@newcount@calcpageno
@newtoks@calcoldeverypar @calcoldeverypar=@everypar
@everypar={@calceverypar@the@calcoldeverypar}
-@ifx@turnoffactive@undefinedzzz@def@turnoffactive{}@fi
@ifx@ninett@undefinedzzz@font@ninett=cmtt9@fi
@catcode`@\=0 \catcode`\@=11
\r@ggedbottomtrue
@end ifnotinfo
Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
-2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
+2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc.
@quotation
Permission is granted to copy, distribute and/or modify this document
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
\afterdisplay
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
\times
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \eqalign{ x &+ a y = 6 \cr
x &+ b y = 10}
@samp{trn(A)*A*X = trn(A)*B}.
@end ifnottex
@tex
-\turnoffactive
$A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
@end tex
Now
@end group
@end ifnottex
@tex
-\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ m = {N \sum x y - \sum x \sum y \over
N \sum x^2 - \left( \sum x \right)^2} $$
@samp{sum(x y)}.)
@end ifnottex
@tex
-\turnoffactive
These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
$\sum x y$.)
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ b = {\sum y - m \sum x \over N} $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \displaylines{
\qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
+ f(a+(n-2)h) + f(a+(n-1)h)) $$
@smallexample
@group
-1: 1 / cos(x) - sin(x) tan(x)
+1: 2 / cos(x)^2 - 2 tan(x)^2
.
- ' 1/cos(x) - sin(x) tan(x) @key{RET} s 1
+ ' 2/cos(x)^2 - 2tan(x)^2 @key{RET} s 1
@end group
@end smallexample
@noindent
If we were simplifying this by hand, we'd probably replace the
@samp{tan} with a @samp{sin/cos} first, then combine over a common
-denominator. There is no Calc command to do the former; the @kbd{a n}
-algebra command will do the latter but we'll do both with rewrite
+denominator. The @kbd{I a s} command will do the former and the @kbd{a n}
+algebra command will do the latter, but we'll do both with rewrite
rules just for practice.
Rewrite rules are written with the @samp{:=} symbol.
@smallexample
@group
-1: 1 / cos(x) - sin(x)^2 / cos(x)
+1: 2 / cos(x)^2 - 2 sin(x)^2 / cos(x)^2
.
a r tan(a) := sin(a)/cos(a) @key{RET}
@smallexample
@group
-1: (1 - sin(x)^2) / cos(x)
+1: (2 - 2 sin(x)^2) / cos(x)^2
.
a r a/x + b/x := (a+b)/x @key{RET}
Second, meta-variable names are independent from variables in the
target formula. Notice that the meta-variable @samp{x} here matches
-the subformula @samp{cos(x)}; Calc never confuses the two meanings of
+the subformula @samp{cos(x)^2}; Calc never confuses the two meanings of
@samp{x}.
And third, rewrite patterns know a little bit about the algebraic
properties of formulas. The pattern called for a sum of two quotients;
Calc was able to match a difference of two quotients by matching
-@samp{a = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.
+@samp{a = 2}, @samp{b = -2 sin(x)^2}, and @samp{x = cos(x)^2}.
@c [fix-ref Algebraic Properties of Rewrite Rules]
We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
One more rewrite will complete the job. We want to use the identity
@samp{sin(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
the identity in a way that matches our formula. The obvious rule
-would be @samp{@w{1 - sin(x)^2} := cos(x)^2}, but a little thought shows
+would be @samp{@w{2 - 2 sin(x)^2} := 2 cos(x)^2}, but a little thought shows
that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The
latter rule has a more general pattern so it will work in many other
situations, too.
@smallexample
@group
-1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x)
- . .
+1: (2 + 2 cos(x)^2 - 2) / cos(x)^2 1: 2
+ . .
a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s
@end group
' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET}
-1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
+1: 2 / cos(x)^2 - 2 tan(x)^2 1: 2
. .
r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr @key{RET} a s
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
\afterdisplay
@end ifnottex
@tex
\beforedisplay
-$$ x_{\rm new} = x - {f(x) \over f'(x)} $$
+$$ x_{\rm new} = x - {f(x) \over f^{\prime}(x)} $$
\afterdisplay
@end tex
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \eqalign{ x &+ a y = 6 \cr
x &+ b y = 10}
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplayh
$$ \openup1\jot \tabskip=0pt plus1fil
\halign to\displaywidth{\tabskip=0pt
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ m \times x + b \times 1 = y $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ 3 (3 a + b - 511 m) + c - 511 n $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ 9 a + 3 b + c - 511\times3 m - 511 n $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
-$$ 9 a + 3 b + c - 511 n' $$
+$$ 9 a + 3 b + c - 511 n^{\prime} $$
\afterdisplay
@end tex
@samp{exp(inf) = inf}. It's tempting to say that the exponential
of infinity must be ``bigger'' than ``regular'' infinity, but as
-far as Calc is concerned all infinities are as just as big.
