Usenet 1993 July/InfoMagic USENET CD-ROM July 1993.ISO/sources/misc/volume24/gnucalc/part44

Text File | 1991-11-01 | 55.5 KB | 1,319 lines

Newsgroups: comp.sources.misc
From: [e-mail] (David Gillespie)
Subject:  v24i092:  gnucalc - GNU Emacs Calculator, v2.00, Part44/56
Message-ID: <[e-mail]>
X-Md4-Signature: 0969d0f8e53ab4b5855eadb3d104aa96
Date: Fri, 1 Nov 1991 18:37:54 GMT
Approved: [e-mail]

Submitted-by: [e-mail] (David Gillespie)
Posting-number: Volume 24, Issue 92
Archive-name: gnucalc/part44
Environment: Emacs
Supersedes: gmcalc: Volume 13, Issue 27-45

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error form.  This includes the estimated error associated with
the mean.  If the inputs are error forms, the error is the square
root of the reciprocal of the sum of the reciprocals of the squares
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
numbers, the error is equal to the standard deviation of the values
divided by the square root of the number of values.  (This works
out to be equivalent to calculating the standard deviation and
then assuming each value's error is equal to this standard
deviation.)@refill
@tex
\turnoffactive
$$ \sigma_\mu = {\sigma \over \sqrt N} $$
@end tex
X
@kindex H u M
@pindex calc-vector-median
@tindex vmedian
@cindex Median of data values
The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}]
command computes the median of the data values.  The values are
first sorted into numerical order; the median is the middle
value after sorting.  (If the number of data values is even,
the median is taken to be the average of the two middle values.)
The median function is different from the other functions in
this section in that the arguments must all be real numbers;
variables are not accepted even when nested inside vectors.
(Otherwise it is not possible to sort the data values.)  If
any of the input values are error forms, their error parts are
ignored.
X
The median function also accepts distributions.  For both normal
(error form) and uniform (interval) distributions, the median is
the same as the mean.
X
@kindex H I u M
@pindex calc-vector-harmonic-mean
@tindex vhmean
@cindex Harmonic mean
The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}]
command computes the harmonic mean of the data values.  This is
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
X
@kindex u G
@pindex calc-vector-geometric-mean
@tindex vgmean
@cindex Geometric mean
The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}]
command computes the geometric mean of the data values.  This
is the @i{N}th root of the product of the values.  This is also
equal to the @code{exp} of the arithmetic mean of the logarithms
of the data values.
@tex
\turnoffactive
$$ \exp \left ( \sum { \ln x_i } \right ) =
X   \left ( \prod { x_i } \right)^{1 / N} $$
@end tex
X
@kindex H u G
@tindex agmean
The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric
mean'' of two numbers taken from the stack.  This is computed by
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
X
@cindex Root-mean-square
Another commonly used mean, the RMS (root-mean-square), can be computed
for a vector of numbers simply by using the @kbd{A} command.
X
@kindex u S
@pindex calc-vector-sdev
@tindex vsdev
@cindex Standard deviation
@cindex Sample statistics
The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command
computes the standard deviation@c{ $\sigma$}
@asis{ }of the data values.  If the
values are error forms, the errors are used as weights just
as for @kbd{u M}.  This is the @emph{sample} standard deviation,
whose value is the square root of the sum of the squares of the
differences between the values and the mean of the @cite{N} values,
divided by @cite{N-1}.
@tex
\turnoffactive
$$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
@end tex
X
This function also applies to distributions.  The standard deviation
of a single error form is simply the error part.  The standard deviation
of a continuous interval happens to equal the difference between the
limits, divided by @c{$\sqrt{12}$}
@cite{sqrt(12)}.  The standard deviation of an
integer interval is the same as the standard deviation of a vector
of those integers.
X
@kindex I u S
@pindex calc-vector-pop-sdev
@tindex vpsdev
@cindex Population statistics
The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}]
command computes the @emph{population} standard deviation.
It is defined by the same formula as above but dividing
by @cite{N} instead of by @cite{N-1}.  The population standard
deviation is used when the input represents the entire set of
data values in the distribution; the sample standard deviation
is used when the input represents a sample of the set of all
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
X
For error forms and continuous intervals, @code{vpsdev} works
exactly like @code{vsdev}.  For integer intervals, it computes the
population standard deviation of the equivalent vector of integers.
X
@kindex H u S
@kindex H I u S
@pindex calc-vector-variance
@pindex calc-vector-pop-variance
@tindex vvar
@tindex vpvar
@cindex Variance of data values
The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and
@kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}]
commands compute the variance of the data values.  The variance
is the square@c{ $\sigma^2$}
@asis{ }of the standard deviation, i.e., the sum of the
squares of the deviations of the data values from the mean.
(This definition also applies when the argument is a distribution.)
X
@tindex vflat
The @code{vflat} algebraic function returns a vector of its
arguments, interpreted in the same way as the other functions
in this section.  For example, @samp{vflat(1, [2, [3, 4]], 5)}
returns @samp{[1, 2, 3, 4, 5]}.
X
@node Paired-Sample Statistics, , Single-Variable Statistics, Statistical Operations
@subsection Paired-Sample Statistics
X
@noindent
The functions in this section take two arguments, which must be
vectors of equal size.  The vectors are each flattened in the same
way as by the single-variable statistical functions.  Given a numeric
prefix argument of 1, these functions instead take one object from
the stack, which must be an @c{$N\times2$}
@asis{Nx2} matrix of data values.  Once
again, variable names can be used in place of actual vectors and
matrices.
