[Emacs-diffs] Changes to emacs/man/calc.texi,v

[Jay Belanger]

CVSROOT:        /cvsroot/emacs
Module name:    emacs
Changes by:     Jay Belanger <jpb>      07/06/21 03:28:16

Index: calc.texi
===================================================================
RCS file: /cvsroot/emacs/emacs/man/calc.texi,v
retrieving revision 1.97
retrieving revision 1.98
diff -u -b -r1.97 -r1.98
--- calc.texi   20 Jun 2007 19:33:19 -0000      1.97
+++ calc.texi   21 Jun 2007 03:28:16 -0000      1.98
@@ -124,28 +124,32 @@
 @end titlepage
 
 @c [begin]
address@hidden
address@hidden
 @node Top, Getting Started, (dir), (dir)
 @chapter The GNU Emacs Calculator
 
 @noindent
 @dfn{Calc} is an advanced desk calculator and mathematical tool
-that runs as part of the GNU Emacs environment.
+written by Dave Gillespie that runs as part of the GNU Emacs environment.
 
-This manual is divided into three major parts: ``Getting Started,''
-the ``Calc Tutorial,'' and the ``Calc Reference.''  The Tutorial
-introduces all the major aspects of Calculator use in an easy,
-hands-on way.  The remainder of the manual is a complete reference to
-the features of the Calculator.
+This manual, also written (mostly) by Dave Gillespie, is divided into
+three major parts: ``Getting Started,'' the ``Calc Tutorial,'' and the
+``Calc Reference.''  The Tutorial introduces all the major aspects of
+Calculator use in an easy, hands-on way.  The remainder of the manual is
+a complete reference to the features of the Calculator.
address@hidden ifnottex
 
address@hidden
 For help in the Emacs Info system (which you are using to read this
 file), type @kbd{?}.  (You can also type @kbd{h} to run through a
 longer Info tutorial.)
-
 @end ifinfo
+
 @menu
 * Getting Started::       General description and overview.
address@hidden
 * Interactive Tutorial::
address@hidden ifinfo
 * Tutorial::              A step-by-step introduction for beginners.
 
 * Introduction::          Introduction to the Calc reference manual.
@@ -179,7 +183,12 @@
 * Lisp Function Index::   Internal Lisp math functions.
 @end menu
 
address@hidden
 @node Getting Started, Interactive Tutorial, Top, Top
address@hidden ifinfo
address@hidden
address@hidden Getting Started, Tutorial, Top, Top
address@hidden ifnotinfo
 @chapter Getting Started
 @noindent
 This chapter provides a general overview of Calc, the GNU Emacs
@@ -267,12 +276,6 @@
 this manual ought to be readable even if you don't know or use Emacs
 regularly.
 
address@hidden
-The manual is divided into three major parts:@: the ``Getting
-Started'' chapter you are reading now, the Calc tutorial (chapter 2),
-and the Calc reference manual (the remaining chapters and appendices).
address@hidden ifinfo
address@hidden
 The manual is divided into three major parts:@: the ``Getting
 Started'' chapter you are reading now, the Calc tutorial (chapter 2),
 and the Calc reference manual (the remaining chapters and appendices).
@@ -280,7 +283,6 @@
 @c This manual has been printed in two volumes, the @dfn{Tutorial} and the
 @c @dfn{Reference}.  Both volumes include a copy of the ``Getting Started''
 @c chapter.
address@hidden iftex
 
 If you are in a hurry to use Calc, there is a brief ``demonstration''
 below which illustrates the major features of Calc in just a couple of
@@ -321,6 +323,7 @@
 function, or variable using @address@hidden k}}, @kbd{h f}, or @kbd{h v},
 respectively.  @xref{Help Commands}.
 
address@hidden
 The Calc manual can be printed, but because the manual is so large, you
 should only make a printed copy if you really need it.  To print the
 manual, you will need the @TeX{} typesetting program (this is a free
@@ -347,7 +350,7 @@
 @example
 dvips calc.dvi
 @end example
-
address@hidden ifnottex
 @c Printed copies of this manual are also available from the Free Software
 @c Foundation.
 