+far as Calc is concerned all infinities are the same size.
In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
to infinity, but the fact the @expr{e^x} grows much faster than
@expr{x} is not relevant here.
@noindent
@cindex Help commands
@kindex ?
+@kindex a ?
+@kindex b ?
+@kindex c ?
+@kindex d ?
+@kindex f ?
+@kindex g ?
+@kindex j ?
+@kindex k ?
+@kindex m ?
+@kindex r ?
+@kindex s ?
+@kindex t ?
+@kindex u ?
+@kindex v ?
+@kindex V ?
+@kindex z ?
+@kindex Z ?
@pindex calc-help
The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs'
@cindex Julian day counting
Another day counting system in common use is, confusingly, also called
-``Julian.'' The Julian day number is the numbers of days since
-12:00 noon (GMT) on Jan 1, 4713 BC, which in Calc's scheme (in GMT)
+``Julian.'' The Julian day number is the numbers of days since
+12:00 noon (GMT) on Jan 1, 4713 BC, which in Calc's scheme (in GMT)
is @mathit{-1721423.5} (recall that Calc starts at midnight instead
of noon). Thus to convert a Calc date code obtained by unpacking a
date form into a Julian day number, simply add 1721423.5 after
compensating for the time zone difference. The built-in @kbd{t J}
command performs this conversion for you.
-The Julian day number is based on the Julian cycle, which was invented
+The Julian day number is based on the Julian cycle, which was invented
in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle
-since it is involves the Julian calendar, but some have suggested that
+since it involves the Julian calendar, but some have suggested that
Scaliger named it in honor of his father, Julius Caesar Scaliger. The
-Julian cycle is based it on three other cycles: the indiction cycle,
-the Metonic cycle, and the solar cycle. The indiction cycle is a 15
-year cycle originally used by the Romans for tax purposes but later
-used to date medieval documents. The Metonic cycle is a 19 year
-cycle; 19 years is close to being a common multiple of a solar year
-and a lunar month, and so every 19 years the phases of the moon will
-occur on the same days of the year. The solar cycle is a 28 year
-cycle; the Julian calendar repeats itself every 28 years. The
-smallest time period which contains multiples of all three cycles is
-the least common multiple of 15 years, 19 years and 28 years, which
-(since they're pairwise relatively prime) is
+Julian cycle is based on three other cycles: the indiction cycle, the
+Metonic cycle, and the solar cycle. The indiction cycle is a 15 year
+cycle originally used by the Romans for tax purposes but later used to
+date medieval documents. The Metonic cycle is a 19 year cycle; 19
+years is close to being a common multiple of a solar year and a lunar
+month, and so every 19 years the phases of the moon will occur on the
+same days of the year. The solar cycle is a 28 year cycle; the Julian
+calendar repeats itself every 28 years. The smallest time period
+which contains multiples of all three cycles is the least common
+multiple of 15 years, 19 years and 28 years, which (since they're
+pairwise relatively prime) is
@texline @math{15\times 19\times 28 = 7980} years.
@infoline 15*19*28 = 7980 years.
This is the length of a Julian cycle. Working backwards, the previous
@cindex Saving mode settings
@cindex Permanent mode settings
@cindex Calc init file, mode settings
-You can save all of the current mode settings in your Calc init file
+You can save all of the current mode settings in your Calc init file
(the file given by the variable @code{calc-settings-file}, typically
-@file{~/.calc.el}) with the @kbd{m m} (@code{calc-save-modes}) command.
-This will cause Emacs to reestablish these modes each time it starts up.
-The modes saved in the file include everything controlled by the @kbd{m}
-and @kbd{d} prefix keys, the current precision and binary word size,
-whether or not the trail is displayed, the current height of the Calc
-window, and more. The current interface (used when you type @kbd{C-x * *})
-is also saved. If there were already saved mode settings in the
-file, they are replaced. Otherwise, the new mode information is
-appended to the end of the file.
+@file{~/.emacs.d/calc.el}) with the @kbd{m m} (@code{calc-save-modes})
+command. This will cause Emacs to reestablish these modes each time
+it starts up. The modes saved in the file include everything
+controlled by the @kbd{m} and @kbd{d} prefix keys, the current
+precision and binary word size, whether or not the trail is displayed,
+the current height of the Calc window, and more. The current
+interface (used when you type @kbd{C-x * *}) is also saved. If there
+were already saved mode settings in the file, they are replaced.
+Otherwise, the new mode information is appended to the end of the
+file.
