@end titlepage
@c [begin]
-@ifinfo
+@ifnottex
@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.
+@end ifnottex
+@ifinfo
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.
+@ifinfo
* Interactive Tutorial::
+@end ifinfo
* Tutorial:: A step-by-step introduction for beginners.
* Introduction:: Introduction to the Calc reference manual.
* Lisp Function Index:: Internal Lisp math functions.
@end menu
+@ifinfo
@node Getting Started, Interactive Tutorial, Top, Top
+@end ifinfo
+@ifnotinfo
+@node Getting Started, Tutorial, Top, Top
+@end ifnotinfo
@chapter Getting Started
@noindent
This chapter provides a general overview of Calc, the GNU Emacs
this manual ought to be readable even if you don't know or use Emacs
regularly.
-@ifinfo
-The manual is divided into three major parts:@: the ``Getting
-Started'' chapter you are reading now, the Calc tutorial (chapter 2),
-and the Calc reference manual (the remaining chapters and appendices).
-@end ifinfo
-@iftex
The manual is divided into three major parts:@: the ``Getting
Started'' chapter you are reading now, the Calc tutorial (chapter 2),
and the Calc reference manual (the remaining chapters and appendices).
@c This manual has been printed in two volumes, the @dfn{Tutorial} and the
@c @dfn{Reference}. Both volumes include a copy of the ``Getting Started''
@c chapter.
-@end iftex
If you are in a hurry to use Calc, there is a brief ``demonstration''
below which illustrates the major features of Calc in just a couple of
function, or variable using @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v},
respectively. @xref{Help Commands}.
+@ifnottex
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
@example
dvips calc.dvi
@end example
-
+@end ifnottex
@c Printed copies of this manual are also available from the Free Software
@c Foundation.
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
+@ifnotinfo
@strong{Help functions.} You can read about any command in the on-line
manual. Type @kbd{C-x * c} to return to Calc after each of these
commands: @kbd{h k t N} to read about the @kbd{t N} command,
@kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
@kbd{h s} to read the Calc summary.
-@end iftex
+@end 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
@menu
* Tutorial::
@end menu
-@end ifinfo
@node Tutorial, Introduction, Interactive Tutorial, Top
+@end ifinfo
+@ifnotinfo
+@node Tutorial, Introduction, Getting Started, Top
+@end ifnotinfo
@chapter Tutorial
@noindent
self-explanatory. @xref{Embedded Mode}, for a description of
the Embedded mode interface.
-@ifinfo
-The easiest way to read this tutorial on-line is to have two windows on
-your Emacs screen, one with Calc and one with the Info system. (If you
-have a printed copy of the manual you can use that instead.) Press
-@kbd{C-x * c} to turn Calc on or to switch into the Calc window, and
-press @kbd{C-x * i} to start the Info system or to switch into its window.
-Or, you may prefer to use the tutorial in printed form.
-@end ifinfo
-@iftex
The easiest way to read this tutorial on-line is to have two windows on
your Emacs screen, one with Calc and one with the Info system. (If you
have a printed copy of the manual you can use that instead.) Press
@kbd{C-x * c} to turn Calc on or to switch into the Calc window, and
press @kbd{C-x * i} to start the Info system or to switch into its window.
-@end iftex
This tutorial is designed to be done in sequence. But the rest of this
manual does not assume you have gone through the tutorial. The tutorial
does not cover everything in the Calculator, but it touches on most
general areas.
-@ifinfo
+@ifnottex
You may wish to print out a copy of the Calc Summary and keep notes on
it as you learn Calc. @xref{About This Manual}, to see how to make a
printed summary. @xref{Summary}.
-@end ifinfo
+@end 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
@subsection RPN Calculations and the Stack
@cindex RPN notation
-@ifinfo
+@ifnottex
@noindent
Calc normally uses RPN notation. You may be familiar with the RPN
system from Hewlett-Packard calculators, FORTH, or PostScript.
(Reverse Polish Notation, RPN, is named after the Polish mathematician
Jan Lukasiewicz.)
