Usenet 1994 October/usenetsourcesnewsgroupsinfomagicoctober1994disk2.iso/misc/volume24/gnucalc/part38

Text File | 1991-10-31 | 55.5 KB | 1,257 lines

Newsgroups: comp.sources.misc
From: [e-mail] (David Gillespie)
Subject:  v24i086:  gnucalc - GNU Emacs Calculator, v2.00, Part38/56
Message-ID: <[e-mail]>
X-Md4-Signature: cc12ad4ca461f596619fbe0135426c34
Date: Thu, 31 Oct 1991 21:46:56 GMT
Approved: [e-mail]

Submitted-by: [e-mail] (David Gillespie)
Posting-number: Volume 24, Issue 86
Archive-name: gnucalc/part38
Environment: Emacs
Supersedes: gmcalc: Volume 13, Issue 27-45

---- Cut Here and unpack ----
#!/bin/sh
# do not concatenate these parts, unpack them in order with /bin/sh
# file calc.texinfo continued
#
if test ! -r _shar_seq_.tmp; then
    echo 'Please unpack part 1 first!'
    exit 1
fi
(read Scheck
 if test "$Scheck" != 38; then
    echo Please unpack part "$Scheck" next!
    exit 1
 else
    exit 0
 fi
) < _shar_seq_.tmp || exit 1
if test ! -f _shar_wnt_.tmp; then
    echo 'x - still skipping calc.texinfo'
else
echo 'x - continuing file calc.texinfo'
sed 's/^X//' << 'SHAR_EOF' >> 'calc.texinfo' &&
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
@w{`@t{(}@i{a} @t{mod} @i{M}@t{)^.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
X
@kindex M (modulo forms)
@tindex mod (operator)
To create a modulo form during numeric entry, press the shift-@kbd{M}
key to enter the word @samp{mod}.  As a special convenience, pressing
shift-@kbd{M} a second time automatically enters the value of @cite{M}
that was most recently used before.  During algebraic entry, either
type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}).
Once again, pressing this a second time enters the current modulo.@refill
X
You can also use @code{calc-pack} and @kbd{%} to modify modulo forms.
@xref{Building Vectors}.  @xref{Basic Arithmetic}.
X
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
@w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo
24 radians!
X
Modulo forms cannot have variables or formulas for components.  If you
enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
to each of the coefficients:  @samp{(1 mod 5) x + (2 mod 5)}.
X
@tindex makemod
The algebraic function @samp{makemod(a, m)} builds the modulo form
@w{@samp{a mod m}}.
X
@node Error Forms, Interval Forms, Modulo Forms, Data Types
@section Error Forms
X
@noindent
@cindex Error forms
@cindex Standard deviations
An @dfn{error form} is a number with an associated standard
deviation, as in @samp{2.3 +/- 0.12}.  The notation
`@i{x} @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 Calculator.@refill
X
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
$$ \eqalign{
X  f(x \hbox{\code{ +/- }} \sigma)
X    &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
X  f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
X    &= f(x,y) \hbox{\code{ +/- }}
X        \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
X                             \right| \right)^2
X             +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
X                             \right| \right)^2 } \cr
} $$
@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 out.@refill
X
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
`@t{sin(}@i{x} @t{+/-} @c{$\sigma$}
@i{sigma}@t{)}' is
`@c{$\sigma$}
@i{sigma} @t{abs(cos(}@i{x}@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 negligible.@refill
X
@kindex p (error forms)
@tindex +/-
To enter an error form during regular numeric entry, use the @kbd{p}
(``plus-or-minus'') key to type the @samp{+/-} symbol.  (If you try actually
typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
@kbd{+} command!)  Within an algebraic formula, you can press @kbd{M-p} to
type the @samp{+/-} symbol, or type it out by hand.
X
Error forms and complex numbers can be mixed; the formulas shown above
are used for complex numbers, too, except that if the error part evaluates
to a complex number its absolute value (or the square root of the sum of
the squares of the absolute values of the two error contributions) is
used.  Mathematically, this corresponds to a radially symmetric Gaussian
distribution of numbers on the complex plane.  However, note that Calc
considers an error form with real components to represent a real number,
not a complex distribution around a real mean.
X
Error forms may also be composed of HMS forms.  For best results, both
the mean and the error should be HMS forms if either one is.
X
@tindex sdev
The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
X
@node Interval Forms, Incomplete Objects, Error Forms, Data Types
@section Interval Forms
X
@noindent
@cindex Interval forms
An @dfn{interval} is a subset of consecutive real numbers.  For example,
the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
inclusive.  If you multiply it by the interval @samp{[0.5 ..@: 2]} you
obtain @samp{[1 ..@: 8]}.  This calculation represents the fact that if
you multiply some number in the range @samp{[2 ..@: 4]} by some other
number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
from 1 to 8.  Interval arithmetic is used to get a worst-case estimate
of the possible range of values a computation will produce, given the
set of possible values of the input.