@kindex m R
@pindex calc-mode-record-mode
their default values, then settings from the file you named are loaded
if this file exists, and this file becomes the one that Calc will
use in the future for commands like @kbd{m m}. The default settings
-file name is @file{~/.calc.el}. You can see the current file name by
+file name is @file{~/.emacs.d/calc.el}. You can see the current file name by
giving a blank response to the @kbd{m F} prompt. See also the
discussion of the @code{calc-settings-file} variable; @pxref{Customizing Calc}.
toggle the Inverse and/or Hyperbolic flags and then execute the
corresponding base command (@code{calc-sin} in this case).
-The Inverse and Hyperbolic flags apply only to the next Calculator
-command, after which they are automatically cleared. (They are also
-cleared if the next keystroke is not a Calc command.) Digits you
-type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
-arguments for the next command, not as numeric entries. The same
-is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
-subtract and keep arguments).
-
-The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
+@kindex O
+@pindex calc-option
+The @kbd{O} key (@code{calc-option}) sets another flag, the
+@dfn{Option Flag}, which also can alter the subsequent Calc command in
+various ways.
+
+The Inverse, Hyperbolic and Option 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}, @kbd{H} or @kbd{O} (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).
+
+Another Calc prefix flag, @kbd{K} (keep-arguments), is discussed
elsewhere. @xref{Keep Arguments}.
@node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
in the current radix. (Larger integers will still be displayed in their
entirety.)
+@cindex Two's complements
+Calc can display @expr{w}-bit integers using two's complement
+notation, although this is most useful with the binary, octal and
+hexadecimal display modes. This option is selected by using the
+@kbd{O} option prefix before setting the display radix, and a negative word
+size might be appropriate (@pxref{Binary Functions}). In two's
+complement notation, the integers in the (nearly) symmetric interval
+from
+@texline @math{-2^{w-1}}
+@infoline @expr{-2^(w-1)}
+to
+@texline @math{2^{w-1}-1}
+@infoline @expr{2^(w-1)-1}
+are represented by the integers from @expr{0} to @expr{2^w-1}:
+the integers from @expr{0} to
+@texline @math{2^{w-1}-1}
+@infoline @expr{2^(w-1)-1}
+are represented by themselves and the integers from
+@texline @math{-2^{w-1}}
+@infoline @expr{-2^(w-1)}
+to @expr{-1} are represented by the integers from
+@texline @math{2^{w-1}}
+@infoline @expr{2^(w-1)}
+to @expr{2^w-1} (the integer @expr{k} is represented by @expr{k+2^w}).
+Calc will display a two's complement integer by the radix (either
+@expr{2}, @expr{8} or @expr{16}), two @kbd{#} symbols, and then its
+representation (including any leading zeros necessary to include all
+@expr{w} bits). In a two's complement display mode, numbers that
+are not displayed in two's complement notation (i.e., that aren't
+integers from
+@texline @math{-2^{w-1}}
+@infoline @expr{-2^(w-1)}
+to
+@c (
+@texline @math{2^{w-1}-1})
+@infoline @expr{2^(w-1)-1})
+will be represented using Calc's usual notation (in the appropriate
+radix).
+
@node Grouping Digits, Float Formats, Radix Modes, Display Modes
@subsection Grouping Digits
@end group
@end example
@tex
-\turnoffactive
$$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
@end tex
@sp 1
@end group
@end example
@tex
-\turnoffactive
$$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
@end tex
@sp 2
@end group
@end example
@tex
-\turnoffactive
$$ 2 + 3 \to 5 $$
$$ 5 $$
@end tex
@end group
@end example
@tex
-\turnoffactive
$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
{\let\to\Rightarrow
$$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
@end group
@end example
@tex
-\turnoffactive
$$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
$$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
@end tex
Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
@item Save
-Record modes in @file{~/.calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
+Record modes in @file{~/.emacs.d/calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
@item Local
Record modes in Embedded buffer (@kbd{m R}).
@mindex v p
@end ignore
@kindex v p (complex)
+@kindex V p (complex)
@pindex calc-pack
The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
the stack into a composite object such as a complex number. With
@mindex v u
@end ignore
@kindex v u (complex)
+@kindex V u (complex)
@pindex calc-unpack
The @kbd{v u} (@code{calc-unpack}) command takes the complex number
(or other composite object) on the top of the stack and unpacks it
@end example
@end ifnottex
@tex
-\turnoffactive
$$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
$$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
$$ \code{fvl}(r, n, p) = p (1 + r)^n $$
particular, negative arguments are converted to positive integers modulo
@expr{2^w} by all binary functions.
-If the word size is negative, binary operations produce 2's complement
+If the word size is negative, binary operations produce twos-complement
integers from
@texline @math{-2^{-w-1}}
@infoline @expr{-(2^(-w-1))}
and @kbd{H I f G} [@code{gammaG}] commands.