X
@kindex u C
@pindex calc-vector-covariance
@tindex vcov
@cindex Covariance
The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command
computes the sample covariance of two vectors.  The covariance
of vectors @var{x} and @var{y} is the sum of the products of the
differences between the elements of @var{x} and the mean of @var{x}
times the differences between the corresponding elements of @var{y}
and the mean of @var{y}, all divided by @cite{N-1}.  Note that
the variance of a vector is just the covariance of the vector
with itself.  Once again, if the inputs are error forms the
errors are used as weight factors.  If both @var{x} and @var{y}
are composed of error forms, the error for a given data point
is taken as the square root of the sum of the squares of the two
input errors.
@tex
\turnoffactive
$$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
$$ \sigma_{x\!y}^2 =
X    {\displaystyle {1 \over N-1}
X                   \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
X     \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
$$
@end tex
X
@kindex I u C
@pindex calc-vector-pop-covariance
@tindex vpcov
The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}]
command computes the population covariance, which is the same as the
sample covariance computed by @kbd{u C} except dividing by @cite{N}
instead of @cite{N-1}.
X
@kindex H u C
@pindex calc-vector-correlation
@tindex vcorr
@cindex Correlation coefficient
@cindex Linear correlation
The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}]
command computes the linear correlation coefficient of two vectors.
This is defined by the covariance of the vectors divided by the
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
X
@node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions
@section Reducing and Mapping Vectors
X
@noindent
The commands in this section allow for more general operations on the
elements of vectors.
X
@kindex V A
@pindex calc-apply
@tindex apply
The simplest of these operations is @kbd{V A} (@code{calc-apply})
[@code{apply}], which applies a given operator to the elements of a vector.
For example, applying the hypothetical function @code{f} to the vector
@samp{[1, 2, 3]} would produce the function call @samp{f(1, 2, 3)}.
Applying the @code{+} function to the vector @samp{[a, b]} gives
@samp{a + b}.  Applying @code{+} to the vector @samp{[a, b, c]} is an
error, since the @code{+} function expects exactly two arguments.
X
While @kbd{V A} is useful in some cases, you will usually find that either
@kbd{V R} or @kbd{V M}, described below, is closer to what you want.
X
@menu
* Specifying Operators::
* Mapping::
* Reducing::
* Nesting and Fixed Points::
* Generalized Products::
@end menu
X
@node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping
@subsection Specifying Operators
X
@noindent
Commands in this section (like @kbd{V A}) prompt you to press the key
corresponding to the desired operator.  Press @kbd{?} for a partial
list of the available operators.  Generally, an operator is any key or
sequence of keys that would normally take one or more arguments from
the stack and replace them with a result.  For example, @kbd{V A H C}
uses the hyperbolic cosine operator, @code{cosh}.  (Since @code{cosh}
expects one argument, @kbd{V A H C} requires a vector with a single
element as its argument.)
X
You can press @kbd{x} at the operator prompt to select any algebraic
function by name to use as the operator.  This includes functions you
have defined yourself using the @kbd{Z F} command.  (@xref{Algebraic
Definitions}.)  If you give a name for which no function has been
defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
Calc will prompt for the number of arguments the function takes if it
can't figure it out on its own (say, because you named a function that
is currently undefined).  It is also possible to type a digit key before
the function name to specify the number of arguments, e.g.,
@kbd{V M 3 x f RET} calls @code{f} with three arguments even if it
looks like it ought to have only two.  This technique may be necessary
if the function allows a variable number of arguments.  For example,
the @kbd{v e} [@code{vexp}] function accepts two or three arguments;
if you want to map with the three-argument version, you will have to
type @kbd{V M 3 v e}.
X
It is also possible to apply any formula to a vector by treating that
formula as a function.  When prompted for the operator to use, press
@kbd{'} (the apostrophe) and type your formula as an algebraic entry.
You will then be prompted for the argument list, which defaults to a
list of all variables that appear in the formula, sorted into alphabetic
order.  For example, suppose you enter the formula @samp{x + 2y^x}.
The default argument list would be @samp{(x y)}, which means that if
this function is applied to the arguments @samp{[3, 10]} the result will
be @samp{3 + 2*10^3}.  (If you plan to use a certain formula in this
way often, you might consider defining it as a function with @kbd{Z F}.)
X
Another way to specify the arguments to the formula you enter is with
@kbd{$}, @kbd{$$}, and so on.  For example, @kbd{V A ' $$ + 2$^$$}
has the same effect as the previous example.  The argument list is
automatically taken to be @samp{($$ $)}.  (The order of the arguments
may seem backwards, but it is analogous to the way normal algebraic
entry interacts with the stack.)
X
If you press @kbd{$} at the operator prompt, the effect is similar to
the apostrophe except that the relevant formula is taken from top-of-stack
instead.  The actual vector arguments of the @kbd{V A $} or related command
then start at the second-to-top stack position.  You will still be
prompted for an argument list.
X
@cindex Nameless functions
@cindex Generic functions
A function can be written without a name using the notation @samp{<#1 - #2>},
which means ``a function of two arguments that computes the first
argument minus the second argument.''  The symbols @samp{#1} and @samp{#2}
are placeholders for the arguments.  You can use any names for these
placeholders if you wish, by including an argument list followed by a
colon:  @samp{<x, y : x - y>}.  When you type @kbd{V M ' $$ + 2$^$$ RET},
Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
to map across the vectors.  When you type @kbd{V M ' x + 2y^x RET RET},
Calc builds the nameless function @samp{<x, y : x + 2 y^x>}.  In both
cases, Calc also writes the nameless function to the Trail so that you
can get it back later if you wish.
X
If there is only one argument, you can write @samp{#} in place of @samp{#1}.
(Note that @samp{< >} notation is also used for date forms.  Calc tells
that @samp{<@var{stuff}>} is a nameless function by the presence of
@samp{#} signs inside @var{stuff}, or by the fact that @var{stuff}
begins with a list of variables followed by a colon.)