X
@ifinfo
Calc supports several varieties of intervals, including @dfn{closed}
intervals of the type shown above, @dfn{open} intervals such as
@samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
@emph{exclusive}, and @dfn{semi-open} intervals in which one end
uses a round parenthesis and the other a square bracket.  In mathematical
terms,
@samp{[2 ..@: 4]} means @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 < 4}.@refill
@end ifinfo
@tex
Calc supports several varieties of intervals, including \dfn{closed}
intervals of the type shown above, \dfn{open} intervals such as
\samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
\emph{exclusive}, and \dfn{semi-open} intervals in which one end
uses a round parenthesis and the other a square bracket.  In mathematical
terms,
$$ \eqalign{
X   [2 \hbox{\cite{..}} 4]  &\quad\hbox{means}\quad  2 \le x \le 4  \cr
X   [2 \hbox{\cite{..}} 4)  &\quad\hbox{means}\quad  2 \le x  <  4  \cr
X   (2 \hbox{\cite{..}} 4]  &\quad\hbox{means}\quad  2  <  x \le 4  \cr
X   (2 \hbox{\cite{..}} 4)  &\quad\hbox{means}\quad  2  <  x  <  4  \cr
} $$
@end tex
X
The lower and upper limits of an interval must be either real numbers
(or HMS or date forms), or symbolic expressions which are assumed to be
real-valued, or @samp{-inf} and @samp{inf}.  In general the lower limit
must be less than the upper limit.  A closed interval containing only
one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
automatically.  An interval containing no values at all (such as
@samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
guaranteed to behave well when used in arithmetic.  Note that the
interval @samp{[3 .. inf)} represents all real numbers greater than
or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
In fact, @samp{[-inf .. inf]} represents all real numbers including
infinity.
X
Intervals are entered in the notation shown here, either as algebraic
formulas, or using incomplete forms.  (@xref{Incomplete Objects}.)
In algebraic formulas, multiple periods in a row are collected from
left to right, so that @samp{1....1e2} is interpreted as @samp{1 ..@: ..@: 1e2}
(a syntax error) rather than @samp{1.@: ..@: .1e2}.  Use spaces or type
@samp{1.0..0.1e2} to make the notation unambiguous.  If you omit
the lower or upper limit, a default of @samp{-inf} or @samp{inf}
(respectively) is furnished.
X
``Infinite mode'' also affects operations on intervals
(@pxref{Infinities}).  Calc will always introduce an open infinity,
as in @samp{1 / (0 .. 2] = [0.5 .. inf)}.  But closed infinities,
@samp{1 / [0 .. 2] = [0.5 .. inf]}, arise only in infinite mode;
otherwise they are left unevaluated.  Note that the ``direction'' of
a zero is not an issue in this case, since the zero is always assumed
to be continuous with the rest of the interval.  For intervals that
contain zero inside them Calc is forced to give the result,
@samp{1 / (-2 .. 2) = [-inf .. inf]}.
X
While it may seem that intervals and error forms are similar, they are
based on entirely different concepts of inexact quantities.  An error
form `@i{x} @t{+/-} @c{$\sigma$}
@i{sigma}' means a variable is random, and its value could
be anything but is ``probably'' within one @c{$\sigma$}
@i{sigma} of the mean value @cite{x}.
An interval `@t{[}@i{a} @t{..@:} @i{b}@t{]}' 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.@refill
X
Intervals may not contain complex numbers, but they may contain
HMS forms or date forms.
X
@xref{Set Operations}, for commands that interpret interval forms
as subsets of the set of real numbers.
X
@tindex intv
The algebraic function @samp{intv(n, a, b)} builds an interval form
from @samp{a} to @samp{b}; @samp{n} is an integer code which must
be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
3 for @samp{[..]}.
X
@node Incomplete Objects, Variables, Interval Forms, Data Types
@section Incomplete Objects
X
@noindent
@kindex [
@kindex ]
@kindex (
@kindex )
@kindex ,
@kindex ;
@cindex Incomplete vectors
@cindex Incomplete complex numbers
@cindex Incomplete interval forms
When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
vector, respectively, the effect is to push an @dfn{incomplete} complex
number or vector onto the stack.  The @kbd{,} key adds the value(s) at
the top of the stack onto the current incomplete object.  The @kbd{)}
and @kbd{]} keys ``close'' the incomplete object after adding any values
on the top of the stack in front of the incomplete object.
X
As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
pushes the vector @samp{[2, 6, 9]} onto the stack.  Likewise, @kbd{( 1 , 2 Q )}
pushes the complex number @samp{(1, 1.414)} (approximately).
X
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}.@refill
X
As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
@kbd{,} adds a zero or duplicates the preceding value in the list being
formed.  Typing @key{DEL} during incomplete entry removes the last item
from the list.
X
The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
numbers:  @kbd{( 1 ; 2 )}.  When entering a vector, @kbd{;} is useful for
creating a matrix.  In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
X
@pindex calc-dots
Incomplete entry is also used to enter intervals.  For example,
@kbd{[ 2 ..@: 4 )} enters a semi-open interval.  Note that when you type
the first period, it will be interpreted as a decimal point, but when
you type a second period immediately afterward, it is re-interpreted as
part of the interval symbol.  Typing @kbd{..} corresponds to executing
the @code{calc-dots} command.
X
If you find incomplete entry distracting, you may wish to enter vectors
and complex numbers as algebraic formulas by pressing the apostrophe key.
X
@node Variables, Formulas, Incomplete Objects, Data Types
@section Variables
X
@noindent
@cindex Variables, in formulas
A @dfn{variable} is somewhere between a storage register on a conventional
calculator, and a variable in a programming language.  (In fact, a Calc
variable is really just an Emacs Lisp variable that contains a Calc number
or formula.)  A variable's name is normally composed of letters and digits.
Calc also allows apostrophes and underscores in variable names.
