@c DO NOT EDIT!  Generated automatically by munge-texi.

@c Copyright (C) 1996, 1997 John W. Eaton
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.

@node Arithmetic
@chapter Arithmetic

Unless otherwise noted, all of the functions described in this chapter
will work for real and complex scalar or matrix arguments.

@menu
* Utility Functions::           
* Complex Arithmetic::          
* Trigonometry::                
* Sums and Products::           
* Special Functions::           
* Coordinate Transformations::
* Mathematical Constants::      
@end menu

@node Utility Functions
@section Utility Functions

The following functions are available for working with complex numbers.
Each expects a single argument.  They are called @dfn{mapping functions}
because when given a matrix argument, they apply the given function to
each element of the matrix.

@anchor{doc-ceil}
@deftypefn {Mapping Function} {} ceil (@var{x})
Return the smallest integer not less than @var{x}.  If @var{x} is
complex, return @code{ceil (real (@var{x})) + ceil (imag (@var{x})) * I}.
@end deftypefn


@anchor{doc-exp}
@deftypefn {Mapping Function} {} exp (@var{x})
Compute the exponential of @var{x}.  To compute the matrix exponential,
see @ref{Linear Algebra}.
@end deftypefn


@anchor{doc-fix}
@deftypefn {Mapping Function} {} fix (@var{x})
Truncate @var{x} toward zero.  If @var{x} is complex, return
@code{fix (real (@var{x})) + fix (imag (@var{x})) * I}.
@end deftypefn


@anchor{doc-floor}
@deftypefn {Mapping Function} {} floor (@var{x})
Return the largest integer not greater than @var{x}.  If @var{x} is
complex, return @code{floor (real (@var{x})) + floor (imag (@var{x})) * I}.
@end deftypefn


@anchor{doc-gcd}
@deftypefn {Loadable Function} {@var{g} =} gcd (@var{a1}, @code{...})
@deftypefnx {Loadable Function} {[@var{g}, @var{v1}, @var{...}] =} gcd (@var{a1}, @code{...})

If a single argument is given then compute the greatest common divisor of
the elements of this argument. Otherwise if more than one argument is
given all arguments must be the same size or scalar. In this case the
greatest common divisor is calculated for element individually. All
elements must be integers. For example,

@example
@group
gcd ([15, 20])
    @result{}  5
@end group
@end example

@noindent
and

@example
@group
gcd ([15, 9], [20 18])
    @result{}  5  9
@end group
@end example

Optional return arguments @var{v1}, etc, contain integer vectors such
that,

@ifinfo
@example
@var{g} = @var{v1} .* @var{a1} + @var{v2} .* @var{a2} + @var{...}
@end example
@end ifinfo
@iftex
@tex
$g = v_1 a_1 + v_2 a_2 + \cdots$
@end tex
@end iftex

For backward compatiability with previous versions of this function, when
all arguments are scalr, a single return argument @var{v1} containing
all of the values of @var{v1}, @var{...} is acceptable.
@end deftypefn
@seealso{lcm, min, max, ceil, and floor}


@anchor{doc-lcm}
@deftypefn {Mapping Function} {} lcm (@var{x}, @code{...})
Compute the least common multiple of the elements elements of @var{x}, or
the list of all the arguments.  For example,

@example
lcm (a1, ..., ak)
@end example

@noindent
is the same as

@example
lcm ([a1, ..., ak]).
@end example
@end deftypefn

All elements must be the same size or scalar.
@seealso{gcd, min, max, ceil, and floor}


@anchor{doc-log}
@deftypefn {Mapping Function} {} log (@var{x})
Compute the natural logarithm for each element of @var{x}.  To compute the
matrix logarithm, see @ref{Linear Algebra}.
@end deftypefn
@seealso{log2, log10, logspace, and exp}


@anchor{doc-log10}
@deftypefn {Mapping Function} {} log10 (@var{x})
Compute the base-10 logarithm for each element of @var{x}.
@end deftypefn
@seealso{log, log2, logspace, and exp}


@anchor{doc-log2}
@deftypefn {Mapping Function} {} log2 (@var{x})
@deftypefnx {Mapping Function} {[@var{f}, @var{e}] =} log2 (@var{x})
Compute the base-2 logarithm of @var{x}.  With two outputs, returns
@var{f} and @var{e} such that
@iftex
@tex
 $1/2 <= |f| < 1$ and $x = f \cdot 2^e$.
@end tex
@end iftex
@ifinfo
 1/2 <= abs(f) < 1 and x = f * 2^e.
@end ifinfo
@end deftypefn
@seealso{log, log10, logspace, and exp}