X
You can type a nameless function directly to @kbd{V M '}, or put one on
the stack and use it with @kbd{V M $}.  Calc will not prompt for an
argument list in this case, since the nameless function specifies the
argument list as well as the function itself.  In @kbd{V M '}, you can
omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
so that @kbd{V M ' #1+#2 RET} is the same as @kbd{V M ' <#1+#2> RET},
which in turn is the same as @kbd{V M ' $$+$ RET}.
X
@cindex Lambda expressions
@tindex lambda
The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
(The word @code{lambda} derives from Lisp notation and the theory of
functions.)  The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
ArgB, ArgA + ArgB)}.  Note that there is no actual Calc function called
@code{lambda}; the whole point is that the @code{lambda} expression is
used in its symbolic form, not evaluated for an answer until it is applied
to specific arguments by a command like @kbd{V A} or @kbd{V M}.
X
(Actually, @code{lambda} does have one special property:  Its arguments
are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
will not simplify the @samp{2/3} until the nameless function is actually
called.)
X
@tindex add
@tindex sub
@tindex mul
@tindex div
@tindex pow
@tindex neg
@tindex mod
@tindex vconcat
As usual, commands like @kbd{V A} have algebraic function name equivalents.
For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
@samp{apply(gcd, v)}.  The first argument specifies the operator name,
and is either a variable whose name is the same as the function name,
or a nameless function like @samp{<#^3+1>}.  Operators that are normally
written as algebraic symbols have the names @code{add}, @code{sub},
@code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
@code{vconcat}.@refill
X
@tindex call
The @code{call} function builds a function call out of several arguments:
@samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
in turn is the same as @samp{gcd(x, y)}.  The first argument of @code{call},
like the other functions described here, may be either a variable naming a
function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
as @samp{x + 2y}).
X
(Experts will notice that it's not quite proper to use a variable to name
a function, since the name @code{gcd} corresponds to the Lisp variable
@code{var-gcd} but to the Lisp function @code{calcFunc-gcd}.  Calc
automatically makes this translation, so you don't have to worry
about it.)
X
@node Mapping, Reducing, Specifying Operators, Reducing and Mapping
@subsection Mapping
X
@noindent
@kindex V M
@pindex calc-map
@tindex map
The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given
operator elementwise to one or more vectors.  For example, mapping
@code{A} [@code{abs}] produces a vector of the absolute values of the
elements in the input vector.  Mapping @code{+} pops two vectors from
the stack, which must be of equal length, and produces a vector of the
pairwise sums of the elements.  If either argument is a non-vector, it
is duplicated for each element of the other vector.  For example,
@kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
With the 2 listed first, it would have computed a vector of powers of
two.  Mapping a user-defined function pops as many arguments from the
stack as the function requires.  If you give an undefined name, you will
be prompted for the number of arguments to use.@refill
X
If any argument to @kbd{V M} is a matrix, the operator is normally mapped
across all elements of the matrix.  For example, given the matrix
@cite{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
produce another @c{$3\times2$}
@asis{3x2} matrix, @cite{[[1, 2, 3], [4, 5, 6]]}.
X
@tindex mapr
The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the
operator prompt) maps by rows instead.  For example, @kbd{V M _ A} views
the above matrix as a vector of two 3-element row vectors.  It produces
a new vector which contains the absolute values of those row vectors,
namely @cite{[3.74, 8.77]}.  (Recall, the absolute value of a vector is
defined as the square root of the sum of the squares of the elements.)
Some operators accept vectors and return new vectors; for example,
@kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row
of the matrix to get a new matrix, @cite{[[3, -2, 1], [-6, 5, -4]]}.
X
Sometimes a vector of vectors (representing, say, strings, sets, or lists)
happens to look like a matrix.  If so, remember to use @kbd{V M _} if you
want to map a function across the whole strings or sets rather than across
their individual elements.
X
@tindex mapc
The command @kbd{V M :} [@code{mapc}] maps by columns.  Basically, it
transposes the input matrix, maps by rows, and then, if the result is a
matrix, transposes again.  For example, @kbd{V M : A} takes the absolute
values of the three columns of the matrix, treating each as a 2-vector,
and @kbd{V M : v v} reverses the columns to get the matrix
@cite{[[-4, 5, -6], [1, -2, 3]]}.
X
(The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
and column-like appearances, and were not already taken by useful
operators.  Also, they appear shifted on most keyboards so they are easy
to type after @kbd{V M}.)
X
The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
not matrices (so if none of the arguments are matrices, they have no
effect at all).  If some of the arguments are matrices and others are
plain numbers, the plain numbers are held constant for all rows of the
matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
a vector takes a dot product of the vector with itself).
X
If some of the arguments are vectors with the same lengths as the
rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
arguments, those vectors are also held constant for every row or
column.
X
Sometimes it is useful to specify another mapping command as the operator
to use with @kbd{V M}.  For example, @kbd{V M _ V A +} applies @kbd{V A +}
to each row of the input matrix, which in turn adds the two values on that
row.  If you give another vector-operator command as the operator for
@kbd{V M}, it automatically uses map-by-rows mode if you don't specify
otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}.  (If
you really want to map-by-elements another mapping command, you can use
a triple-nested mapping command:  @kbd{V M V M V A +} means to map
@kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
mapped over the elements of each row.)
X
@tindex mapa
@tindex mapd
Previous versions of Calc had ``map across'' and ``map down'' modes
that are now considered obsolete; the old ``map across'' is now simply
@kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}.  The algebraic
functions @code{mapa} and @code{mapd} are still supported, though.
Note also that, while the old mapping modes were persistent (once you
set the mode, it would apply to later mapping commands until you reset
it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
mapping command.  The default @kbd{V M} always means map-by-elements.
X
@xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like
@kbd{V M} but for equations and inequalities instead of vectors.
@xref{Storing Variables}, for the @kbd{s m} command which modifies a
variable's stored value using a @kbd{V M}-like operator.