@end ifnottex
@tex
-\turnoffactive
The functions corresponding to the integrals that define $P(a,x)$
and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
vectors.
@kindex v p
+@kindex V p
@pindex calc-pack
The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several
elements from the stack into a matrix, complex number, HMS form, error
by the mode.
@kindex v u
+@kindex V u
@pindex calc-unpack
The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
number, HMS form, or other composite object on the top of the stack and
to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster.
@kindex v d
+@kindex V d
@pindex calc-diag
@tindex diag
The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
@kindex v i
+@kindex V i
@pindex calc-ident
@tindex idn
The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity
dimensions.
@kindex v x
+@kindex V x
@pindex calc-index
@tindex index
The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
is one for positive @var{n} or two for negative @var{n}.
@kindex v b
+@kindex V b
@pindex calc-build-vector
@tindex cvec
The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
to build a matrix of copies of that row.)
@kindex v h
+@kindex V h
@kindex I v h
+@kindex I V h
@pindex calc-head
@pindex calc-tail
@tindex head
cases, the argument must be a non-empty vector.
@kindex v k
+@kindex V k
@pindex calc-cons
@tindex cons
The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h}
whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
@kindex H v h
+@kindex H V h
@tindex rhead
@ignore
@mindex @idots
@end ignore
@kindex H I v h
+@kindex H I V h
@ignore
@mindex @null
@end ignore
@kindex H v k
+@kindex H V k
@ignore
@mindex @null
@end ignore
@noindent
@kindex v r
+@kindex V r
@pindex calc-mrow
@tindex mrow
The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
function is called @code{getdiag}.
@kindex v c
+@kindex V c
@pindex calc-mcol
@tindex mcol
@tindex mrcol
of matrix @expr{m}.
@kindex v s
+@kindex V s
@pindex calc-subvector
@tindex subvec
The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts
has this effect when used as the ending index.
@kindex I v s
+@kindex I V s
@tindex rsubvec
With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector
from a vector. The arguments are interpreted the same as for the
@noindent
@kindex v l
+@kindex V l
@pindex calc-vlength
@tindex vlen
The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
command.
@kindex H v l
+@kindex H V l
@tindex mdims
With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector
of the dimensions of a vector, matrix, or higher-order object. For
matrix.
@kindex v f
+@kindex V f
@pindex calc-vector-find
@tindex find
The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches
allows you to select any starting index for the search.
@kindex v a
+@kindex V a
@pindex calc-arrange-vector
@tindex arrange
@cindex Arranging a matrix
@samp{[1, 2, @w{3, 4}]}.
@cindex Sorting data
+@kindex v S
@kindex V S
+@kindex I v S
@kindex I V S
@pindex calc-sort
@tindex sort
@cindex Inverse of permutation
@cindex Index tables
@cindex Rank tables
+@kindex v G
@kindex V G
+@kindex I v G
@kindex I V G
@pindex calc-grade
@tindex grade
phone numbers will remain sorted by name even after the second sort.
@cindex Histograms
+@kindex v H
@kindex V H
@pindex calc-histogram
@ignore
that the counts in the result vector don't add up to the length of the
input vector.)
+If no prefix is given, then you will be prompted for a vector which
+will be used to determine the bins. (If a positive integer is given at
+this prompt, it will be still treated as if it were given as a
+prefix.) Each bin will consist of the interval of numbers closest to
+the corresponding number of this new vector; if the vector
+@expr{[a, b, c, ...]} is entered at the prompt, the bins will be
+@expr{(-inf, (a+b)/2]}, @expr{((a+b)/2, (b+c)/2]}, etc. The result of
+this command will be a vector counting how many elements of the
+original vector are in each bin.
+
+The result will then be a vector with the same length as this new vector;
+each element of the new vector will be replaced by the number of
+elements of the original vector which are closest to it.
+
+@kindex H v H
@kindex H V H
With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
The second-to-top vector is the list of numbers as before. The top
vector. Without the hyperbolic flag, every element has a weight of one.
@kindex v t
+@kindex V t
@pindex calc-transpose
@tindex trn
The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
a one-column matrix.
@kindex v v
+@kindex V v
@pindex calc-reverse-vector
@tindex rev
The @kbd{v v} (@code{calc-reverse-vector}) [@code{rev}] command reverses
a matrix.)
@kindex v m
+@kindex V m
@pindex calc-mask-vector
@tindex vmask
The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses
@xref{Logical Operations}.
@kindex v e
+@kindex V e
@pindex calc-expand-vector
@tindex vexp
The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command
produces @samp{[a, 0, b, 0, 7]}.