The Calc variable @code{foo} corresponds to the Emacs Lisp variable
@code{var-foo}.  The @kbd{s s} (@code{calc-store}) command and other
commands that operate on variables can
be made to use any arbitrary Lisp variable simply by backspacing over the
@samp{var-} prefix in the minibuffer.@refill
X
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
@w{@kbd{3 s s foo @key{RET}}} stores that number in variable
@code{var-foo}.@refill
X
To push a variable itself (as opposed to the variable's value) on the
stack, enter its name as an algebraic expression using the apostrophe
(@key{'}) key.  Variable names in algebraic formulas implicitly have
@samp{var-} prefixed to their names.  The @samp{#} character in variable
names used in algebraic formulas corresponds to a dash @samp{-} in the
Lisp variable name.  If the name contains any dashes, the prefix @samp{var-}
is @emph{not} automatically added.  Thus the two formulas @samp{foo + 1}
and @samp{var#foo + 1} both refer to the same variable.
X
@kindex =
@pindex calc-evaluate
@cindex Evaluation of variables in a formula
@cindex Variables, evaluation
@cindex Formulas, evaluation
The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
replacing all variables in the formula which have been given values by a
@code{calc-store} or @code{calc-let} command by their stored values.
Other variables are left alone.  Thus a variable that has not been
stored acts like an abstract variable in algebra; a variable that has
been stored acts more like a register in a traditional calculator.
With a positive numeric prefix argument, @kbd{=} evaluates the top
@var{n} stack entries; with a negative argument, @kbd{=} evaluates
the @var{n}th stack entry.
X
@cindex @code{e} variable
@cindex @code{pi} variable
@cindex @code{i} variable
@cindex @code{phi} variable
@cindex @code{gamma} variable
@vindex e
@vindex pi
@vindex i
@vindex phi
@vindex gamma
A few variables are called @dfn{special constants}.  Their names are
@samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
(@xref{Scientific Functions}.)  When they are evaluated with @kbd{=},
their values are calculated if necessary according to the current precision
or complex polar mode.  If you wish to use these symbols for other purposes,
simply undefine or redefine them using @code{calc-store}.@refill
X
The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
infinite or indeterminate values.  It's best not to use them for
regular variables, since Calc uses special algebraic rules when
it manipulates them.  Calc displays a warning message if you store
a value into any of these special variables.
X
@xref{Store and Recall}, for a discussion of commands dealing with variables.
X
@node Formulas, , Variables, Data Types
@section Formulas
X
@noindent
@cindex Formulas
@cindex Expressions
@cindex Operators in formulas
@cindex Precedence of operators
When you press the apostrophe key you may enter any expression or formula
in algebraic form.  (Calc uses the terms ``expression'' and ``formula''
interchangeably.)  An expression is built up of numbers, variable names,
and function calls, combined with various arithmetic operators.
Parentheses may
be used to indicate grouping.  Spaces are ignored within formulas, except
that spaces are not permitted within variable names or numbers.
Arithmetic operators, in order from highest to lowest precedence, and
with their equivalent function names, are:
X
@display
@samp{_} [@code{subscr}] (subscripts);
X
postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25});
X
prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x})
X  and prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x});
X
@samp{+/-} [@code{sdev}] (the standard deviation symbol) and
X  @samp{mod} [@code{makemod}] (the symbol for modulo forms);
X
postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!})
X  and postfix @samp{!!} [@code{dfact}] (double factorial);
X
@samp{^} [@code{pow}] (raised-to-the-power-of);
X
@samp{*} [@code{mul}];
X
@samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and
X  @samp{\} [@code{idiv}] (integer division);
X
infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y});
X
@samp{|} [@code{vconcat}] (vector concatenation);
X
relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}], @samp{>} [@code{gt}],
X  @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}];
X
@samp{&&} [@code{land}] (logical ``and'');
X
@samp{||} [@code{lor}] (logical ``or'');
X
the C-style ``if'' operator @samp{a?b:c} [@code{if}];
X
@samp{!!!} [@code{pnot}] (rewrite pattern ``not'');
X
@samp{&&&} [@code{pand}] (rewrite pattern ``and'');
X
@samp{|||} [@code{por}] (rewrite pattern ``or'');
X
@samp{:=} [@code{assign}] (for assignments and rewrite rules);
X
@samp{::} [@code{condition}] (rewrite pattern condition);
X
@samp{=>} [@code{evalto}].
@end display
X
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)}.
X
@cindex Multiplication, implicit
@cindex Implicit multiplication
The multiplication sign @samp{*} may be omitted in many cases.  In particular,
if the righthand side is a number, variable name, or parenthesized
expression, the @samp{*} may be omitted.  Implicit multiplication has the
same precedence as the explicit @samp{*} operator.  The one exception to
the rule is that a variable name followed by a parenthesized expression,
as in @samp{f(x)},
is interpreted as a function call, not an implicit @samp{*}.  In many
cases you must use a space if you omit the @samp{*}:  @samp{2a} is the
same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
is a variable called @code{ab}, @emph{not} the product of @samp{a} and
@samp{b}!  Also note that @samp{f (x)} is still a function call.@refill
X
@cindex Implicit comma in vectors
The rules are slightly different for vectors written with square brackets.
There, the space character is interpreted (like the comma) as a separator
of elements of the vector.  Thus @samp{[ 2a b+c d ]} is equivalent to
@samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent to
@samp{2*a*b + c*d}.
Note that spaces around the brackets, and around explicit commas, are
ignored.  To force spaces to be interpreted as multiplication you can
enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
interpreted as @samp{[a*b, 2*c*d]}.  An implicit comma is also inserted
between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.@refill
X
Vectors that contain commas (not embedded within nested parentheses or
brackets) do not treat spaces specially:  @samp{[a b, 2 c d]} is a vector
of two elements.  Also, vectors entered with curly braces instead of
square brackets do not give spaces any special treatment.  When Calc
displays a vector that does not contain any commas, it will insert
parentheses if necessary to make the meaning clear:  @samp{[(a b)]}.