X
@node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
@subsection Reducing
X
@noindent
@kindex V R
@pindex calc-reduce
@tindex reduce
The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given
binary operator across all the elements of a vector.  A binary operator is
a function such as @code{+} or @code{max} which takes two arguments.  For
example, reducing @code{+} over a vector computes the sum of the elements
of the vector.  Reducing @code{-} computes the first element minus each of
the remaining elements.  Reducing @code{max} computes the maximum element
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)}.
X
@kindex I V R
@tindex rreduce
The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
that works from right to left through the vector.  For example, plain
@kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d}
but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
or @samp{a - b + c - d}.  This ``alternating sum'' occurs frequently
in power series expansions.
X
@kindex V U
@tindex accum
The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
accumulation operation.  Here Calc does the corresponding reduction
operation, but instead of producing only the final result, it produces
a vector of all the intermediate results.  Accumulating @code{+} over
the vector @samp{[a, b, c, d]} produces the vector
@samp{[a, a + b, a + b + c, a + b + c + d]}.
X
@kindex I V U
@tindex raccum
The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
X
@tindex reducea
@tindex rreducea
@tindex reduced
@tindex rreduced
As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise.  For
example, given the matrix @cite{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
compute @cite{a + b + c + d + e + f}.  You can type @kbd{V R _} or
@kbd{V R :} to modify this behavior.  The @kbd{V R _} [@code{reducea}]
command reduces ``across'' the matrix; it reduces each row of the matrix
as a vector, then collects the results.  Thus @kbd{V R _ +} of this
matrix would produce @cite{[a + b + c, d + e + f]}.  Similarly, @kbd{V R :}
[@code{reduced}] reduces down; @kbd{V R : +} would produce @cite{[a + d,
b + e, c + f]}.
X
@tindex reducer
@tindex rreducer
There is a third ``by rows'' mode for reduction that is occasionally
useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over
the rows of the matrix themselves.  Thus @kbd{V R = +} on the above
matrix would get the same result as @kbd{V R : +}, since adding two
row vectors is equivalent to adding their elements.  But @kbd{V R = *}
would multiply the two rows (to get a single number, their dot product),
while @kbd{V R : *} would produce a vector of the products of the columns.
X
These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
but they are not currently supported with @kbd{V U} or @kbd{I V U}.
X
@tindex reducec
@tindex rreducec
The obsolete reduce-by-columns function, @code{reducec}, is still
supported but there is no way to get it through the @kbd{V R} command.
X
The commands @kbd{M-# :} and @kbd{M-# _} are equivalent to typing
@kbd{M-# r} to grab a rectangle of data into Calc, and then typing
@kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or
rows of the matrix.  @xref{Grabbing From Buffers}.
X
@node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping
@subsection Nesting and Fixed Points
X
@noindent
@kindex H V R
@tindex nest
The @kbd{H V R} [@code{nest}] command applies a function to a given
argument repeatedly.  It takes two values, @samp{a} and @samp{n}, from
the stack, where @samp{n} must be an integer.  It then applies the
function nested @samp{n} times; if the function is @samp{f} and @samp{n}
is 3, the result is @samp{f(f(f(a)))}.  The number @samp{n} may be
negative if Calc knows an inverse for the function @samp{f}; for
example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
X
@kindex H V U
@tindex anest
The @kbd{H V U} [@code{anest}] command is an accumulating version of
@code{nest}:  It returns a vector of @samp{n+1} values, e.g.,
@samp{[a, f(a), f(f(a)), f(f(f(a)))]}.  If @samp{n} is negative and
@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)))]}.
X
@kindex H I V R
@tindex fixp
@cindex Fixed points
The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except
that it takes only an @samp{a} value from the stack; the function is
applied until it reaches a ``fixed point,'' i.e., until the result
no longer changes.
X
@kindex H I V U
@tindex afixp
The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
The first element of the return vector will be the initial value @samp{a};
the last element will be the final result that would have been returned
by @code{fixp}.
X
For example, 0.739085 is a fixed point of the cosine function (in radians):
@samp{cos(0.739085) = 0.739085}.  You can find this value by putting, say,
1.0 on the stack and typing @kbd{H I V U C}.  (We use the accumulating
version so we can see the intermediate results:  @samp{[1, 0.540302, 0.857553,
0.65329, ...]}.  With a precision of six, this command will take 36 steps
to converge to 0.739085.)
X
Newton's method for finding roots is a classic example of iteration
to a fixed point.  To find the square root of five starting with an
initial guess, Newton's method would look for a fixed point of the
function @samp{(x + 5/x) / 2}.  Putting a guess of 1 on the stack
and typing @kbd{H I V R ' ($ + 5/$)/2 RET} quickly yields the result
2.23607.  This is equivalent to using the @kbd{a R} (@code{calc-find-root})
command to find a root of the equation @samp{x^2 = 5}.
X
These examples used numbers for @samp{a} values.  Calc keeps applying
the function until two successive results are equal to within the
current precision.  For complex numbers, both the real parts and the
imaginary parts must be equal to within the current precision.  If
@samp{a} is a formula (say, a variable name), then the function is
applied until two successive results are exactly the same formula.
It is up to you to ensure that the function will eventually converge;
if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
X
The algebraic @code{fixp} function takes two optional arguments, @samp{n}
and @samp{tol}.  The first is the maximum number of steps to be allowed,
and must be either an integer or the symbol @samp{inf} (infinity, the
default).  The second is a convergence tolerance.  If a tolerance is
specified, all results during the calculation must be numbers, not
formulas, and the iteration stops when the magnitude of the difference
between two successive results is less than or equal to the tolerance.
(This implies that a tolerance of zero iterates until the results are
exactly equal.)
X
Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
computes the square root of @samp{A} given the initial guess @samp{B},
stopping when the result is correct within the specified tolerance, or
when 20 steps have been taken, whichever is sooner.