@kindex H v e
+@kindex H V e
With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
top of the stack; the mask and target vectors come from the third and
second elements of the stack. This filler is used where the mask is
@code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
@code{float}, @code{frac}. @xref{Function Index}.
+@kindex v J
@kindex V J
@pindex calc-conj-transpose
@tindex ctrn
from that point to the origin.
@kindex v n
+@kindex V n
@pindex calc-rnorm
@tindex rnorm
The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes the
a matrix, this is the maximum of the row-absolute-value-sums, i.e., of
the sums of the absolute values of the elements along the various rows.
+@kindex v N
@kindex V N
@pindex calc-cnorm
@tindex cnorm
not provided. However, the 2-norm (or Frobenius norm) is provided for
vectors by the @kbd{A} (@code{calc-abs}) command.
+@kindex v C
@kindex V C
@pindex calc-cross
@tindex cross
@samp{/} operator also does a matrix inversion when dividing one
by a matrix.
+@kindex v D
@kindex V D
@pindex calc-mdet
@tindex det
The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
determinant of a square matrix.
+@kindex v L
@kindex V L
@pindex calc-mlud
@tindex lud
algorithm, the second is lower-triangular with ones on the diagonal,
and the third is upper-triangular.
+@kindex v T
@kindex V T
@pindex calc-mtrace
@tindex tr
trace of a square matrix. This is defined as the sum of the diagonal
elements of the matrix.
+@kindex v K
@kindex V K
@pindex calc-kron
@tindex kron
a certain value is a member of a given set. To test if the set @expr{A}
is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
+@kindex v +
@kindex V +
@pindex calc-remove-duplicates
@tindex rdup
other set-based commands apply @kbd{V +} to their inputs before using
them.
+@kindex v V
@kindex V V
@pindex calc-set-union
@tindex vunion
accomplish the same thing by concatenating the sets with @kbd{|},
then using @kbd{V +}.)
+@kindex v ^
@kindex V ^
@pindex calc-set-intersect
@tindex vint
@texline intersection@tie{}(@math{A \cap B}).
@infoline intersection.
+@kindex v -
@kindex V -
@pindex calc-set-difference
@tindex vdiff
your problem is small enough to list in a Calc vector (or simple
enough to express in a few intervals).
+@kindex v X
@kindex V X
@pindex calc-set-xor
@tindex vxor
if it is in one, but @emph{not} both, of the sets. Objects that
occur in both sets ``cancel out.''
+@kindex v ~
@kindex V ~
@pindex calc-set-complement
@tindex vcompl
For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
@samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}.
+@kindex v F
@kindex V F
@pindex calc-set-floor
@tindex vfloor
the complement with respect to the set of integers you could type
@kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
+@kindex v E
@kindex V E
@pindex calc-set-enumerate
@tindex venum
the intervals. This only works for finite sets (i.e., sets which
do not involve @samp{-inf} or @samp{inf}).
+@kindex v :
@kindex V :
@pindex calc-set-span
@tindex vspan
limit will be the largest element. For an empty set, @samp{vspan([])}
returns the empty interval @w{@samp{[0 .. 0)}}.
+@kindex v #
@kindex V #
@pindex calc-set-cardinality
@tindex vcard
@texline @math{1 /\sigma^2}.
@infoline @expr{1 / s^2}.
@tex
-\turnoffactive
$$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
\displaystyle \sum { 1 \over \sigma_i^2 } } $$
@end tex
of the input errors. (I.e., the variance is the reciprocal of the
sum of the reciprocals of the variances.)
@tex
-\turnoffactive
$$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
@end tex
If the inputs are plain
then assuming each value's error is equal to this standard
deviation.)
@tex
-\turnoffactive
$$ \sigma_\mu^2 = {\sigma^2 \over N} $$
@end tex
defined as the reciprocal of the arithmetic mean of the reciprocals
of the values.
@tex
-\turnoffactive
$$ { N \over \displaystyle \sum {1 \over x_i} } $$
@end tex
equal to the @code{exp} of the arithmetic mean of the logarithms
of the data values.
@tex
-\turnoffactive
$$ \exp \left ( \sum { \ln x_i } \right ) =
\left ( \prod { x_i } \right)^{1 / N} $$
@end tex
replacing the two numbers with their arithmetic mean and geometric
mean, then repeating until the two values converge.
@tex
-\turnoffactive
$$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
@end tex
the differences between the values and the mean of the @expr{N} values,
divided by @expr{N-1}.
@tex
-\turnoffactive
$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
@end tex
data values, so that the mean computed from the input is itself
only an estimate of the true mean.
@tex
-\turnoffactive
$$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
@end tex
is taken as the square root of the sum of the squares of the two
input errors.