X
The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
or five modulo minus-two?  Calc always interprets the leftmost symbol as
an infix operator preferentially (modulo, in this case), so you would
need to write @samp{(5%)-2} to get the former interpretation.
X
@cindex Function call notation
A function call is, e.g., @samp{sin(1+x)}.  Function names follow the same
rules as variable names except that the default prefix @samp{calcFunc-} is
used (instead of @samp{var-}) for the internal Lisp form.
Most mathematical Calculator commands like
@code{calc-sin} have function equivalents like @code{sin}.
If no Lisp function is defined for a function called by a formula, the
call is left as it is during algebraic manipulation: @samp{f(x+y)} is
left alone.  Beware that many innocent-looking short names like @code{in}
and @code{re} have predefined meanings which could surprise you; however,
single letters or single letters followed by digits are always safe.
@xref{Function Index}.@refill
X
In the documentation for particular commands, the notation @kbd{H S}
(@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the
command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
represent the same operation.@refill
X
Commands that interpret (``parse'') text as algebraic formulas include
algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
the contents of the editing buffer when you finish, the @kbd{M-# g}
and @kbd{M-# r} commands, the @kbd{C-y} command, the X window system
``paste'' mouse operation, and Embedded Mode.  All of these operations
use the same rules for parsing formulas; in particular, language modes
(@pxref{Language Modes}) affect them all in the same way.
X
When you read a large amount of text into the Calculator (say a vector
which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
you may wish to include comments in the text.  Calc's formula parser
ignores the symbol @samp{%%} and anything following it on a line:
X
@example
[ a + b,   %% the sum of "a" and "b"
X  c + d,
X  %% last line is coming up:
X  e + f ]
@end example
X
@noindent
This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
X
@xref{Algebra}, for commands for manipulating formulas symbolically.
X
@node Stack and Trail, Mode Settings, Data Types, Top
@chapter Stack and Trail Commands
X
@noindent
This chapter describes the Calc commands for manipulating objects in the
stack and in the trail buffer.  (These commands operate on objects of any
type, such as numbers, vectors, formulas, and incomplete objects.)
X
@menu
* Stack Manipulation::
* Trail Commands::
* Keep Arguments::
@end menu
X
@node Stack Manipulation, Trail Commands, Stack and Trail, Stack and Trail
@section Stack Manipulation Commands
X
@noindent
@kindex RET
@kindex SPC
@pindex calc-enter
@cindex Duplicating stack entries
To duplicate the top object on the stack, press @key{RET} or @key{SPC}
(two equivalent keys for the @code{calc-enter} command).
Given a positive numeric prefix argument, these commands duplicate
several elements at the top of the stack.
Given a negative argument,
these commands duplicate the specified element of the stack.
Given an argument of zero, they duplicate the entire stack.
For example, with @samp{10 20 30} on the stack,
@key{RET} creates @samp{10 20 30 30},
@kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30},
@kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and
@kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.@refill
X
@kindex LFD
@pindex calc-over
The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
have it, else on @kbd{C-j}) is like @code{calc-enter}
except that the sign of the numeric prefix argument is interpreted
oppositely.  Also, with no prefix argument the default argument is 2.
Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
@samp{10 20 30 20}.@refill
X
@kindex DEL
@kindex C-d
@pindex calc-pop
@cindex Removing stack entries
@cindex Deleting stack entries
To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
The @kbd{C-d} key is a synonym for @key{DEL}.
(If the top element is an incomplete object with at least one element, the
last element is removed from it.)  Given a positive numeric prefix argument,
several elements are removed.  Given a negative argument, the specified
element of the stack is deleted.  Given an argument of zero, the entire
stack is emptied.
For example, with @samp{10 20 30} on the stack,
@key{DEL} creates @samp{10 20},
@kbd{C-u 2 @key{DEL}} creates @samp{10},
@kbd{C-u - 2 @key{DEL}} creates @samp{10 30}, and
@kbd{C-u 0 @key{DEL}} creates an empty stack.@refill
X
@kindex M-DEL
@pindex calc-pop-above
The @kbd{M-DEL} (@code{calc-pop-above}) command is to @kbd{DEL} what
@kbd{LFD} is to @kbd{RET}, i.e., it interprets the sign of the numeric
prefix argument in the opposite way, and the default argument is 2.
Thus @kbd{M-DEL} by itself removes the second-from-top stack element,
leaving the first, third, fourth, and so on; @kbd{M-3 M-DEL} deletes
the third stack element.
X
@kindex TAB
@pindex calc-roll-down
To exchange the top two elements of the stack, press @key{TAB}
(@code{calc-roll-down}).  Given a positive numeric prefix argument, the
specified number of elements at the top of the stack are rotated downward.
Given a negative argument, the entire stack is rotated downward the specified
number of times.  Given an argument of zero, the entire stack is reversed
top-for-bottom.
For example, with @samp{10 20 30 40 50} on the stack,
@key{TAB} creates @samp{10 20 30 50 40},
@kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40},
@kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
@kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.@refill
X
@kindex M-TAB
@pindex calc-roll-up
The command @key{M-TAB} (@code{calc-roll-up}) is analogous to @key{TAB}
except that it rotates upward instead of downward.  Also, the default
with no prefix argument is to rotate the top 3 elements.