X
@node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping
@subsection Generalized Products
X
@kindex V O
@pindex calc-outer-product
@tindex outer
The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies
a given binary operator to all possible pairs of elements from two
vectors, to produce a matrix.  For example, @kbd{V O *} with @samp{[a, b]}
and @samp{[x, y, z]} on the stack produces a multiplication table:
@samp{[[a x, a y, a z], [b x, b y, b z]]}.  Element @var{r},@var{c} of
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.
X
@kindex V I
@pindex calc-inner-product
@tindex inner
The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes
the generalized inner product of two vectors or matrices, given a
``multiplicative'' operator and an ``additive'' operator.  These can each
actually be any binary operators; if they are @samp{*} and @samp{+},
respectively, the result is a standard matrix multiplication.  Element
@var{r},@var{c} of the result matrix is obtained by mapping the
multiplicative operator across row @var{r} of the lefthand matrix and
column @var{c} of the righthand matrix, and then reducing with the additive
operator.  Just as for the standard @kbd{*} command, this can also do a
vector-matrix or matrix-vector inner product, or a vector-vector
generalized dot product.
X
Since @kbd{V I} requires two operators, it prompts twice.  In each case,
you can use any of the usual methods for entering the operator.  If you
use @kbd{$} twice to take both operator formulas from the stack, the
first (multiplicative) operator is taken from the top of the stack
and the second (additive) operator is taken from second-to-top.
X
@node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions
@section Vector and Matrix Display Formats
X
@noindent
Commands for controlling vector and matrix display use the @kbd{v} prefix
instead of the usual @kbd{d} prefix.  But they are display modes; in
particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
in the same way (@pxref{Display Modes}).  Matrix display is also
influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
@pxref{Normal Language Modes}.
X
@kindex v <
@pindex calc-matrix-left-justify
@kindex v =
@pindex calc-matrix-center-justify
@kindex v >
@pindex calc-matrix-right-justify
The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
(@code{calc-matrix-right-justify}), and @kbd{v =}
(@code{calc-matrix-center-justify}) control whether matrix elements
are justified to the left, right, or center of their columns.@refill
X
@kindex v [
@pindex calc-vector-brackets
@kindex v @{
@pindex calc-vector-braces
@kindex v (
@pindex calc-vector-parens
The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
brackets that surround vectors and matrices displayed in the stack on
and off.  The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
(@code{calc-vector-parens}) commands use curly braces or parentheses,
respectively, instead of square brackets.  For example, @kbd{v @{} might
be used in preparation for yanking a matrix into a buffer running
Mathematica.  (In fact, the Mathematica language mode uses this mode;
@pxref{Mathematica Language Mode}.)  Note that, regardless of the
display mode, either brackets or braces may be used to enter vectors,
and parentheses may never be used for this purpose.@refill
X
@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:
X
@group
@example
[ [ 123,  0,   0  ]       [ [ 123,  0,   0  ],
X  [  0,  123,  0  ]         [  0,  123,  0  ],
X  [  0,   0,  123 ] ]       [  0,   0,  123 ] ]
X
X         RO                        ROC
X
@end example
@end group
@noindent
@group
@example
X  [ 123,  0,   0            [ 123,  0,   0 ;
X     0,  123,  0               0,  123,  0 ;
X     0,   0,  123 ]            0,   0,  123 ]
X
X          O                        OC
X
@end example
@end group
@noindent
@group
@example
X  [ 123,  0,   0  ]           123,  0,   0
X  [  0,  123,  0  ]            0,  123,  0
X  [  0,   0,  123 ]            0,   0,  123
X
X          R                       @r{blank}
@end example
@end group
X
@noindent
Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
@samp{OC} are all recognized as matrices during reading, while
the others are useful for display only.
X
@kindex v ,
@pindex calc-vector-commas
The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
off in vector and matrix display.@refill
X
In vectors of length one, and in all vectors when commas have been
turned off, Calc adds extra parentheses around formulas that might
otherwise be ambiguous.  For example, @samp{[a b]} could be a vector
of the one formula @samp{a b}, or it could be a vector of two
variables with commas turned off.  Calc will display the former
case as @samp{[(a b)]}.  You can disable these extra parentheses
(to make the output less cluttered at the expense of allowing some
ambiguity) by adding the letter @code{P} to the control string you
give to @kbd{v ]} (as described above).
X
@kindex v .
@pindex calc-full-vectors
The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
display of long vectors on and off.  In this mode, vectors of six
or more elements, or matrices of six or more rows or columns, will
be displayed in an abbreviated form that displays only the first
three elements and the last element:  @samp{[a, b, c, ..., z]}.
When very large vectors are involved this will substantially
improve Calc's display speed.
X
@kindex t .
@pindex calc-full-trail-vectors
The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
similar mode for recording vectors in the Trail.  If you turn on
this mode, vectors of six or more elements and matrices of six or
more rows or columns will be abbreviated when they are put in the
Trail.  The @kbd{t y} (@code{calc-trail-yank}) command will be
unable to recover those vectors.  If you are working with very
large vectors, this mode will improve the speed of all operations
that involve the trail.
X
@kindex v /
@pindex calc-break-vectors
The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
vector display on and off.  Normally, matrices are displayed with one
row per line but all other types of vectors are displayed in a single
line.  This mode causes all vectors, whether matrices or not, to be
displayed with a single element per line.  Sub-vectors within the
vectors will still use the normal linear form.
X
@node Algebra, Units, Matrix Functions, Top
@chapter Algebra
X
@noindent
This section covers the Calc features that help you work with
algebraic formulas.  First, the general sub-formula selection
mechanism is described; this works in conjunction with any Calc
commands.  Then, commands for specific algebraic operations are
described.  Finally, the flexible @dfn{rewrite rule} mechanism
is discussed.