@anchor{doc-max}
@deftypefn {Mapping Function} {} max (@var{x}, @var{y}, @var{dim})
@deftypefnx {Mapping Function} {[@var{w}, @var{iw}] =} max (@var{x})
@cindex Utility Functions
For a vector argument, return the maximum value.  For a matrix
argument, return the maximum value from each column, as a row
vector, or over the dimension @var{dim} if defined. For two matrices
(or a matrix and scalar), return the pair-wise maximum.
Thus,

@example
max (max (@var{x}))
@end example

@noindent
returns the largest element of @var{x}, and

@example
@group
max (2:5, pi)
    @result{}  3.1416  3.1416  4.0000  5.0000
@end group
@end example
@noindent
compares each element of the range @code{2:5} with @code{pi}, and
returns a row vector of the maximum values.

For complex arguments, the magnitude of the elements are used for
comparison.

If called with one input and two output arguments,
@code{max} also returns the first index of the
maximum value(s). Thus,

@example
@group
[x, ix] = max ([1, 3, 5, 2, 5])
    @result{}  x = 5
        ix = 3
@end group
@end example
@end deftypefn


@anchor{doc-min}
@deftypefn {Mapping Function} {} min (@var{x}, @var{y}, @var{dim})
@deftypefnx {Mapping Function} {[@var{w}, @var{iw}] =} min (@var{x})
@cindex Utility Functions
For a vector argument, return the minimum value.  For a matrix
argument, return the minimum value from each column, as a row
vector, or over the dimension @var{dim} if defined. For two matrices
(or a matrix and scalar), return the pair-wise minimum.
Thus,

@example
min (min (@var{x}))
@end example

@noindent
returns the smallest element of @var{x}, and

@example
@group
min (2:5, pi)
    @result{}  2.0000  3.0000  3.1416  3.1416
@end group
@end example
@noindent
compares each element of the range @code{2:5} with @code{pi}, and
returns a row vector of the minimum values.

For complex arguments, the magnitude of the elements are used for
comparison.

If called with one input and two output arguments,
@code{min} also returns the first index of the
minimum value(s). Thus,

@example
@group
[x, ix] = min ([1, 3, 0, 2, 5])
    @result{}  x = 0
        ix = 3
@end group
@end example
@end deftypefn


@anchor{doc-mod}
@deftypefn {Mapping Function} {} mod (@var{x}, @var{y})
Compute modulo function, using

@example
x - y .* floor (x ./ y)
@end example

Note that this handles negative numbers correctly:
@code{mod (-1, 3)} is 2, not -1 as @code{rem (-1, 3)} returns.
Also, @code{mod (@var{x}, 0)} returns @var{x}.

An error message is printed if the dimensions of the arguments do not
agree, or if either of the arguments is complex.
@end deftypefn
@seealso{rem, round}


@anchor{doc-nextpow2}
@deftypefn {Function File} {} nextpow2 (@var{x})
If @var{x} is a scalar, returns the first integer @var{n} such that
@iftex
@tex
 $2^n \ge |x|$.
@end tex
@end iftex
@ifinfo
 2^n >= abs (x).
@end ifinfo

If @var{x} is a vector, return @code{nextpow2 (length (@var{x}))}.
@end deftypefn
@seealso{pow2}


@anchor{doc-pow2}
@deftypefn {Mapping Function} {} pow2 (@var{x})
@deftypefnx {Mapping Function} {} pow2 (@var{f}, @var{e})
With one argument, computes
@iftex
@tex
 $2^x$
@end tex
@end iftex
@ifinfo
 2 .^ x
@end ifinfo
for each element of @var{x}.  With two arguments, returns
@iftex
@tex
 $f \cdot 2^e$.
@end tex
@end iftex
@ifinfo
 f .* (2 .^ e).
@end ifinfo
@end deftypefn
@seealso{nextpow2}


@anchor{doc-rem}
@deftypefn {Mapping Function} {} rem (@var{x}, @var{y})
Return the remainder of @code{@var{x} / @var{y}}, computed using the
expression