-@end ifinfo
+@end ifnottex
@tex
\noindent
Calc normally uses RPN notation. You may be familiar with the RPN
@noindent
or, in large mathematical notation,
-@ifinfo
+@ifnottex
@example
@group
3 * 4 * 5
6 * 7
@end group
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
Matrix inverses are related to systems of linear equations in algebra.
Suppose we had the following set of equations:
-@ifinfo
+@ifnottex
@group
@example
a + 2b + 3c = 6
7a + 6b = 3
@end example
@end group
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplayh
@noindent
This can be cast into the matrix equation,
-@ifinfo
+@ifnottex
@group
@example
[ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
[ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
@end example
@end group
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
system of equations to get expressions for @expr{x} and @expr{y}
in terms of @expr{a} and @expr{b}.
-@ifinfo
+@ifnottex
@group
@example
x + a y = 6
x + b y = 10
@end example
@end group
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
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
+@ifnottex
@samp{trn(A)*A*X = trn(A)*B}.
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
$A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
of equations. Use Calc to solve the following over-determined
system:
-@ifinfo
+@ifnottex
@group
@example
a + 2b + 3c = 6
2a + 4b + 6c = 11
@end example
@end group
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplayh
In a least squares fit, the slope @expr{m} is given by the formula
-@ifinfo
+@ifnottex
@example
m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
@end group
@end smallexample
-@ifinfo
+@ifnottex
@noindent
These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
@samp{sum(x y)}.)
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
That gives us the slope @expr{m}. The y-intercept @expr{b} can now
be found with the simple formula,
-@ifinfo
+@ifnottex
@example
b = (sum(y) - m sum(x)) / N
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
with or without surrounding vector brackets.
@xref{List Answer 3, 3}. (@bullet{})
-@ifinfo
+@ifnottex
As another example, a theorem about binomial coefficients tells
us that the alternating sum of binomial coefficients
@var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
on up to @var{n}-choose-@var{n},
always comes out to zero. Let's verify this
for @expr{n=6}.
-@end ifinfo
+@end ifnottex
@tex
As another example, a theorem about binomial coefficients tells
us that the alternating sum of binomial coefficients
that the steps are not required to be flat. Simpson's rule boils
down to the formula,
-@ifinfo
+@ifnottex
@example
(h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
+ 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
For reference, here is the corresponding formula for the stairstep
method:
-@ifinfo
+@ifnottex
@example
h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
+ f(a+(n-2)*h) + f(a+(n-1)*h))
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
infinite series that exactly equals the value of that function at
values of @expr{x} near zero.
-@ifinfo
+@ifnottex
@example
cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
Mathematicians often write a truncated series using a ``big-O'' notation
that records what was the lowest term that was truncated.
-@ifinfo
+@ifnottex
@example
cos(x) = 1 - x^2 / 2! + O(x^3)
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
@expr{x_0} which is reasonably close to the desired solution, apply
this formula over and over:
-@ifinfo
+@ifnottex
@example
new_x = x - f(x)/f'(x)
@end example
-@end ifinfo
+@end ifnottex
@tex
\beforedisplay
$$ x_{\rm new} = x - {f(x) \over f'(x)} $$
@infoline @expr{ln(gamma(z))}.
For large values of @expr{z}, it can be approximated by the infinite sum
-@ifinfo
+@ifnottex
@example
psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
@end example
-@end ifinfo
+@end ifnottex
@tex
\beforedisplay
$$ \psi(z) \approx \ln z - {1\over2z} -
(@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
first kind} are defined by the recurrences,
-@ifinfo
+@ifnottex
@example
s(n,n) = 1 for n >= 0,
s(n,0) = 0 for n > 0,
s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
@node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
@subsection Matrix Tutorial Exercise 2
-@ifinfo
+@ifnottex
@example
@group
x + a y = 6
x + b y = 10
@end group
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
@infoline @expr{A2 * X = B2}
which we can solve using Calc's @samp{/} command.
-@ifinfo
+@ifnottex
@example
@group
a + 2b + 3c = 6
2a + 4b + 6c = 11
@end group
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplayh
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
+@ifnottex
@example
m*x + b*1 = y
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
subtracting off enough 511's to put the result in the desired range.