X
The algebraic commands use the @kbd{a} key prefix; selection
commands use the @kbd{j} (for ``just a letter that wasn't used
for anything else'') prefix.
X
@xref{Editing Stack Entries}, to see how to manipulate formulas
using regular Emacs editing commands.@refill
X
When doing algebraic work, you may find several of the Calculator's
modes to be helpful, including algebraic-simplification mode (@kbd{m A})
or no-simplification mode (@kbd{m O}),
algebraic-entry mode (@kbd{m a}), fraction mode (@kbd{m f}), and
symbolic mode (@kbd{m s}).  @xref{Mode Settings}, for discussions
of these modes.  You may also wish to select ``big'' display mode (@kbd{d B}).
@xref{Normal Language Modes}.@refill
X
@menu
* Selecting Subformulas::
* Algebraic Manipulation::
* Simplifying Formulas::
* Polynomials::
* Calculus::
* Solving Equations::
* Numerical Solutions::
* Curve Fitting::
* Summations::
* Logical Operations::
* Rewrite Rules::
@end menu
X
@node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
@section Selecting Sub-Formulas
X
@noindent
@cindex Selections
@cindex Sub-formulas
@cindex Parts of formulas
When working with an algebraic formula it is often necessary to
manipulate a portion of the formula rather than the formula as a
whole.  Calc allows you to ``select'' a portion of any formula on
the stack.  Commands which would normally operate on that stack
entry will now operate only on the sub-formula, leaving the
surrounding part of the stack entry alone.
X
One common non-algebraic use for selection involves vectors.  To work
on one element of a vector in-place, simply select that element as a
``sub-formula'' of the vector.
X
@menu
* Making Selections::
* Changing Selections::
* Displaying Selections::
* Operating on Selections::
* Rearranging with Selections::
@end menu
X
@node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas
@subsection Making Selections
X
@noindent
@kindex j s
@pindex calc-select-here
To select a sub-formula, move the Emacs cursor to any character in that
sub-formula, and press @kbd{j s} (@code{calc-select-here}).  Calc will
highlight the smallest portion of the formula that contains that
character.  By default the sub-formula is highlighted by blanking out
all of the rest of the formula with dots.  Selection works in any
display mode but is perhaps easiest in ``big'' (@kbd{d B}) mode.
Suppose you enter the following formula:
X
@group
@smallexample
X           3    ___
X    (a + b)  + V c
1:  ---------------
X        2 x + 1
@end smallexample
@end group
X
@noindent
(by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}).  If you move the
cursor to the letter @samp{b} and press @kbd{j s}, the display changes
to
X
@group
@smallexample
X           .    ...
X    .. . b.  . . .
1*  ...............
X        . . . .
@end smallexample
@end group
X
@noindent
Every character not part of the sub-formula @samp{b} has been changed
to a dot.  The @samp{*} next to the line number is to remind you that
the formula has a portion of it selected.  (In this case, it's very
obvious, but it might not always be.  If Embedded Mode is enabled,
the word @samp{Sel} also appears in the mode line because the stack
may not be visible.  @pxref{Embedded Mode}.)
X
If you had instead placed the cursor on the parenthesis immediately to
the right of the @samp{b}, the selection would have been:
X
@group
@smallexample
X           .    ...
X    (a + b)  . . .
1*  ...............
X        . . . .
@end smallexample
@end group
X
@noindent
The portion selected is always large enough to be considered a complete
formula all by itself, so selecting the parenthesis selects the whole
formula that it encloses.  Putting the cursor on the the @samp{+} sign
would have had the same effect.
X
(Strictly speaking, the Emacs cursor is really the manifestation of
the Emacs ``point,'' which is a position @emph{between} two characters
in the buffer.  So purists would say that Calc selects the smallest
sub-formula which contains the character to the right of ``point.'')
X
If you supply a numeric prefix argument @var{n}, the selection is
expanded to the @var{n}th enclosing sub-formula.  Thus, positioning
the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
@samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
and so on.
X
If the cursor is not on any part of the formula, or if you give a
numeric prefix that is too large, the entire formula is selected.
X
If the cursor is on the @samp{.} line that marks the top of the stack
(i.e., its normal ``rest position''), this command selects the entire
formula at stack level 1.  Most selection commands similarly operate
on the formula at the top of the stack if you haven't positioned the
cursor on any stack entry.
X
@kindex j a
@pindex calc-select-additional
The @kbd{j a} (@code{calc-select-additional}) command enlarges the
current selection to encompass the cursor.  To select the smallest
sub-formula defined by two different points, move to the first and
press @kbd{j s}, then move to the other and press @kbd{j a}.  This
is roughly analogous to using @code{set-mark-command} to select the
two ends of a region of text during normal Emacs editing.
X
@kindex j o
@pindex calc-select-once
The @kbd{j o} (@code{calc-select-once}) command selects a formula in
exactly the same way as @kbd{j s}, except that the selection will
last only as long as the next command that uses it.  For example,
@kbd{j o 1 +} is a handy way to add one to the sub-formula indicated
by the cursor.
X
(A somewhat more precise definition: The @kbd{j o} command sets a flag
such that the next command involving selected stack entries will clear
the selections on those stack entries afterwards.  All other selection
commands except @kbd{j a} and @kbd{j O} clear this flag.)
X
@kindex j S
@kindex j O
@pindex calc-select-here-maybe
@pindex calc-select-once-maybe
The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
(@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
and @kbd{j o}, respectively, except that if the formula already
has a selection they have no effect.  This is analogous to the
behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
@pxref{Selections with Rewrite Rules}) and is mainly intended to be
used in keyboard macros that implement your own selection-oriented
commands.@refill
X
Selection of sub-formulas normally treats associative terms like
@samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula.
If you place the cursor anywhere inside @samp{a + b - c + d} except
on one of the variable names and use @kbd{j s}, you will select the
entire four-term sum.
