to return ! @var{expr} evaluated at @var{old} = @address@hidden @end defun @defun multi-subst expr old new --- 34071,34077 ---- Returns a copy of @var{expr}, with all occurrences of @var{old} replaced by @var{new}. This treats @code{lambda} forms specially with respect to the dummy argument variables, so that the effect is always to return ! @var{expr} evaluated at @var{old} = @var{new}. @end defun @defun multi-subst expr old new *************** *** 33763,33769 **** @defun expr-height expr Returns the height'' of @var{expr}, which is the deepest level to which function calls are nested. (Note that @address@hidden + @var{b}} ! counts as a function call.) For primitive objects, this returns address@hidden @end defun @defun polynomial-p expr var --- 34090,34096 ---- @defun expr-height expr Returns the height'' of @var{expr}, which is the deepest level to which function calls are nested. (Note that @address@hidden + @var{b}} ! counts as a function call.) For primitive objects, this returns zero. @end defun @defun polynomial-p expr var *************** *** 33775,33781 **** (@code{calc-expand}), would consist of a sum of terms in which @var{var} appears only raised to nonnegative integer powers. Note that if @var{var} does not occur in @var{expr}, then @var{expr} is considered ! a polynomial of degree address@hidden @end defun @defun is-polynomial expr var degree loose --- 34102,34108 ---- (@code{calc-expand}), would consist of a sum of terms in which @var{var} appears only raised to nonnegative integer powers. Note that if @var{var} does not occur in @var{expr}, then @var{expr} is considered ! a polynomial of degree 0. @end defun @defun is-polynomial expr var degree loose *************** *** 33797,33803 **** themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin x))}. The result will never be @code{nil} in loose mode, since any ! expression can be interpreted as a constant'' loose address@hidden @end defun @defun polynomial-base expr pred --- 34124,34130 ---- themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin x))}. The result will never be @code{nil} in loose mode, since any ! expression can be interpreted as a constant'' loose polynomial. @end defun @defun polynomial-base expr pred *************** *** 33810,33816 **** The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})}; you can use @var{pred} to specify additional conditions. Or, you could have @var{pred} build up a list of every suitable @var{subexpr} that ! is address@hidden @end defun @defun poly-simplify poly --- 34137,34143 ---- The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})}; you can use @var{pred} to specify additional conditions. Or, you could have @var{pred} build up a list of every suitable @var{subexpr} that ! is found. @end defun @defun poly-simplify poly *************** *** 33822,33828 **** Mix two polynomial lists @var{a} and @var{b} (in the form returned by @code{is-polynomial}) in a linear combination with coefficient expressions @var{ac} and @var{bc}. The result is a (not necessarily simplified) ! polynomial list representing @address@hidden @var{a} + @var{bc} @address@hidden @end defun @defun poly-mul a b --- 34149,34155 ---- Mix two polynomial lists @var{a} and @var{b} (in the form returned by @code{is-polynomial}) in a linear combination with coefficient expressions @var{ac} and @var{bc}. The result is a (not necessarily simplified) ! polynomial list representing @address@hidden @var{a} + @var{bc} @var{b}}. @end defun @defun poly-mul a b *************** *** 33835,33841 **** list @var{poly} applied to variable @var{var}. The @kbd{a c} (@code{calc-collect}) command uses @code{is-polynomial} to turn an expression into a coefficient list, then @code{build-polynomial-expr} ! to turn the list back into an expression in regular address@hidden @end defun @defun check-unit-name var --- 34162,34168 ---- list @var{poly} applied to variable @var{var}. The @kbd{a c} (@code{calc-collect}) command uses @code{is-polynomial} to turn an expression into a coefficient list, then @code{build-polynomial-expr} ! to turn the list back into an expression in regular form. @end defun @defun check-unit-name var *************** *** 33852,33858 **** Return true if @var{expr} contains any variables which can be interpreted as units. If @var{sub-exprs} is @code{t}, the entire expression is searched. If @var{sub-exprs} is @code{nil}, this ! checks whether @var{expr} is directly a units address@hidden @end defun @defun single-units-in-expr-p expr --- 34179,34185 ---- Return true if @var{expr} contains any variables which can be interpreted as units. If @var{sub-exprs} is @code{t}, the entire expression is searched. If @var{sub-exprs} is @code{nil}, this ! checks whether @var{expr} is directly a units expression. @end defun @defun single-units-in-expr-p expr *************** *** 33867,33873 **** is @code{nil}, use Calc's native base units. Otherwise, @var{which} can specify a units system, which is a list of two-element lists, where the first element is a Calc base symbol name and the second ! is an expression to substitute for address@hidden @end defun @defun remove-units expr --- 34194,34200 ---- is @code{nil}, use Calc's native base units. Otherwise, @var{which} can specify a units system, which is a list of two-element lists, where the first element is a Calc base symbol name and the second ! is an expression to substitute for it. @end defun @defun remove-units expr *************** *** 33903,33909 **** not have the form of a valid expression, return a list of the form @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index into @var{str} of the general location of the error, and @var{msg} is ! a string describing the address@hidden @end defun @defun read-exprs str --- 34230,34236 ---- not have the form of a valid expression, return a list of the form @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index into @var{str} of the general location of the error, and @var{msg} is ! a string describing the problem. @end defun @defun read-exprs str *************** *** 33922,33935 **** If @var{prompt} is given, it is the prompt string to use; the default is Algebraic:''. If @var{no-norm} is @code{t}, the formulas will be returned exactly as parsed; otherwise, they will be passed through ! @code{calc-normalize} address@hidden To support the use of @kbd{} characters in the algebraic entry, use @code{let} to bind @code{calc-dollar-values} to a list of the values to be substituted for @kbd{}, @kbd{$$}, and so on, and bind
@code{calc-dollar-used} to 0.  Upon return, @code{calc-dollar-used}
will have been changed to the highest number of consecutive @kbd{$}s ! that actually appeared in the address@hidden @end defun @defun format-number a --- 34249,34262 ---- If @var{prompt} is given, it is the prompt string to use; the default is Algebraic:''. If @var{no-norm} is @code{t}, the formulas will be returned exactly as parsed; otherwise, they will be passed through ! @code{calc-normalize} first. To support the use of @kbd{$} characters in the algebraic entry, use
@code{let} to bind @code{calc-dollar-values} to a list of the values
to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind @code{calc-dollar-used} to 0. Upon return, @code{calc-dollar-used} will have been changed to the highest number of consecutive @kbd{$}s
! that actually appeared in the input.
@end defun