@tex
-\turnoffactive
$$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
$$ \sigma_{x\!y}^2 =
{\displaystyle {1 \over N-1}
product of their standard deviations. (There is no difference
between sample or population statistics here.)
@tex
-\turnoffactive
$$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
@end tex
The commands in this section allow for more general operations on the
elements of vectors.
+@kindex v A
@kindex V A
@pindex calc-apply
@tindex apply
@subsection Mapping
@noindent
+@kindex v M
@kindex V M
@pindex calc-map
@tindex map
@subsection Reducing
@noindent
+@kindex v R
@kindex V R
@pindex calc-reduce
@tindex reduce
and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
produces @samp{f(f(f(a, b), c), d)}.
+@kindex I v R
@kindex I V R
@tindex rreduce
The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
in power series expansions.
+@kindex v U
@kindex V U
@tindex accum
The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
the vector @samp{[a, b, c, d]} produces the vector
@samp{[a, a + b, a + b + c, a + b + c + d]}.
+@kindex I v U
@kindex I V U
@tindex raccum
The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
@subsection Nesting and Fixed Points
@noindent
+@kindex H v R
@kindex H V R
@tindex nest
The @kbd{H V R} [@code{nest}] command applies a function to a given
negative if Calc knows an inverse for the function @samp{f}; for
example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
+@kindex H v U
@kindex H V U
@tindex anest
The @kbd{H V U} [@code{anest}] command is an accumulating version of
@samp{F} is the inverse of @samp{f}, then the result is of the
form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
+@kindex H I v R
@kindex H I V R
@tindex fixp
@cindex Fixed points
applied until it reaches a ``fixed point,'' i.e., until the result
no longer changes.
+@kindex H I v U
@kindex H I V U
@tindex afixp
The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
@node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping
@subsection Generalized Products
+@kindex v O
@kindex V O
@pindex calc-outer-product
@tindex outer
the result matrix is obtained by applying the operator to element @var{r}
of the lefthand vector and element @var{c} of the righthand vector.
+@kindex v I
@kindex V I
@pindex calc-inner-product
@tindex inner
influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
@pxref{Normal Language Modes}.
+@kindex v <
@kindex V <
@pindex calc-matrix-left-justify
+@kindex v =
@kindex V =
@pindex calc-matrix-center-justify
+@kindex v >
@kindex V >
@pindex calc-matrix-right-justify
The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
(@code{calc-matrix-center-justify}) control whether matrix elements
are justified to the left, right, or center of their columns.
+@kindex v [
@kindex V [
@pindex calc-vector-brackets
+@kindex v @{
@kindex V @{
@pindex calc-vector-braces
+@kindex v (
@kindex V (
@pindex calc-vector-parens
The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
and parentheses may never be used for this purpose.
@kindex V ]
+@kindex v ]
+@kindex V )
+@kindex v )
+@kindex V @}
+@kindex v @}
@pindex calc-matrix-brackets
The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
-``big'' style display of matrices. It prompts for a string of code
-letters; currently implemented letters are @code{R}, which enables
-brackets on each row of the matrix; @code{O}, which enables outer
-brackets in opposite corners of the matrix; and @code{C}, which
-enables commas or semicolons at the ends of all rows but the last.
-The default format is @samp{RO}. (Before Calc 2.00, the format
-was fixed at @samp{ROC}.) Here are some example matrices:
+``big'' style display of matrices, for matrices which have more than
+one row. It prompts for a string of code letters; currently
+implemented letters are @code{R}, which enables brackets on each row
+of the matrix; @code{O}, which enables outer brackets in opposite
+corners of the matrix; and @code{C}, which enables commas or
+semicolons at the ends of all rows but the last. The default format
+is @samp{RO}. (Before Calc 2.00, the format was fixed at @samp{ROC}.)
+Here are some example matrices:
@example
@group
@samp{OC} are all recognized as matrices during reading, while
the others are useful for display only.
+@kindex v ,
@kindex V ,
@pindex calc-vector-commas
The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
ambiguity) by adding the letter @code{P} to the control string you
give to @kbd{v ]} (as described above).
+@kindex v .
@kindex V .
@pindex calc-full-vectors
The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
large vectors, this mode will improve the speed of all operations
that involve the trail.
+@kindex v /
@kindex V /
@pindex calc-break-vectors
The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
@noindent
@kindex a s
+@kindex I a s
+@kindex H a s
@pindex calc-simplify
@tindex simplify
The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies
simplification occurs automatically. Normally only the ``default
simplifications'' occur.