@example
x - y .* fix (x ./ y)
@end example

An error message is printed if the dimensions of the arguments do not
agree, or if either of the arguments is complex.
@end deftypefn
@seealso{mod, round}


@anchor{doc-round}
@deftypefn {Mapping Function} {} round (@var{x})
Return the integer nearest to @var{x}.  If @var{x} is complex, return
@code{round (real (@var{x})) + round (imag (@var{x})) * I}.
@end deftypefn
@seealso{rem}


@anchor{doc-sign}
@deftypefn {Mapping Function} {} sign (@var{x})
Compute the @dfn{signum} function, which is defined as
@iftex
@tex
$$
{\rm sign} (@var{x}) = \cases{1,&$x>0$;\cr 0,&$x=0$;\cr -1,&$x<0$.\cr}
$$
@end tex
@end iftex
@ifinfo

@example
           -1, x < 0;
sign (x) =  0, x = 0;
            1, x > 0.
@end example
@end ifinfo

For complex arguments, @code{sign} returns @code{x ./ abs (@var{x})}.
@end deftypefn


@anchor{doc-sqrt}
@deftypefn {Mapping Function} {} sqrt (@var{x})
Compute the square root of @var{x}.  If @var{x} is negative, a complex
result is returned.  To compute the matrix square root, see
@ref{Linear Algebra}.
@end deftypefn


@node Complex Arithmetic
@section Complex Arithmetic

The following functions are available for working with complex
numbers.  Each expects a single argument.  Given a matrix they work on
an element by element basis.  In the descriptions of the following
functions,
@iftex
@tex
$z$ is the complex number $x + iy$, where $i$ is defined as
$\sqrt{-1}$.
@end tex
@end iftex
@ifinfo
@var{z} is the complex number @var{x} + @var{i}@var{y}, where @var{i} is
defined as @code{sqrt (-1)}.
@end ifinfo

@anchor{doc-abs}
@deftypefn {Mapping Function} {} abs (@var{z})
Compute the magnitude of @var{z}, defined as
@iftex
@tex
$|z| = \sqrt{x^2 + y^2}$.
@end tex
@end iftex
@ifinfo
|@var{z}| = @code{sqrt (x^2 + y^2)}.
@end ifinfo

For example,

@example
@group
abs (3 + 4i)
     @result{} 5
@end group
@end example
@end deftypefn


@anchor{doc-arg}
@deftypefn {Mapping Function} {} arg (@var{z})
@deftypefnx {Mapping Function} {} angle (@var{z})
Compute the argument of @var{z}, defined as
@iftex
@tex
$\theta = \tan^{-1}(y/x)$.
@end tex
@end iftex
@ifinfo
@var{theta} = @code{atan (@var{y}/@var{x})}.
@end ifinfo
@noindent
in radians. 

For example,

@example
@group
arg (3 + 4i)
     @result{} 0.92730
@end group
@end example
@end deftypefn


@anchor{doc-conj}
@deftypefn {Mapping Function} {} conj (@var{z})
Return the complex conjugate of @var{z}, defined as
@iftex
@tex
$\bar{z} = x - iy$.
@end tex
@end iftex
@ifinfo
@code{conj (@var{z})} = @var{x} - @var{i}@var{y}.
@end ifinfo
@end deftypefn
@seealso{real and imag}


@anchor{doc-imag}
@deftypefn {Mapping Function} {} imag (@var{z})
Return the imaginary part of @var{z} as a real number.
@end deftypefn

@seealso{real and conj}


@anchor{doc-real}
@deftypefn {Mapping Function} {} real (@var{z})
Return the real part of @var{z}.
@end deftypefn
@seealso{imag and conj}


@node Trigonometry
@section Trigonometry

Octave provides the following trigonometric functions.  Angles are
specified in radians.  To convert from degrees to radians multipy by
@iftex
@tex
$\pi/180$
@end tex
@end iftex
@ifinfo
@code{pi/180}
@end ifinfo
 (e.g. @code{sin (30 * pi/180)} returns the sine of 30 degrees).