For example, with @samp{10 20 30 40 50} on the stack,
@key{M-TAB} creates @samp{10 20 40 50 30},
@kbd{C-u 4 @key{M-TAB}} creates @samp{10 30 40 50 20},
@kbd{C-u - 2 @key{M-TAB}} creates @samp{30 40 50 10 20}, and
@kbd{C-u 0 @key{M-TAB}} creates @samp{50 40 30 20 10}.@refill
X
A good way to view the operation of @key{TAB} and @key{M-TAB} is in
terms of moving a particular element to a new position in the stack.
With a positive argument @i{n}, @key{TAB} moves the top stack
element down to level @i{n}, making room for it by pulling all the
intervening stack elements toward the top.  @key{M-TAB} moves the
element at level @i{n} up to the top.  (Compare with @key{LFD},
which copies instead of moving the element in level @i{n}.)
X
With a negative argument @i{-n}, @key{TAB} rotates the stack
to move the object in level @i{n} to the deepest place in the
stack, and the object in level @i{n+1} to the top.  @key{M-TAB}
rotates the deepest stack element to be in level @i{n}, also
putting the top stack element in level @i{n+1}.
X
@xref{Selecting Subformulas}, for a way to apply these commands to
any portion of a vector or formula on the stack.
X
@node Trail Commands, Keep Arguments, Stack Manipulation, Stack and Trail
@section Trail Commands
X
@noindent
@cindex Trail buffer
The commands for manipulating the Calc Trail buffer are two-key sequences
beginning with the @kbd{t} prefix.
X
@kindex t d
@pindex calc-trail-display
The @kbd{t d} (@code{calc-trail-display}) command turns display of the
trail on and off.  Normally the trail display is toggled on if it was off,
off if it was on.  With a numeric prefix of zero, this command always
turns the trail off; with a prefix of one, it always turns the trail on.
The other trail-manipulation commands described here automatically turn
the trail on.  Note that when the trail is off values are still recorded
there; they are simply not displayed.  To set Emacs to turn the trail
off by default, turn it off with @kbd{t d}, then save the mode settings
with @kbd{m m} (@code{calc-save-modes}).
X
@kindex t i
@pindex calc-trail-in
@kindex t o
@pindex calc-trail-out
The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
(@code{calc-trail-out}) commands switch the cursor into and out of the
Calc Trail window.  In practice they are rarely used, since the commands
shown below are a more convenient way to move around in the
trail, and they work ``by remote control'' when the cursor is still
in the Calculator window.@refill
X
@cindex Trail pointer
There is a @dfn{trail pointer} which selects some entry of the trail at
any given time.  The trail pointer looks like a @samp{>} symbol right
before the selected number.  The following commands operate on the
trail pointer in various ways.
X
@kindex t y
@pindex calc-trail-yank
@cindex Retrieving previous results
The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
the trail and pushes it onto the Calculator stack.  It allows you to
re-use any previously computed value without retyping.  With a numeric
prefix argument @var{n}, it yanks the value @var{n} lines above the current
trail pointer.
X
@kindex t <
@pindex calc-trail-scroll-left
@kindex t >
@pindex calc-trail-scroll-right
The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
(@code{calc-trail-scroll-right}) commands horizontally scroll the trail
window left or right by one half of its width.@refill
X
@kindex t n
@pindex calc-trail-next
@kindex t p
@pindex calc-trail-previous
@kindex t f
@pindex calc-trail-forward
@kindex t b
@pindex calc-trail-backward
The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
(@code{calc-trail-previous)} commands move the trail pointer down or up
one line.  The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
(@code{calc-trail-backward}) commands move the trail pointer down or up
one screenful at a time.  All of these commands accept numeric prefix
arguments to move several lines or screenfuls at a time.@refill
X
@kindex t [
@pindex calc-trail-first
@kindex t ]
@pindex calc-trail-last
@kindex t h
@pindex calc-trail-here
The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
(@code{calc-trail-last}) commands move the trail pointer to the first or
last line of the trail.  The @kbd{t h} (@code{calc-trail-here}) command
moves the trail pointer to the cursor position; unlike the other trail
commands, @kbd{t h} works only when Calc Trail is the selected window.@refill
X
@kindex t s
@pindex calc-trail-isearch-forward
@kindex t r
@pindex calc-trail-isearch-backward
@ifinfo
The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
(@code{calc-trail-isearch-backward}) commands perform an incremental
search forward and 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 does not
move.@refill
@end ifinfo
@tex
The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
(@code{calc-trail-isearch-backward}) com\-mands perform an incremental
search forward and 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 does not
move.
@end tex
X
@kindex t m
@pindex calc-trail-marker
The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
line of text of your own choosing into the trail.  The text is inserted
after the line containing the trail pointer; this usually means it is
added to the end of the trail.  Trail markers are useful mainly as the
targets for later incremental searches in the trail.
X
@kindex t k
@pindex calc-trail-kill
The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
from the trail.  The line is saved in the Emacs kill ring suitable for
yanking into another buffer, but it is not easy to yank the text back
into the trail buffer.  With a numeric prefix argument, this command
kills the @var{n} lines below or above the selected one.
X
The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
elsewhere; @pxref{Vector and Matrix Formats}.
X
@node Keep Arguments, , Trail Commands, Stack and Trail
@section Keep Arguments
X
@noindent
@kindex K
@pindex calc-keep-args
The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
the following command that prevents that command from removing its
arguments from the stack.  For example, after @kbd{2 @key{RET} 3 +},
the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
the stack contains the arguments and the result: @samp{2 3 5}.
X
This works for all commands that take arguments off the stack.  As
another example, @kbd{K a s} simplifies a formula, pushing the
simplified version of the formula onto the stack after the original
formula (rather than replacing the original formula).