@defun format-number a
***************
*** 33945,33951 ****
complex number format, and point character, are ignored to ensure the
result will be re-readable.  The @var{prec} parameter is normally 0; if
you pass a large integer like 1000 instead, the expression will be
! surrounded by parentheses unless it is a plain number or variable
@end defun

@defun format-nice-expr a width
--- 34272,34278 ----
complex number format, and point character, are ignored to ensure the
result will be re-readable.  The @var{prec} parameter is normally 0; if
you pass a large integer like 1000 instead, the expression will be
! surrounded by parentheses unless it is a plain number or variable name.
@end defun

@defun format-nice-expr a width
***************
*** 33964,33970 ****
contain newline characters to separate the lines.  The @var{w}
parameter, if given, is the target window size for which to format
the expressions.  If @var{w} is omitted, the width of the Calculator
@end defun

@defun compose-expr a prec
--- 34291,34297 ----
contain newline characters to separate the lines.  The @var{w}
parameter, if given, is the target window size for which to format
the expressions.  If @var{w} is omitted, the width of the Calculator
! window is used.
@end defun

@defun compose-expr a prec
***************
*** 34012,34018 ****
@defun comp-first-char c
If composition @var{c} is a flat'' composition, return the first
(leftmost) character of the composition as an integer.  Otherwise,
@end defun

@defun comp-last-char c
--- 34339,34345 ----
@defun comp-first-char c
If composition @var{c} is a flat'' composition, return the first
(leftmost) character of the composition as an integer.  Otherwise,
! return @code{nil}.
@end defun

@defun comp-last-char c
***************
*** 34192,34198 ****
to the appropriate file name.  You may also need to change the variables
@code{calc-gnuplot-plot-command} and @code{calc-gnuplot-print-command} in
order to get correct displays and hardcopies, respectively, of your

@ifinfo
@example
--- 34519,34525 ----
to the appropriate file name.  You may also need to change the variables
@code{calc-gnuplot-plot-command} and @code{calc-gnuplot-print-command} in
order to get correct displays and hardcopies, respectively, of your
! plots.

@ifinfo
@example
***************
*** 34207,34213 ****
@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} address@hidden

To print the Calc manual in one huge 470 page tome, you will need the
source code to this manual, @file{calc.texi}, available as part of the
--- 34534,34540 ----
@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 470 page tome, you will need the
source code to this manual, @file{calc.texi}, available as part of the
***************
*** 34253,34259 ****
@code{".emacs"} as a substring, and if the variable
@code{calc-loaded-settings-file} is @code{nil}, then Calc will

@ifinfo
@example
--- 34580,34586 ----
@code{".emacs"} as a substring, and if the variable
@code{calc-loaded-settings-file} is @code{nil}, then Calc will
! Calc is invoked.