@@ -543,13 +546,13 @@
 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}.)
 
address@hidden
address@hidden
 @strong{Help functions.}  You can read about any command in the on-line
 manual.  Type @kbd{C-x * c} to return to Calc after each of these
 commands: @kbd{h k t N} to read about the @kbd{t N} command,
 @kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
 @kbd{h s} to read the Calc summary.
address@hidden iftex
address@hidden ifnotinfo
 @ifinfo
 @strong{Help functions.}  You can read about any command in the on-line
 manual.  Remember to type the letter @kbd{l}, then @kbd{C-x * c}, to
@@ -1251,9 +1254,12 @@
 @menu
 * Tutorial::
 @end menu
address@hidden ifinfo
 
 @node Tutorial, Introduction, Interactive Tutorial, Top
address@hidden ifinfo
address@hidden
address@hidden Tutorial, Introduction, Getting Started, Top
address@hidden ifnotinfo
 @chapter Tutorial
 
 @noindent
@@ -1272,32 +1278,22 @@
 self-explanatory.  @xref{Embedded Mode}, for a description of
 the Embedded mode interface.
 
address@hidden
-The easiest way to read this tutorial on-line is to have two windows on
-your Emacs screen, one with Calc and one with the Info system.  (If you
-have a printed copy of the manual you can use that instead.)  Press
address@hidden * c} to turn Calc on or to switch into the Calc window, and
-press @kbd{C-x * i} to start the Info system or to switch into its window.
-Or, you may prefer to use the tutorial in printed form.
address@hidden ifinfo
address@hidden
 The easiest way to read this tutorial on-line is to have two windows on
 your Emacs screen, one with Calc and one with the Info system.  (If you
 have a printed copy of the manual you can use that instead.)  Press
 @kbd{C-x * c} to turn Calc on or to switch into the Calc window, and
 press @kbd{C-x * i} to start the Info system or to switch into its window.
address@hidden iftex
 
 This tutorial is designed to be done in sequence.  But the rest of this
 manual does not assume you have gone through the tutorial.  The tutorial
 does not cover everything in the Calculator, but it touches on most
 general areas.
 
address@hidden
address@hidden
 You may wish to print out a copy of the Calc Summary and keep notes on
 it as you learn Calc.  @xref{About This Manual}, to see how to make a
 printed summary.  @xref{Summary}.
address@hidden ifinfo
address@hidden ifnottex
 @iftex
 The Calc Summary at the end of the reference manual includes some blank
 space for your own use.  You may wish to keep notes there as you learn
@@ -1334,13 +1330,13 @@
 @subsection RPN Calculations and the Stack
 
 @cindex RPN notation
address@hidden
address@hidden
 @noindent
 Calc normally uses RPN notation.  You may be familiar with the RPN
 system from Hewlett-Packard calculators, FORTH, or PostScript.
 (Reverse Polish Notation, RPN, is named after the Polish mathematician
 Jan Lukasiewicz.)
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \noindent
 Calc normally uses RPN notation.  You may be familiar with the RPN
@@ -1769,7 +1765,7 @@
 @noindent
 or, in large mathematical notation,
 
address@hidden
address@hidden
 @example
 @group
     3 * 4 * 5
@@ -1778,7 +1774,7 @@
      6 * 7
 @end group
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -3325,7 +3321,7 @@
 Matrix inverses are related to systems of linear equations in algebra.
 Suppose we had the following set of equations:
 
address@hidden
address@hidden
 @group
 @example
     a + 2b + 3c = 6
@@ -3333,7 +3329,7 @@
    7a + 6b      = 3
 @end example
 @end group
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplayh
@@ -3352,7 +3348,7 @@
 @noindent
 This can be cast into the matrix equation,
 
address@hidden
address@hidden
 @group
 @example
    [ [ 1, 2, 3 ]     [ [ a ]     [ [ 6 ]
@@ -3360,7 +3356,7 @@
      [ 7, 6, 0 ] ]     [ c ] ]     [ 3 ] ]
 @end example
 @end group
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -3425,14 +3421,14 @@
 system of equations to get expressions for @expr{x} and @expr{y}
 in terms of @expr{a} and @expr{b}.
 
address@hidden
address@hidden
 @group
 @example
    x + a y = 6
    x + b y = 10
 @end example
 @end group
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -3456,9 +3452,9 @@
 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}:
address@hidden
address@hidden
 @samp{trn(A)*A*X = trn(A)*B}.
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 $A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
@@ -3472,7 +3468,7 @@
 of equations.  Use Calc to solve the following over-determined
 system:
 
address@hidden
address@hidden
 @group
 @example
     a + 2b + 3c = 6
@@ -3481,7 +3477,7 @@
    2a + 4b + 6c = 11
 @end example
 @end group
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplayh
@@ -3749,11 +3745,11 @@
 