X
Note that you could get the same effect by typing @kbd{RET a s},
copying the formula and then simplifying the copy.  One difference
is that for a very large formula the time taken to format the
intermediate copy in @kbd{RET a s} could be noticeable; @kbd{K a s}
would avoid this extra work.
X
Even stack manipulation commands are affected.  @key{TAB} works by
popping two values and pushing them back in the opposite order,
so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
X
A few Calc commands provide other ways of doing the same thing.
For example, @kbd{' sin($)} replaces the number on the stack with
its sine using algebraic entry; to push the sine and keep the
original argument you could use either @kbd{' sin($1)} or
@kbd{K ' sin($)}.  @xref{Algebraic Entry}.  Also, the @kbd{s s}
command is effectively the same as @kbd{K s t}.  @xref{Storing Variables}.
X
Keyboard macros may interact surprisingly with the @kbd{K} prefix.
If you have defined a keyboard macro to be, say, @samp{Q +} to add
one number to the square root of another, then typing @kbd{K X} will
execute @kbd{K Q +}, probably not what you expected.  The @kbd{K}
prefix will apply to just the first command in the macro rather than
the whole macro.
X
If you execute a command and then decide you really wanted to keep
the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}).
This command pushes the last arguments that were popped by any command
onto the stack.  Note that the order of things on the stack will be
different than with @kbd{K}:  @kbd{2 @key{RET} 3 + M-@key{RET}} leaves
@samp{5 2 3} on the stack instead of @samp{2 3 5}.  @xref{Undo}.
X
@node Mode Settings, Arithmetic, Stack and Trail, Top
@chapter Mode Settings
X
@noindent
This chapter describes commands that set modes in the Calculator.
They do not affect the contents of the stack, although they may change
the @emph{appearance} or @emph{interpretation} of the stack's contents.
X
@menu
* General Mode Commands::
* Precision::
* Inverse and Hyperbolic::
* Calculation Modes::
* Simplification Modes::
* Declarations::
* Display Modes::
* Language Modes::
* Calc Mode Line::
@end menu
X
@node General Mode Commands, Precision, Mode Settings, Mode Settings
@section General Mode Commands
X
@noindent
@kindex m m
@pindex calc-save-modes
@cindex Continuous memory
@cindex Saving mode settings
@cindex Permanent mode settings
@cindex @file{.emacs} file, mode settings
You can save all of the current mode settings in your @file{.emacs} file
with the @kbd{m m} (@code{calc-save-modes}) command.  This will cause
Emacs to reestablish these modes each time it starts up.  The modes saved
in the file include everything controlled by the @kbd{m} and @kbd{d}
prefix keys, the current precision and binary word size, whether or not
the trail is displayed, and the current height of the Calc window.
The current interface (used when you type @kbd{M-# M-#}) is also saved.
If there were already saved mode settings in the file, they are replaced.
Otherwise, the new mode information is appended to the end of the file.
X
@kindex m R
@pindex calc-mode-record-mode
The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
record the new mode settings (as if by pressing @kbd{m m}) every
time a mode setting changes.  If Embedded Mode is enabled, other
options are available; @pxref{Mode Settings in Embedded Mode}.
X
@kindex m F
@pindex calc-settings-file-name
The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
choose a different place than your @file{.emacs} file for @kbd{m m},
@kbd{Z P}, and similar commands to save permanent information.
You are prompted for a file name.  All Calc modes are then reset to
their default values, settings from the file you named are loaded
if this file exists, and this file becomes the one that Calc will
use in the future for commands like @kbd{m m}.  The default settings
file name is @file{~/.emacs}.  You can see the current file name by
giving a blank response to the @kbd{m F} prompt.  See also the
discussion of the @code{calc-settings-file} variable; @pxref{Installation}.
X
If the file name you give contains the string @samp{.emacs} anywhere
inside it, @kbd{m F} will not automatically load the new file.  This
is because you are presumably switching to your @file{~/.emacs} file,
which may contain other things you don't want to reread.  You can give
a numeric prefix argument of 1 to @kbd{m F} to force it to read the
file no matter what its name.  Conversely, an argument of @i{-1} tells
@kbd{m F} @emph{not} to read the new file.  An argument of 2 or @i{-2}
tells @kbd{m F} not to reset the modes to their defaults beforehand,
which is useful if you intend your new file to have a variant of the
modes present in the file you were using before.
X
@kindex m x
@pindex calc-always-load-extensions
The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
in which the first use of Calc loads the entire program, including all
extensions modules.  Otherwise, the extensions modules will not be loaded
until the various advanced Calc features are used.  Since this mode only
has effect when Calc is first loaded, @kbd{m x} is usually followed by
@kbd{m m} to make the mode-setting permanent.
X
@kindex m X
@kindex M-# L
@pindex calc-load-everything
The @kbd{m X} or @kbd{M-# L} (@code{calc-load-everything}) command loads
all parts of Calc that have not yet been loaded.  This is the command
that Calc executes upon startup if @kbd{m x} mode has been enabled.
X
@kindex m S
@pindex calc-shift-prefix
The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
all of Calc's letter prefix keys may be typed shifted as well as unshifted.
If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
you might find it easier to turn this mode on so that you can type
@kbd{A S} instead.  When this mode is enabled, the commands that used to
be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
now be invoked by pressing the shifted letter twice: @kbd{A A}.  Note
that the @kbd{v} prefix key always works both shifted and unshifted, and
the @kbd{z} and @kbd{Z} prefix keys are always distinct.  Also, the @kbd{h}
prefix is not affected by this mode.  Press @kbd{m S} again to disable
shifted-prefix mode.