@anchor{doc-sin}
@deftypefn {Mapping Function} {} sin (@var{x})
Compute the sine of each element of @var{x}.
@end deftypefn

@anchor{doc-cos}
@deftypefn {Mapping Function} {} cos (@var{x})
Compute the cosine of each element of @var{x}.
@end deftypefn

@anchor{doc-tan}
@deftypefn {Mapping Function} {} tan (@var{z})
Compute tangent of each element of @var{x}.
@end deftypefn

@anchor{doc-sec}
@deftypefn {Mapping Function} {} sec (@var{x})
Compute the secant of each element of @var{x}.
@end deftypefn

@anchor{doc-csc}
@deftypefn {Mapping Function} {} csc (@var{x})
Compute the cosecant of each element of @var{x}.
@end deftypefn

@anchor{doc-cot}
@deftypefn {Mapping Function} {} cot (@var{x})
Compute the cotangent of each element of @var{x}.
@end deftypefn


@anchor{doc-asin}
@deftypefn {Mapping Function} {} asin (@var{x})
Compute the inverse sine of each element of @var{x}.
@end deftypefn

@anchor{doc-acos}
@deftypefn {Mapping Function} {} acos (@var{x})
Compute the inverse cosine of each element of @var{x}.
@end deftypefn

@anchor{doc-atan}
@deftypefn {Mapping Function} {} atan (@var{x})
Compute the inverse tangent of each element of @var{x}.
@end deftypefn

@anchor{doc-asec}
@deftypefn {Mapping Function} {} asec (@var{x})
Compute the inverse secant of each element of @var{x}.
@end deftypefn

@anchor{doc-acsc}
@deftypefn {Mapping Function} {} acsc (@var{x})
Compute the inverse cosecant of each element of @var{x}.
@end deftypefn

@anchor{doc-acot}
@deftypefn {Mapping Function} {} acot (@var{x})
Compute the inverse cotangent of each element of @var{x}.
@end deftypefn


@anchor{doc-sinh}
@deftypefn {Mapping Function} {} sinh (@var{x})
Compute the inverse hyperbolic sine of each element of @var{x}.
@end deftypefn

@anchor{doc-cosh}
@deftypefn {Mapping Function} {} cosh (@var{x})
Compute the hyperbolic cosine of each element of @var{x}.
@end deftypefn

@anchor{doc-tanh}
@deftypefn {Mapping Function} {} tanh (@var{x})
Compute hyperbolic tangent of each element of @var{x}.
@end deftypefn

@anchor{doc-sech}
@deftypefn {Mapping Function} {} sech (@var{x})
Compute the hyperbolic secant of each element of @var{x}.
@end deftypefn

@anchor{doc-csch}
@deftypefn {Mapping Function} {} csch (@var{x})
Compute the hyperbolic cosecant of each element of @var{x}.
@end deftypefn

@anchor{doc-coth}
@deftypefn {Mapping Function} {} coth (@var{x})
Compute the hyperbolic cotangent of each element of @var{x}.
@end deftypefn


@anchor{doc-asinh}
@deftypefn {Mapping Function} {} asinh (@var{x})
Compute the inverse hyperbolic sine of each element of @var{x}.
@end deftypefn

@anchor{doc-acosh}
@deftypefn {Mapping Function} {} acosh (@var{x})
Compute the inverse hyperbolic cosine of each element of @var{x}.
@end deftypefn

@anchor{doc-atanh}
@deftypefn {Mapping Function} {} atanh (@var{x})
Compute the inverse hyperbolic tangent of each element of @var{x}.
@end deftypefn

@anchor{doc-asech}
@deftypefn {Mapping Function} {} asech (@var{x})
Compute the inverse hyperbolic secant of each element of @var{x}.
@end deftypefn

@anchor{doc-acsch}
@deftypefn {Mapping Function} {} acsch (@var{x})
Compute the inverse hyperbolic cosecant of each element of @var{x}.
@end deftypefn

@anchor{doc-acoth}
@deftypefn {Mapping Function} {} acoth (@var{x})
Compute the inverse hyperbolic cotangent of each element of @var{x}.
@end deftypefn


Each of these functions expect a single argument.  For matrix arguments,
they work on an element by element basis.  For example,

@example
@group
sin ([1, 2; 3, 4])
     @result{}  0.84147   0.90930
         0.14112  -0.75680
@end group
@end example

@anchor{doc-atan2}
@deftypefn {Mapping Function} {} atan2 (@var{y}, @var{x})
Compute atan (@var{y} / @var{x}) for corresponding elements of @var{y}
and @var{x}.  The result is in range -pi to pi.
@end deftypefn