So the result when we take the modulo after every step is,
-@ifinfo
+@ifnottex
@example
3 (3 a + b - 511 m) + c - 511 n
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
for some suitable integers @expr{m} and @expr{n}. Expanding out by
the distributive law yields
-@ifinfo
+@ifnottex
@example
9 a + 3 b + c - 511*3 m - 511 n
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
term. So we can take it out to get an equivalent formula with
@expr{n' = 3m + n},
-@ifinfo
+@ifnottex
@example
9 a + 3 b + c - 511 n'
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
of the possible range of values a computation will produce, given the
set of possible values of the input.
-@ifinfo
+@ifnottex
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
@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
+@end ifnottex
@tex
Calc supports several varieties of intervals, including \dfn{closed}
intervals of the type shown above, \dfn{open} intervals such as
@pindex calc-trail-isearch-forward
@kindex t r
@pindex calc-trail-isearch-backward
-@ifinfo
+@ifnottex
The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
(@code{calc-trail-isearch-backward}) commands perform an incremental
search forward or backward through the trail. You can press @key{RET}
to terminate the search; the trail pointer moves to the current line.
If you cancel the search with @kbd{C-g}, the trail pointer stays where
it was when the search began.
-@end ifinfo
+@end 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
Also, the ``discretionary multiplication sign'' @samp{\*} is read
the same as @samp{*}.
-@ifinfo
+@ifnottex
The @TeX{} version of this manual includes some printed examples at the
end of this section.
-@end ifinfo
+@end ifnottex
@iftex
Here are some examples of how various Calc formulas are formatted in @TeX{}:
(@code{calc-expand-formula}) command, or when taking derivatives or
integrals or solving equations involving the functions.
-@ifinfo
+@ifnottex
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}.
ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
life
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
$$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
You can think of this as taking the other half of the integral, from
@expr{x} to infinity.
-@ifinfo
+@ifnottex
The functions corresponding to the integrals that define @expr{P(a,x)}
and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
(where @expr{g} and @expr{G} represent the lower- and upper-case Greek
letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}]
and @kbd{H I f G} [@code{gammaG}] commands.
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
The functions corresponding to the integrals that define $P(a,x)$
@kindex H k c
@pindex calc-perm
@tindex perm
-@ifinfo
+@ifnottex
The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the
number-of-permutations function @expr{N! / (N-M)!}.
-@end ifinfo
+@end ifnottex
@tex
The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
number-of-perm\-utations function $N! \over (N-M)!\,$.
command will again prompt for an integration variable, then prompt for a
lower limit and an upper limit.
-@ifinfo
+@ifnottex
If you use the @code{integ} function directly in an algebraic formula,
you can also write @samp{integ(f,x,v)} which expresses the resulting
indefinite integral in terms of variable @code{v} instead of @code{x}.
With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
integral from @code{a} to @code{b}.
-@end ifinfo
+@end 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
For example, suppose the data matrix
-@ifinfo
+@ifnottex
@example
@group
[ [ 1, 2, 3, 4, 5 ]
[ 5, 7, 9, 11, 13 ] ]
@end group
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\turnoffactive
the method of least squares. The idea is to define the @dfn{chi-square}
error measure
-@ifinfo
+@ifnottex
@example
chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
@infoline @expr{chi^2}
statistic is now,
-@ifinfo
+@ifnottex
@example
chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
@end example
-@end ifinfo
+@end ifnottex
@tex
\turnoffactive
\beforedisplay
@tex
for \AA ngstroms.
@end tex
-@ifinfo
+@ifnottex
for Angstroms.
-@end ifinfo
+@end 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
@iftex
@unnumberedsec TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
@end iftex
-@ifinfo
+@ifnottex
@center TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
-@end ifinfo
+@end ifnottex
@enumerate 0
@item
@iftex
@heading NO WARRANTY
@end iftex
-@ifinfo
+@ifnottex
@center NO WARRANTY
-@end ifinfo
+@end ifnottex
@item
BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY
@iftex
@heading END OF TERMS AND CONDITIONS
@end iftex
-@ifinfo
+@ifnottex
@center END OF TERMS AND CONDITIONS
-@end ifinfo
+@end ifnottex
@page
@unnumberedsec Appendix: How to Apply These Terms to Your New Programs
projects / emacs.git / commitdiff