X
@kindex j b
@pindex calc-break-selections
The @kbd{j b} (@code{calc-break-selections}) command controls a mode
in which the ``deep structure'' of these associative formulas shows
through.  Calc actually stores the above formulas as @samp{((a + b) - c) + d}
and @samp{x * (y * z)} (note that for certain obscure reasons, Calc
treats multiplication as right-associative).  Once you have enabled
@kbd{j b} mode, selecting with the cursor on the @samp{-} sign would
only select the @samp{a + b - c} portion, which makes sense when the
deep structure of the sum is considered.  There is no way to select
the @samp{b - c + d} portion; although this might initially look
like just as legitimate a sub-formula as @samp{a + b - c}, the deep
structure shows that it isn't.  The @kbd{d U} command can be used
to view the deep structure of any formula (@pxref{Normal Language Modes}).
X
When @kbd{j b} mode has not been enabled, the deep structure is
generally hidden by the selection commands---what you see is what
you get.
X
@kindex j u
@pindex calc-unselect
The @kbd{j u} (@code{calc-unselect}) command unselects the formula
that the cursor is on.  If there was no selection in the formula,
this command has no effect.  With a numeric prefix argument, it
unselects that stack element rather than using the cursor position.
X
@kindex j c
@pindex calc-clear-selections
The @kbd{j c} (@code{calc-clear-selections}) command unselects all
stack elements.
X
@node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas
@subsection Changing Selections
X
@noindent
@kindex j m
@pindex calc-select-more
Once you have selected a sub-formula, you can expand it using the
@w{@kbd{j m}} (@code{calc-select-more}) command.  If @samp{a + b} is
selected, pressing @w{@kbd{j m}} repeatedly works as follows:
X
@group
@smallexample
X           3    ...                3    ___                3    ___
X    (a + b)  . . .          (a + b)  + V c          (a + b)  + V c
1*  ...............     1*  ...............     1*  ---------------
X        . . . .                 . . . .                 2 x + 1
@end smallexample
@end group
X
@noindent
In the last example, the entire formula is selected.  This is roughly
the same as having no selection at all, but because there are subtle
differences the @samp{*} character is still there on the line number.
X
With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
times (or until the entire formula is selected).  Note that @kbd{j s}
with argument @var{n} is equivalent to plain @kbd{j s} followed by
@kbd{j m} with argument @var{n}.  If @w{@kbd{j m}} is used when there
is no current selection, it is equivalent to @w{@kbd{j s}}.
X
Even though @kbd{j m} does not explicitly use the location of the
cursor within the formula, it nevertheless uses the cursor to determine
which stack element to operate on.  As usual, @kbd{j m} when the cursor
is not on any stack element operates on the top stack element.
X
@kindex j l
@pindex calc-select-less
The @kbd{j l} (@code{calc-select-less}) command reduces the current
selection around the cursor position.  That is, it selects the
immediate sub-formula of the current selection which contains the
cursor, the opposite of @kbd{j m}.  If the cursor is not inside the
current selection, the command de-selects the formula.
X
@kindex j 1-9
@pindex calc-select-part
The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
select the @var{n}th sub-formula of the current selection.  They are
like @kbd{j l} (@code{calc-select-less}) except they use counting
rather than the cursor position to decide which sub-formula to select.
For example, if the current selection is @kbd{a + b + c} or
@kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
@kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
these cases, @kbd{j 4} through @kbd{j 9} would be errors.
X
If there is no current selection, @kbd{j 1} through @kbd{j 9} select
the @var{n}th top-level sub-formula.  (In other words, they act as if
the entire stack entry were selected first.)  To select the @var{n}th
sub-formula where @var{n} is greater than nine, you must instead invoke
@w{@kbd{j 1}} with @var{n} as a numeric prefix argument.@refill
X
@kindex j n
@kindex j p
@pindex calc-select-next
@pindex calc-select-previous
The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
(@code{calc-select-previous}) commands change the current selection
to the next or previous sub-formula at the same level.  For example,
if @samp{b} is selected in @samp{2 + a*b*c + x}, then @kbd{j n}
selects @samp{c}.  Further @kbd{j n} commands would be in error because,
even though there is something to the right of @samp{c} (namely, @samp{x}),
it is not at the same level; in this case, it is not a term of the
same product as @samp{b} and @samp{c}.  However, @kbd{j m} (to select
the whole product @samp{a*b*c} as a term of the sum) followed by
@w{@kbd{j n}} would successfully select the @samp{x}.
X
Similarly, @kbd{j p} moves the selection from the @samp{b} in this
sample formula to the @samp{a}.  Both commands accept numeric prefix
arguments to move several steps at a time.
X
It is interesting to compare Calc's selection commands with the
Emacs Info system's commands for navigating through hierarchically
organized documentation.  Calc's @kbd{j n} command is completely
analogous to Info's @kbd{n} command.  Likewise, @kbd{j p} maps to
@kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
(Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
@kbd{j l}; in each case, you can jump directly to a sub-component
of the hierarchy simply by pointing to it with the cursor.
X
@node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas
@subsection Displaying Selections
X
@noindent
@kindex j d
@pindex calc-show-selections
The @kbd{j d} (@code{calc-show-selections}) command controls how
selected sub-formulas are displayed.  One of the alternatives is
illustrated in the above examples; if we press @kbd{j d} we switch
to the other style in which the selected portion itself is obscured
by @samp{#} signs:
X
@group
@smallexample
X           3    ...                  #    ___
X    (a + b)  . . .            ## # ##  + V c
1*  ...............       1*  ---------------
X        . . . .                   2 x + 1
@end smallexample
@end group
X
@node Operating on Selections, Rearranging with Selections, Displaying Selections, Selecting Subformulas
@subsection Operating on Selections
X
@noindent
Once a selection is made, all Calc commands that manipulate items
on the stack will operate on the selected portions of the items
instead.  (Note that several stack elements may have selections
at once, though there can be only one selection at a time in any
given stack element.)