 In a least squares fit, the slope @expr{m} is given by the formula
 
address@hidden
address@hidden
 @example
 m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -3790,12 +3786,12 @@
 @end group
 @end smallexample
 
address@hidden
address@hidden
 @noindent
 These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
 respectively.  (We could have used @kbd{*} to compute @samp{sum(x^2)} and
 @samp{sum(x y)}.)
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
@@ -3845,11 +3841,11 @@
 That gives us the slope @expr{m}.  The y-intercept @expr{b} can now
 be found with the simple formula,
 
address@hidden
address@hidden
 @example
 b = (sum(y) - m sum(x)) / N
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -3987,14 +3983,14 @@
 with or without surrounding vector brackets.
 @xref{List Answer 3, 3}. (@bullet{})
 
address@hidden
address@hidden
 As another example, a theorem about binomial coefficients tells
 us that the alternating sum of binomial coefficients
 @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
 on up to @address@hidden,
 always comes out to zero.  Let's verify this
 for @expr{n=6}.
address@hidden ifinfo
address@hidden ifnottex
 @tex
 As another example, a theorem about binomial coefficients tells
 us that the alternating sum of binomial coefficients
@@ -5193,12 +5189,12 @@
 that the steps are not required to be flat.  Simpson's rule boils
 down to the formula,
 
address@hidden
address@hidden
 @example
 (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
               + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -5215,12 +5211,12 @@
 For reference, here is the corresponding formula for the stairstep
 method:
 
address@hidden
address@hidden
 @example
 h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
           + f(a+(n-2)*h) + f(a+(n-1)*h))
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -5657,11 +5653,11 @@
 infinite series that exactly equals the value of that function at
 values of @expr{x} near zero.
 
address@hidden
address@hidden
 @example
 cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -5675,11 +5671,11 @@
 Mathematicians often write a truncated series using a ``big-O'' notation
 that records what was the lowest term that was truncated.
 
address@hidden
address@hidden
 @example
 cos(x) = 1 - x^2 / 2! + O(x^3)
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -6204,11 +6200,11 @@
 @expr{x_0} which is reasonably close to the desired solution, apply
 this formula over and over:
 
address@hidden
address@hidden
 @example
 new_x = x - f(x)/f'(x)
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \beforedisplay
 $$ x_{\rm new} = x - {f(x) \over f'(x)} $$
@@ -6242,11 +6238,11 @@
 @infoline @expr{ln(gamma(z))}.  
 For large values of @expr{z}, it can be approximated by the infinite sum
 
address@hidden
address@hidden
 @example
 psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \beforedisplay
 $$ \psi(z) \approx \ln z - {1\over2z} -
@@ -6305,13 +6301,13 @@
 (@bullet{}) @strong{Exercise 11.}  The @dfn{Stirling numbers of the
 first kind} are defined by the recurrences,
 
address@hidden
address@hidden
 @example
 s(n,n) = 1   for n >= 0,
 s(n,0) = 0   for n > 0,
 s(n+1,m) = s(n,m-1) - n s(n,m)   for n >= m >= 1.
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -6843,14 +6839,14 @@
 @node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
 @subsection Matrix Tutorial Exercise 2
 
address@hidden
address@hidden
 @example
 @group
    x + a y = 6
    x + b y = 10
 @end group
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -6905,7 +6901,7 @@
 @infoline @expr{A2 * X = B2} 
 which we can solve using Calc's @samp{/} command.
 
address@hidden
address@hidden
 @example
 @group
     a + 2b + 3c = 6
@@ -6914,7 +6910,7 @@
    2a + 4b + 6c = 11
 @end group
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplayh
@@ -7045,11 +7041,11 @@
 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.
 
address@hidden
address@hidden
 @example
    m*x + b*1 = y
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -7836,11 +7832,11 @@
 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,
 
address@hidden
address@hidden
 @example
 3 (3 a + b - 511 m) + c - 511 n
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -7852,11 +7848,11 @@
 for some suitable integers @expr{m} and @expr{n}.  Expanding out by
 the distributive law yields
 
address@hidden
address@hidden
 @example
 9 a + 3 b + c - 511*3 m - 511 n
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -7870,11 +7866,11 @@
 term.  So we can take it out to get an equivalent formula with
 @expr{n' = 3m + n},
 
address@hidden
address@hidden
 @example
 9 a + 3 b + c - 511 n'
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -11285,7 +11281,7 @@
 of the possible range of values a computation will produce, given the
 set of possible values of the input.
 