+There are some simplifications that, while sometimes useful, are never
+done automatically. For example, the @kbd{I} prefix can be given to
+@kbd{a s}; the @kbd{I a s} command will change any trigonometric
+function to the appropriate combination of @samp{sin}s and @samp{cos}s
+before simplifying. This can be useful in simplifying even mildly
+complicated trigonometric expressions. For example, while @kbd{a s}
+can reduce @samp{sin(x) csc(x)} to @samp{1}, it will not simplify
+@samp{sin(x)^2 csc(x)}. The command @kbd{I a s} can be used to
+simplify this latter expression; it will transform @samp{sin(x)^2
+csc(x)} into @samp{sin(x)}. However, @kbd{I a s} will also perform
+some ``simplifications'' which may not be desired; for example, it
+will transform @samp{tan(x)^2} into @samp{sin(x)^2 / cos(x)^2}. The
+Hyperbolic prefix @kbd{H} can be used similarly; the @kbd{H a s} will
+replace any hyperbolic functions in the formula with the appropriate
+combinations of @samp{sinh}s and @samp{cosh}s before simplifying.
+
+
@menu
* Default Simplifications::
* Algebraic Simplifications::
@end example
@end ifnottex
@tex
-\turnoffactive
-\turnoffactive
\beforedisplay
$$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
5 & 7 & 9 & 11 & 13 }
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
\afterdisplay
@end example
@end ifnottex
@tex
-\turnoffactive
\beforedisplay
$$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
\afterdisplay
the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
produces the result 55.
@tex
-\turnoffactive
$$ \sum_{k=1}^5 k^2 = 55 $$
@end tex
@cindex Calc init file, user-defined units
The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
units in your Calc init file (the file given by the variable
-@code{calc-settings-file}, typically @file{~/.calc.el}), so that the
+@code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}), so that the
units will still be available in subsequent Emacs sessions. If there
was already a set of user-defined units in your Calc init file, it
is replaced by the new set. (@xref{General Mode Commands}, for a way to
@cindex Calc init file, variables
The @kbd{s p} (@code{calc-permanent-variable}) command saves a
variable's value permanently in your Calc init file (the file given by
-the variable @code{calc-settings-file}, typically @file{~/.calc.el}), so
+the variable @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}), so
that its value will still be available in future Emacs sessions. You
can re-execute @w{@kbd{s p}} later on to update the saved value, but the
only way to remove a saved variable is to edit your calc init file
@vindex calc-gnuplot-name
If you have GNUPLOT installed on your system but Calc is unable to
-find it, you may need to set the @code{calc-gnuplot-name} variable
-in your Calc init file or @file{.emacs}. You may also need to set some Lisp
-variables to show Calc how to run GNUPLOT on your system; these
-are described under @kbd{g D} and @kbd{g O} below. If you are
-using the X window system, Calc will configure GNUPLOT for you
-automatically. If you have GNUPLOT 3.0 or later and you are not using X,
-Calc will configure GNUPLOT to display graphs using simple character
-graphics that will work on any terminal.
+find it, you may need to set the @code{calc-gnuplot-name} variable in
+your Calc init file or @file{.emacs}. You may also need to set some
+Lisp variables to show Calc how to run GNUPLOT on your system; these
+are described under @kbd{g D} and @kbd{g O} below. If you are using
+the X window system or MS-Windows, Calc will configure GNUPLOT for you
+automatically. If you have GNUPLOT 3.0 or later and you are using a
+Unix or GNU system without X, Calc will configure GNUPLOT to display
+graphs using simple character graphics that will work on any
+Posix-compatible terminal.
@menu
* Basic Graphics::
blank line this command shows you the current default. The special
name @code{default} signifies that Calc should choose @code{x11} if
the X window system is in use (as indicated by the presence of a
-@code{DISPLAY} environment variable), or otherwise @code{dumb} under
-GNUPLOT 3.0 and later, or @code{postscript} under GNUPLOT 2.0.
-This is the initial default value.
+@code{DISPLAY} environment variable), @code{windows} on MS-Windows, or
+otherwise @code{dumb} under GNUPLOT 3.0 and later, or
+@code{postscript} under GNUPLOT 2.0. This is the initial default
+value.
The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
terminals with no special graphics facilities. It writes a crude
@kindex g O
@pindex calc-graph-output
-The @kbd{g O} (@code{calc-graph-output}) command sets the name of
-the output file used by GNUPLOT. For some devices, notably @code{x11},
-there is no output file and this information is not used. Many other
-``devices'' are really file formats like @code{postscript}; in these
-cases the output in the desired format goes into the file you name
-with @kbd{g O}. Type @kbd{g O stdout @key{RET}} to set GNUPLOT to write
-to its standard output stream, i.e., to @samp{*Gnuplot Trail*}.
-This is the default setting.