@ifinfo
@example
***************
*** 35194,35214 ****
@enumerate
@c 1
@item
! Positive prefix arguments apply to @cite{n} stack entries.
! Negative prefix arguments apply to the @cite{-n}th stack entry.
A prefix of zero applies to the entire stack.  (For @key{LFD} and
@address@hidden, the meaning of the sign is reversed.)

@c 2
@item
! Positive prefix arguments apply to @cite{n} stack entries.
Negative prefix arguments apply to the top stack entry
! and the next @cite{-n} stack entries.

@c 3
@item
! Positive prefix arguments rotate top @cite{n} stack entries by one.
! Negative prefix arguments rotate the entire stack by @cite{-n}.
A prefix of zero reverses the entire stack.

@c 4
--- 35521,35541 ----
@enumerate
@c 1
@item
! Positive prefix arguments apply to @expr{n} stack entries.
! Negative prefix arguments apply to the @expr{-n}th stack entry.
A prefix of zero applies to the entire stack.  (For @key{LFD} and
@address@hidden, the meaning of the sign is reversed.)

@c 2
@item
! Positive prefix arguments apply to @expr{n} stack entries.
Negative prefix arguments apply to the top stack entry
! and the next @expr{-n} stack entries.

@c 3
@item
! Positive prefix arguments rotate top @expr{n} stack entries by one.
! Negative prefix arguments rotate the entire stack by @expr{-n}.
A prefix of zero reverses the entire stack.

@c 4
***************
*** 35217,35224 ****

@c 5
@item
! Positive prefix arguments specify a precision @cite{p}.
! Negative prefix arguments reduce the current precision by @cite{-p}.

@c 6
@item
--- 35544,35551 ----

@c 5
@item
! Positive prefix arguments specify a precision @expr{p}.
! Negative prefix arguments reduce the current precision by @expr{-p}.

@c 6
@item
***************
*** 35236,35252 ****

@c 9
@item
! Positive prefix arguments specify a word size of @cite{w} bits, unsigned.
! Negative prefix arguments specify a word size of @cite{w} bits, signed.

@c 10
@item
! Prefix arguments specify the shift amount @cite{n}.  The @cite{w} argument
cannot be specified in the keyboard version of this command.

@c 11
@item
! From the keyboard, @cite{d} is omitted and defaults to zero.

@c 12
@item
--- 35563,35579 ----

@c 9
@item
! Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
! Negative prefix arguments specify a word size of @expr{w} bits, signed.

@c 10
@item
! Prefix arguments specify the shift amount @expr{n}.  The @expr{w} argument
cannot be specified in the keyboard version of this command.

@c 11
@item
! From the keyboard, @expr{d} is omitted and defaults to zero.

@c 12
@item
***************
*** 35259,35274 ****

@c 14
@item
! A prefix argument, if any, is used for @cite{m} instead of taking
! @cite{m} from the stack.  @cite{M} may take any of these values:
@iftex
@end iftex
@table @asis
@item Integer
! Random integer in the interval @cite{[0 .. m)}.
@item Float
! Random floating-point number in the interval @cite{[0 .. m)}.
@item 0.0
Gaussian with mean 1 and standard deviation 0.
@item Error form
--- 35586,35601 ----

@c 14
@item
! A prefix argument, if any, is used for @expr{m} instead of taking
! @expr{m} from the stack.  @expr{M} may take any of these values:
@iftex
@end iftex
@table @asis
@item Integer
! Random integer in the interval @expr{[0 .. m)}.
@item Float
! Random floating-point number in the interval @expr{[0 .. m)}.
@item 0.0
Gaussian with mean 1 and standard deviation 0.
@item Error form
***************
*** 35308,35327 ****

@c 20
@item
! With a prefix argument of 1, take a single @address@hidden
! stack instead of two separate data vectors.

@c 21
@item
! The row or column number @cite{n} may be given as a numeric prefix
! argument instead.  A plain @kbd{C-u} prefix says to take @cite{n}
! from the top of the stack.  If @cite{n} is a vector or interval,
a subvector/submatrix of the input is created.

@c 22
@item
! The @cite{op} prompt can be answered with the key sequence for the
desired function, or with @kbd{x} or @kbd{z} followed by a function name,
or with @kbd{$} to take a formula from the top of the stack, or with @kbd{'} and a typed formula. In the last two cases, the formula may --- 35635,35655 ---- @c 20 @item ! With a prefix argument of 1, take a single ! @texline @address@hidden ! @infoline @address@hidden ! matrix from the stack instead of two separate data vectors. @c 21 @item ! The row or column number @expr{n} may be given as a numeric prefix ! argument instead. A plain @kbd{C-u} prefix says to take @expr{n} ! from the top of the stack. If @expr{n} is a vector or interval, a subvector/submatrix of the input is created. @c 22 @item ! The @expr{op} prompt can be answered with the key sequence for the desired function, or with @kbd{x} or @kbd{z} followed by a function name, or with @kbd{$} to take a formula from the top of the stack, or with
@kbd{'} and a typed formula.  In the last two cases, the formula may
***************
*** 35336,35342 ****
One of the mapping direction keys @kbd{_} (horizontal, i.e., map
by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
reduce down), or @kbd{=} (map or reduce by rows) may be used before
! entering @cite{op}; these modify the function name by adding the letter
@code{r} for rows,'' @code{c} for columns,'' @code{a} for across,''
or @code{d} for down.''