address@hidden
address@hidden
 Calc supports several varieties of intervals, including @dfn{closed}
 intervals of the type shown above, @dfn{open} intervals such as
 @samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
@@ -11296,7 +11292,7 @@
 @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}.
address@hidden ifinfo
address@hidden ifnottex
 @tex
 Calc supports several varieties of intervals, including \dfn{closed}
 intervals of the type shown above, \dfn{open} intervals such as
@@ -11929,14 +11925,14 @@
 @pindex calc-trail-isearch-forward
 @kindex t r
 @pindex calc-trail-isearch-backward
address@hidden
address@hidden
 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
 (@code{calc-trail-isearch-backward}) commands perform an incremental
 search forward or backward through the trail.  You can press @key{RET}
 to terminate the search; the trail pointer moves to the current line.
 If you cancel the search with @kbd{C-g}, the trail pointer stays where
 it was when the search began.
address@hidden ifinfo
address@hidden ifnottex
 @tex
 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
 (@code{calc-trail-isearch-backward}) com\-mands perform an incremental
@@ -14237,10 +14233,10 @@
 Also, the ``discretionary multiplication sign'' @samp{\*} is read
 the same as @samp{*}.
 
address@hidden
address@hidden
 The @TeX{} version of this manual includes some printed examples at the
 end of this section.
address@hidden ifinfo
address@hidden ifnottex
 @iftex
 Here are some examples of how various Calc formulas are formatted in @TeX{}:
 
@@ -17656,7 +17652,7 @@
 (@code{calc-expand-formula}) command, or when taking derivatives or
 integrals or solving equations involving the functions.
 
address@hidden
address@hidden
 These formulas are shown using the conventions of Big display
 mode (@kbd{d B}); for example, the formula for @code{fv} written
 linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
@@ -17736,7 +17732,7 @@
 ddb(cost, salv, life, per) = --------,  book = cost - depreciation so far
                                life
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
@@ -18385,14 +18381,14 @@
 You can think of this as taking the other half of the integral, from
 @expr{x} to infinity.
 
address@hidden
address@hidden
 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.
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 The functions corresponding to the integrals that define $P(a,x)$
@@ -18908,10 +18904,10 @@
 @kindex H k c
 @pindex calc-perm
 @tindex perm
address@hidden
address@hidden
 The @kbd{H k c} (@code{calc-perm}) address@hidden command computes the
 number-of-permutations function @expr{N! / (N-M)!}.
address@hidden ifinfo
address@hidden ifnottex
 @tex
 The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
 number-of-perm\-utations function $N! \over (N-M)!\,$.
@@ -23151,13 +23147,13 @@
 command will again prompt for an integration variable, then prompt for a
 lower limit and an upper limit.
 
address@hidden
address@hidden
 If you use the @code{integ} function directly in an algebraic formula,
 you can also write @samp{integ(f,x,v)} which expresses the resulting
 indefinite integral in terms of variable @code{v} instead of @code{x}.
 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
 integral from @code{a} to @code{b}.
address@hidden ifinfo
address@hidden ifnottex
 @tex
 If you use the @code{integ} function directly in an algebraic formula,
 you can also write @samp{integ(f,x,v)} which expresses the resulting
@@ -24038,14 +24034,14 @@
 
 For example, suppose the data matrix
 
address@hidden
address@hidden
 @example
 @group
 [ [ 1, 2, 3, 4,  5  ]
   [ 5, 7, 9, 11, 13 ] ]
 @end group
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \turnoffactive
@@ -24102,11 +24098,11 @@
 the method of least squares.  The idea is to define the @dfn{chi-square}
 error measure
 
address@hidden
address@hidden
 @example
 chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -24291,11 +24287,11 @@
 @infoline @expr{chi^2}
 statistic is now,
 
address@hidden
address@hidden
 @example
 chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
 @end example
address@hidden ifinfo
address@hidden ifnottex
 @tex
 \turnoffactive
 \beforedisplay
@@ -27613,9 +27609,9 @@
 @tex
 for \AA ngstroms.
 @end tex
address@hidden
address@hidden
 for Angstroms.
address@hidden ifinfo
address@hidden ifnottex
 
 The unit @code{pt} stands for pints; the name @code{point} stands for
 a typographical point, defined by @samp{72 point = 1 in}.  This is
@@ -34535,9 +34531,9 @@
 @iftex
 @unnumberedsec TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
 @end iftex
address@hidden
address@hidden
 @center TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
address@hidden ifinfo
address@hidden ifnottex
 
 @enumerate 0
 @item
@@ -34760,9 +34756,9 @@
 @iftex
 @heading NO WARRANTY
 @end iftex
address@hidden
address@hidden
 @center NO WARRANTY
address@hidden ifinfo
address@hidden ifnottex
 
 @item
 BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY
@@ -34790,9 +34786,9 @@
 @iftex
 @heading END OF TERMS AND CONDITIONS
 @end iftex
address@hidden
address@hidden
 @center END OF TERMS AND CONDITIONS
address@hidden ifinfo
address@hidden ifnottex
 
 @page
 @unnumberedsec Appendix: How to Apply These Terms to Your New Programs