+The @kbd{g O} (@code{calc-graph-output}) command sets the name of the
+output file used by GNUPLOT. For some devices, notably @code{x11} and
+@code{windows}, there is no output file and this information is not
+used. Many other ``devices'' are really file formats like
+@code{postscript}; in these cases the output in the desired format
+goes into the file you name with @kbd{g O}. Type @kbd{g O stdout
+@key{RET}} to set GNUPLOT to write to its standard output stream,
+i.e., to @samp{*Gnuplot Trail*}. This is the default setting.
Another special output name is @code{tty}, which means that GNUPLOT
is going to write graphics commands directly to its standard output,
The normal value is @code{default}, which generally means your
window manager will let you place the window interactively.
Entering @samp{800x500+0+0} would create an 800-by-500 pixel
-window in the upper-left corner of the screen.
+window in the upper-left corner of the screen. This command has no
+effect if the current device is @code{windows}.
The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
session with GNUPLOT. This shows the commands Calc has ``typed'' to
GNUPLOT and the responses it has received. Calc tries to notice when an
error message has appeared here and display the buffer for you when
this happens. You can check this buffer yourself if you suspect
-something has gone wrong.
+something has gone wrong@footnote{
+On MS-Windows, due to the peculiarities of how the Windows version of
+GNUPLOT (called @command{wgnuplot}) works, the GNUPLOT responses are
+not communicated back to Calc. Instead, you need to look them up in
+the GNUPLOT command window that is displayed as in normal interactive
+usage of GNUPLOT.
+}.
@kindex g C
@pindex calc-graph-command
This happens automatically when Calc thinks there is something you
will want to see in either of these buffers. If you type @kbd{g v}
or @kbd{g V} when the relevant buffer is already displayed, the
-buffer is hidden again.
+buffer is hidden again. (Note that on MS-Windows, the @samp{*Gnuplot
+Trail*} buffer will usually show nothing of interest, because
+GNUPLOT's responses are not communicated back to Calc.)
One reason to use @kbd{g v} is to add your own commands to the
@samp{*Gnuplot Commands*} buffer. Press @kbd{g v}, then use
which are also available outside of Embedded mode.
(@pxref{General Mode Commands}.) They are @code{Save}, in which mode
settings are recorded permanently in your Calc init file (the file given
-by the variable @code{calc-settings-file}, typically @file{~/.calc.el})
+by the variable @code{calc-settings-file}, typically @file{~/.emacs.d/calc.el})
rather than by annotating the current document, and no-recording
mode (where there is no symbol like @code{Save} or @code{Local} in
the mode line), in which mode-changing commands do not leave any
binding permanent so that it will remain in effect even in future Emacs
sessions. (It does this by adding a suitable bit of Lisp code into
your Calc init file; that is, the file given by the variable
-@code{calc-settings-file}, typically @file{~/.calc.el}.) For example,
+@code{calc-settings-file}, typically @file{~/.emacs.d/calc.el}.) For example,
@kbd{Z P s} would register our @code{sincos} command permanently. If
you later wish to unregister this command you must edit your Calc init
file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
A good place to put your @code{defmath} commands is your Calc init file
(the file given by @code{calc-settings-file}, typically
-@file{~/.calc.el}), which will not be loaded until Calc starts.
+@file{~/.emacs.d/calc.el}), which will not be loaded until Calc starts.
If a file named @file{.emacs} exists in your home directory, Emacs reads
and executes the Lisp forms in this file as it starts up. While it may
seem reasonable to put your favorite @code{defmath} commands there,
@smallexample
(let ((calc-command-flags nil))
(unwind-protect
- (save-excursion
+ (save-current-buffer
(calc-select-buffer)
@emph{body of function}
@emph{renumber stack}
@code{nil}, then Calc will automatically load your settings file (if it
exists) the first time Calc is invoked.
-The default value for this variable is @code{"~/.calc.el"}.
+The default value for this variable is @code{"~/.emacs.d/calc.el"}
+unless the file @file{~/.calc.el} exists, in which case the default
+value will be @code{"~/.calc.el"}.
@end defvar
@defvar calc-gnuplot-name
so any efforts can be coordinated.
The latest version of Calc is available from Savannah, in the Emacs
-CVS tree. See @uref{http://savannah.gnu.org/projects/emacs}.
+repository. See @uref{http://savannah.gnu.org/projects/emacs}.
@c [summary]
@node Summary, Key Index, Reporting Bugs, Top
@r{ @: M @: @: @:calc-more-recursion-depth@:}
@r{ @: I M @: @: @:calc-less-recursion-depth@:}
@r{ a@: N @: @: 5 @:evalvn@:(a)}
+@r{ @: O @:command @: 32 @:@:Option}
@r{ @: P @: @: @:@:pi}
@r{ @: I P @: @: @:@:gamma}
@r{ @: H P @: @: @:@:e}
HCoop / bpt / emacs.git / blobdiff