X
@kindex j e
@pindex calc-enable-selections
The @kbd{j e} (@code{calc-enable-selections}) command disables the
effect that selections have on Calc commands.  The current selections
still exist, but Calc commands operate on whole stack elements anyway.
This mode can be identified by the fact that the @samp{*} markers on
the line numbers are gone, even though selections are visible.  To
reactivate the selections, press @kbd{j e} again.
X
To extract a sub-formula as a new formula, simply press @key{RET}.
This normally duplicates the top stack element; here it duplicates
only the selected portion of that element.
X
To replace a sub-formula with something different, you can enter the
new value onto the stack and press @key{TAB}.  This normally exchanges
the top two stack elements; here it swaps the value you entered into
the selected portion of the formula, returning the old selected
portion to the top of the stack.
X
@group
@smallexample
X           3    ...                    ...                    ___
X    (a + b)  . . .           17 x y . . .           17 x y + V c
2*  ...............      2*  .............      2:  -------------
X        . . . .                 . . . .                2 x + 1
X
X                                    3                      3
1:  17 x y               1:  (a + b)            1:  (a + b)
@end smallexample
@end group
X
In this example we select a sub-formula of our original example,
enter a new formula, @key{TAB} it into place, then deselect to see
the complete, edited formula.
X
If you want to swap whole formulas around even though they contain
selections, just use @kbd{j e} before and after.
X
@kindex j '
@pindex calc-enter-selection
The @kbd{j '} (@code{calc-enter-selection}) command is another way
to replace a selected sub-formula.  This command does an algebraic
entry just like the regular @kbd{'} key.  When you press @key{RET},
the formula you type replaces the original selection.  You can use
the @samp{$} symbol in the formula to refer to the original
selection.  If there is no selection in the formula under the cursor,
the cursor is used to make a temporary selection for the purposes of
the command.  Thus, to change a term of a formula, all you have to
do is move the Emacs cursor to that term and press @kbd{j '}.
X
@kindex j `
@pindex calc-edit-selection
The @kbd{j `} (@code{calc-edit-selection}) command is a similar
analogue of the @kbd{`} (@code{calc-edit}) command.  It edits the
selected sub-formula in a separate buffer.  If there is no
selection, it edits the sub-formula indicated by the cursor.
X
To delete a sub-formula, press @key{DEL}.  This generally replaces
the sub-formula with the constant zero, but in a few suitable contexts
it uses the constant one instead.  The @key{DEL} key automatically
deselects and re-simplifies the entire formula afterwards.  Thus:
X
@group
@smallexample
X              ###
X    17 x y + # #          17 x y         17 # y          17 y
1*  -------------     1:  -------    1*  -------    1:  -------
X       2 x + 1            2 x + 1        2 x + 1        2 x + 1
@end smallexample
@end group
X
In this example, we first delete the @samp{sqrt(c)} term; Calc
accomplishes this by replacing @samp{sqrt(c)} with zero and
resimplifying.  We then delete the @kbd{x} in the numerator;
since this is part of a product, Calc replaces it with @samp{1}
and resimplifies.
X
If you select an element of a vector and press @key{DEL}, that
element is deleted from the vector.  If you delete one side of
an equation or inequality, only the opposite side remains.
X
@kindex j DEL
@pindex calc-del-selection
The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
@key{DEL} but with the auto-selecting behavior of @kbd{j '} and
@kbd{j `}.  It deletes the selected portion of the formula
indicated by the cursor, or, in the absence of a selection, it
deletes the sub-formula indicated by the cursor position.
X
@kindex j RET
@pindex calc-grab-selection
(There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
command.)
X
Normal arithmetic operations also apply to sub-formulas.  Here we
select the denominator, press @kbd{5 -} to subtract five from the
denominator, press @kbd{n} to negate the denominator, then
press @kbd{Q} to take the square root.
X
@group
@smallexample
X     .. .           .. .           .. .             .. .
1*  .......    1*  .......    1*  .......    1*  ..........
X    2 x + 1        2 x - 4        4 - 2 x         _________
X                                                 V 4 - 2 x
@end smallexample
@end group
X
Certain types of operations on selections are not allowed.  For
example, for an arithmetic function like @kbd{-} no more than one of
the arguments may be a selected sub-formula.  (As the above example
shows, the result of the subtraction is spliced back into the argument
which had the selection; if there were more than one selection involved,
this would not be well-defined.)  If you try to subtract two selections,
the command will abort with an error message.
X
Operations on sub-formulas sometimes leave the formula as a whole
in an ``un-natural'' state.  Consider negating the @samp{2 x} term
of our sample formula by selecting it and pressing @kbd{n}
(@code{calc-change-sign}).@refill
X
@group
@smallexample
X       .. .                .. .
1*  ..........      1*  ...........
X     .........           ..........
X    . . . 2 x           . . . -2 x
@end smallexample
@end group
X
Unselecting the sub-formula reveals that the minus sign, which would
normally have cancelled out with the subtraction automatically, has
not been able to do so because the subtraction was not part of the
selected portion.  Pressing @kbd{=} (@code{calc-evaluate}) or doing
any other mathematical operation on the whole formula will cause it
to be simplified.
X
@group
@smallexample
X       17 y                17 y
1:  -----------     1:  ----------
X     __________          _________
X    V 4 - -2 x          V 4 + 2 x
@end smallexample
SHAR_EOF
true || echo 'restore of calc.texinfo failed'
fi
echo 'End of  part 44'
echo 'File calc.texinfo is continued in part 45'
echo 45 > _shar_seq_.tmp
exit 0
exit 0 # Just in case...
--
Kent Landfield                   INTERNET: [e-mail]
Sterling Software, IMD           UUCP:     uunet!sparky!kent
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