X
@node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings
@section Precision
X
@noindent
@kindex p
@pindex calc-precision
@cindex Precision of calculations
The @kbd{p} (@code{calc-precision}) command controls the precision to
which floating-point calculations are carried.  The precision must be
at least 3 digits and may be arbitrarily high, within the limits of
memory and time.  This affects only floats:  Integer and rational
calculations are always carried out with as many digits as necessary.
X
The @kbd{p} key prompts for the current precision.  If you wish you
can instead give the precision as a numeric prefix argument.
X
Many internal calculations are carried to one or two digits higher
precision than normal.  Results are rounded down afterward to the
current precision.  Unless a special display mode has been selected,
floats are always displayed with their full stored precision, i.e.,
what you see is what you get.  Reducing the current precision does not
round values already on the stack, but those values will be rounded
down before being used in any calculation.  The @kbd{c 0} through
@kbd{c 9} commands (@pxref{Conversions}) can be used to round an
existing value to a new precision.@refill
X
@cindex Accuracy of calculations
It is important to distinguish the concepts of @dfn{precision} and
@dfn{accuracy}.  In the normal usage of these words, the number
123.4567 has a precision of 7 digits but an accuracy of 4 digits;
the precision is the total number of digits not counting leading
or trailing zeros (regardless of the position of the decimal point).
The accuracy is simply the number of digits after the decimal point
(again not counting trailing zeros).  In Calc you control the precision,
not the accuracy of computations.  If you were to set the accuracy
instead, then calculations like @samp{exp(100)} would generate many
more digits than you would typically need, while @samp{exp(-100)} would
probably round to zero!  In Calc, both these computations give you
exactly 12 (or the requested number of) significant digits.
X
The only Calc features that deal with accuracy instead of precision
are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
and the rounding functions like @code{floor} and @code{round}
(@pxref{Integer Truncation}).  Also, @kbd{c 0} through @kbd{c 9}
deal with both precision and accuracy depending on the magnitudes
of the numbers involved.
X
If you need to work with a particular fixed accuracy (say, dollars and
cents with two digits after the decimal point), one solution is to work
with integers and an ``implied'' decimal point.  For example, $8.99
divided by 6 would be entered @kbd{899 RET 6 /}, yielding 149.833
(actually $1.49833 with our implied decimal point); pressing @kbd{R}
would round this to 150 cents, i.e., $1.50.
X
@node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings
@section Inverse and Hyperbolic Flags
X
@noindent
@kindex I
@pindex calc-inverse
There is no single-key equivalent to the @code{calc-arcsin} function.
Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
The @kbd{I} key actually toggles the Inverse Flag.  When this flag
is set, the word @samp{Inv} appears in the mode line.@refill
X
@kindex H
@pindex calc-hyperbolic
Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
If both of these flags are set at once, the effect will be
@code{calc-arcsinh}.  (The Hyperbolic flag is also used by some
non-trigonometric commands; for example @kbd{H L} computes a base-10,
instead of base-@i{e}, logarithm.)@refill
X
Command names like @code{calc-arcsin} are provided for completeness, and
may be executed with @kbd{x} or @kbd{M-x}.  Their effect is simply to
toggle the Inverse and/or Hyperbolic flags and then execute the
corresponding base command (@code{calc-sin} in this case).
X
The Inverse and Hyperbolic flags apply only to the next Calculator
command, after which they are automatically cleared.  (They are also
cleared if the next keystroke is not a Calc command.)  Digits you
type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
arguments for the next command, not as numeric entries.  The same
is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
subtract and keep arguments).
X
The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
elsewhere.  @xref{Keep Arguments}.
X
@node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
@section Calculation Modes
X
@noindent
The commands in this section are two-key sequences beginning with
the @kbd{m} prefix.  (That's the letter @kbd{m}, not the @key{META} key.)
X
The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
(@pxref{Algebraic Entry}).
X
@menu
* Angular Modes::
* Polar Mode::
* Fraction Mode::
* Infinite Mode::
* Symbolic Mode::
* Matrix Mode::
* Automatic Recomputation::
* Working Message::
@end menu
X
@node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
@subsection Angular Modes
X
@noindent
@cindex Angular mode
The Calculator supports three notations for angles: radians, degrees,
and degrees-minutes-seconds.  When a number is presented to a function
like @code{sin} that requires an angle, the current angular mode is
used to interpret the number as either radians or degrees.  If an HMS
form is presented to @code{sin}, it is always interpreted as
degrees-minutes-seconds.
X
Functions that compute angles produce a number in radians, a number in
degrees, or an HMS form depending on the current angular mode.  If the
result is a complex number and the current mode is HMS, the number is
instead expressed in degrees.  (Complex-number calculations would
normally be done in radians mode, though.  Complex numbers are converted
to degrees by calculating the complex result in radians and then
multiplying by 180 over @c{$\pi$}
@cite{pi}.)
X
@kindex m r
@pindex calc-radians-mode
@kindex m d
@pindex calc-degrees-mode
@kindex m h
@pindex calc-hms-mode
The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
The current angular mode is displayed on the Emacs mode line.
The default angular mode is degrees.@refill
X
@node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
@subsection Polar Mode
X
@noindent
@cindex Polar mode
The Calculator normally ``prefers'' rectangular complex numbers in the
sense that rectangular form is used when the proper form can not be
decided from the input.  This might happen by multiplying a rectangular
number by a polar one, by taking the square root of a negative real
number, or by entering @kbd{( 2 @key{SPC} 3 )}.
X
@kindex m p
@pindex calc-polar-mode
The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
preference between rectangular and polar forms.  In polar mode, all
of the above example situations would produce polar complex numbers.