--- 35664,35670 ----
One of the mapping direction keys @kbd{_} (horizontal, i.e., map
by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
reduce down), or @kbd{=} (map or reduce by rows) may be used before
! entering @expr{op}; these modify the function name by adding the letter
@code{r} for rows,'' @code{c} for columns,'' @code{a} for across,''
or @code{d} for down.''

***************
*** 35374,35380 ****
@item -11
(@var{2})  Float with integer mantissa.
@item -12
! (@var{2})  Float with mantissa in @cite{[1 .. 10)}.
@item -13
(@var{1})  Date form (using date numbers).
@item -14
--- 35702,35708 ----
@item -11
(@var{2})  Float with integer mantissa.
@item -12
! (@var{2})  Float with mantissa in @expr{[1 .. 10)}.
@item -13
(@var{1})  Date form (using date numbers).
@item -14
***************
*** 35388,35400 ****

@c 25
@item
! A prefix argument specifies the size @cite{n} of the matrix.  With no
! prefix argument, @cite{n} is omitted and the size is inferred from
the input vector.

@c 26
@item
! The prefix argument specifies the starting position @cite{n} (default 1).

@c 27
@item
--- 35716,35728 ----

@c 25
@item
! A prefix argument specifies the size @expr{n} of the matrix.  With no
! prefix argument, @expr{n} is omitted and the size is inferred from
the input vector.

@c 26
@item
! The prefix argument specifies the starting position @expr{n} (default 1).

@c 27
@item
***************
*** 35460,35475 ****

@c 39
@item
! Answer for @cite{v} may also be of the form @cite{v = v_0} or
! @cite{v - v_0}.

@c 40
@item
! With a positive prefix argument, stack contains many @cite{y}'s and one
! common @cite{x}.  With a zero prefix, stack contains a vector of
! @cite{y}s and a common @cite{x}.  With a negative prefix, stack
! contains many @cite{[x,y]} vectors.  (For 3D plots, substitute
! @cite{z} for @cite{y} and @cite{x,y} for @cite{x}.)

@c 41
@item
--- 35788,35803 ----

@c 39
@item
! Answer for @expr{v} may also be of the form @expr{v = v_0} or
! @expr{v - v_0}.

@c 40
@item
! With a positive prefix argument, stack contains many @expr{y}'s and one
! common @expr{x}.  With a zero prefix, stack contains a vector of
! @expr{y}s and a common @expr{x}.  With a negative prefix, stack
! contains many @expr{[x,y]} vectors.  (For 3D plots, substitute
! @expr{z} for @expr{y} and @expr{x,y} for @expr{x}.)

@c 41
@item
***************
*** 35504,35522 ****
stack levels.  The notation @kbd{$3} refers to stack level three without causing that value to be removed from the stack. Use @key{LFD} in place of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET} ! to evaluate address@hidden @c 47 @item The variable is replaced by the formula shown on the right. The Inverse flag reverses the order of the operands, e.g., @kbd{I s - x} ! assigns @c{$x \coloneq a-x$} ! @cite{x := a-x}. @c 48 @item Press @kbd{?} repeatedly to see how to choose a model. Answer the ! variables prompt with @cite{iv} or @cite{iv;pv} to specify independent and parameter variables. A positive prefix argument takes @address@hidden vectors from the stack; a zero prefix takes a matrix and a vector from the stack. --- 35832,35851 ---- stack levels. The notation @kbd{$3} refers to stack level three without
causing that value to be removed from the stack.  Use @key{LFD} in place
of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
! to evaluate variables.

@c 47
@item
The variable is replaced by the formula shown on the right.  The
Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
! assigns
! @texline @tmath{x \coloneq a-x}.
! @infoline @expr{x := a-x}.

@c 48
@item
Press @kbd{?} repeatedly to see how to choose a model.  Answer the
! variables prompt with @expr{iv} or @expr{iv;pv} to specify
independent and parameter variables.  A positive prefix argument
takes @address@hidden vectors from the stack; a zero prefix takes a matrix
and a vector from the stack.