X
@node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
@subsection Fraction Mode
X
@noindent
@cindex Fraction mode
@cindex Division of integers
Division of two integers normally yields a floating-point number if the
result cannot be expressed as an integer.  In some cases you would
rather get an exact fractional answer.  One way to accomplish this is
to multiply fractions instead:  @kbd{6 @key{RET} 1:4 *} produces @cite{3:2}
even though @kbd{6 @key{RET} 4 /} produces @cite{1.5}.
X
@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, but @cite{1.5} in
Float Mode.@refill
X
At any time you can use @kbd{c f} (@code{calc-float}) to convert a
fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
float to a fraction.  @xref{Conversions}.
X
@node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
@subsection Infinite Mode
X
@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.
X
@kindex m i
@pindex calc-infinite-mode
The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
on and off.  When the mode is off, infinities do not arise except
in calculations that already had infinities as inputs.  (One exception
is that infinite open intervals like @samp{[0 .. inf)} can be
generated; however, intervals closed at infinity (@samp{[0 .. inf]})
will not be generated when infinite mode is off.)
X
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,
note that @samp{exp(inf) = inf} regardless of infinite mode because
this calculation has infinity as an input.
X
@cindex Positive infinite mode
The @kbd{m i} command with a numeric prefix argument of zero,
i.e., @kbd{C-u 0 m i}, turns on a ``positive infinite mode'' in
which zero is treated as positive instead of being directionless.
Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
Note that zero never actually has a sign in Calc; there are no
separate representations for @i{+0} and @i{-0}.  Positive
infinite mode merely changes the interpretation given to the
single symbol, @samp{0}.  One consequence of this is that, while
you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
X
@node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
@subsection Symbolic Mode
X
@noindent
@cindex Symbolic mode
@cindex Inexact results
Calculations are normally performed numerically wherever possible.
For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
algebraic expression, produces a numeric answer if the argument is a
number or a symbolic expression if the argument is an expression:
@kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}.
X
@kindex m s
@pindex calc-symbolic-mode
In @dfn{symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
command, functions which would produce inexact, irrational results are
left in symbolic form.  Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
@samp{sqrt(2)}.
X
@kindex N
@pindex calc-eval-num
The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically
the expression at the top of the stack, by temporarily disabling
@code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}).
Given a numeric prefix argument, it also
sets the floating-point precision to the specified value for the duration
of the command.@refill
X
To evaluate a formula numerically without expanding the variables it
contains, you can use the key sequence @kbd{m s a v m s} (this uses
@code{calc-alg-evaluate}, which resimplifies but doesn't evaluate
variables.)
X
@node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes
@subsection Matrix and Scalar Modes
X
@noindent
@cindex Matrix mode
@cindex Scalar mode
Calc sometimes makes assumptions during algebraic manipulation that
are awkward or incorrect when vectors and matrices are involved.
Calc has two modes, @dfn{matrix mode} and @dfn{scalar mode}, which
modify its behavior around vectors in useful ways.
X
@kindex m v
@pindex calc-matrix-mode
Press @kbd{m v} (@code{calc-matrix-mode}) once to enter matrix mode.
In this mode, all objects are assumed to be matrices unless provably
otherwise.  One major effect is that Calc will no longer consider
multiplication to be commutative.  (Recall that in matrix arithmetic,
@samp{A*B} is not the same as @samp{B*A}.)  This assumption affects
rewrite rules and algebraic simplification.  Another effect of this
mode is that calculations that would normally produce constants like
0 and 1 (e.g., @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 @var{n}x@var{n}
identity matrix; if @var{n} is omitted, it doesn't know what
dimension to use and so the @code{idn} call remains in symbolic
form.  However, if this generic identity matrix is later combined
with a matrix whose size is known, it will be converted into
a true identity matrix of the appropriate size.  On the other hand,
if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
will assume it really was a scalar after all and produce, e.g., 3.
X
Press @kbd{m v} a second time to get scalar mode.  Here, objects are
assumed @emph{not} to be vectors or matrices unless provably so.
For example, normally adding a variable to a vector, as in
@samp{[x, y, z] + a}, will leave the sum in symbolic form because
as far as Calc knows, @samp{a} could represent either a number or
another 3-vector.  In scalar mode, @samp{a} is assumed to be a
non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
X
Press @kbd{m v} a third time to return to the normal mode of operation.
X
If you press @kbd{m v} with a numeric prefix argument @var{n}, you
get a special ``dimensioned matrix mode'' in which matrices of
unknown size are assumed to be @var{n}x@var{n} square matrices.
Then, the function call @samp{idn(1)} will expand into an actual
matrix rather than representing a ``generic'' matrix.
X
@cindex Declaring scalar variables
Of course these modes are approximations to the true state of
affairs, which is probably that some quantities will be matrices
and others will be scalars.  One solution is to ``declare''
certain variables or functions to be scalar-valued.
@xref{Declarations}, to see how to make declarations in Calc.
X
There is nothing stopping you from declaring a variable to be
scalar and then storing a matrix in it; however, if you do, the
results you get from Calc may not be valid.  Suppose you let Calc
get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
@samp{[1, 2, 3]} in @samp{a}.  The result would not be the same as
for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
your earlier promise to Calc that @samp{a} would be scalar.
X
Another way to mix scalars and matrices is to use selections
SHAR_EOF
true || echo 'restore of calc.texinfo failed'
fi
echo 'End of  part 38'
echo 'File calc.texinfo is continued in part 39'
echo 39 > _shar_seq_.tmp
exit 0
exit 0 # Just in case...
--
Kent Landfield                   INTERNET: [e-mail]
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