@node Sums and Products
@section Sums and Products

@anchor{doc-sum}
@deftypefn {Built-in Function} {} sum (@var{x}, @var{dim})
Sum of elements along dimension @var{dim}.  If @var{dim} is
omitted, it defaults to 1 (column-wise sum).
@end deftypefn


@anchor{doc-prod}
@deftypefn {Built-in Function} {} prod (@var{x}, @var{dim})
Product of elements along dimension @var{dim}.  If @var{dim} is
omitted, it defaults to 1 (column-wise products).
@end deftypefn


@anchor{doc-cumsum}
@deftypefn {Built-in Function} {} cumsum (@var{x}, @var{dim})
Cumulative sum of elements along dimension @var{dim}.  If @var{dim}
is omitted, it defaults to 1 (column-wise cumulative sums).
@end deftypefn


@anchor{doc-cumprod}
@deftypefn {Built-in Function} {} cumprod (@var{x}, @var{dim})
Cumulative product of elements along dimension @var{dim}.  If
@var{dim} is omitted, it defaults to 1 (column-wise cumulative
products).
@end deftypefn


@anchor{doc-sumsq}
@deftypefn {Built-in Function} {} sumsq (@var{x}, @var{dim})
Sum of squares of elements along dimension @var{dim}.  If @var{dim}
is omitted, it defaults to 1 (column-wise sum of squares).

This function is equivalent to computing
@example
sum (x .* conj (x), dim)
@end example
but it uses less memory and avoids calling conj if @var{x} is real.
@end deftypefn


@node Special Functions
@section Special Functions

@anchor{doc-besselj}
@deftypefn {Loadable Function} {[@var{j}, @var{ierr}] =} besselj (@var{alpha}, @var{x}, @var{opt})
@deftypefnx {Loadable Function} {[@var{y}, @var{ierr}] =} bessely (@var{alpha}, @var{x}, @var{opt})
@deftypefnx {Loadable Function} {[@var{i}, @var{ierr}] =} besseli (@var{alpha}, @var{x}, @var{opt})
@deftypefnx {Loadable Function} {[@var{k}, @var{ierr}] =} besselk (@var{alpha}, @var{x}, @var{opt})
@deftypefnx {Loadable Function} {[@var{h}, @var{ierr}] =} besselh (@var{alpha}, @var{k}, @var{x}, @var{opt})
Compute Bessel or Hankel functions of various kinds:

@table @code
@item besselj
Bessel functions of the first kind.
@item bessely
Bessel functions of the second kind.
@item besseli
Modified Bessel functions of the first kind.
@item besselk
Modified Bessel functions of the second kind.
@item besselh
Compute Hankel functions of the first (@var{k} = 1) or second (@var{k}
 = 2) kind.
@end table

If the argument @var{opt} is supplied, the result is scaled by the
@code{exp (-I*@var{x})} for @var{k} = 1 or @code{exp (I*@var{x})} for
 @var{k} = 2.

If @var{alpha} is a scalar, the result is the same size as @var{x}.
If @var{x} is a scalar, the result is the same size as @var{alpha}.
If @var{alpha} is a row vector and @var{x} is a column vector, the
result is a matrix with @code{length (@var{x})} rows and
@code{length (@var{alpha})} columns.  Otherwise, @var{alpha} and
@var{x} must conform and the result will be the same size.

The value of @var{alpha} must be real.  The value of @var{x} may be
complex.

If requested, @var{ierr} contains the following status information
and is the same size as the result.

@enumerate 0
@item
Normal return.
@item
Input error, return @code{NaN}.
@item
Overflow, return @code{Inf}.
@item
Loss of significance by argument reduction results in less than
half of machine accuracy.
@item
Complete loss of significance by argument reduction, return @code{NaN}.
@item
Error---no computation, algorithm termination condition not met,
return @code{NaN}.
@end enumerate
@end deftypefn


@anchor{doc-airy}
@deftypefn {Loadable Function} {[@var{a}, @var{ierr}] =} airy (@var{k}, @var{z}, @var{opt})
Compute Airy functions of the first and second kind, and their
derivatives.

@example
  K   Function   Scale factor (if a third argument is supplied)
 ---  --------   ----------------------------------------------
  0   Ai (Z)     exp ((2/3) * Z * sqrt (Z))
  1   dAi(Z)/dZ  exp ((2/3) * Z * sqrt (Z))
  2   Bi (Z)     exp (-abs (real ((2/3) * Z *sqrt (Z))))
  3   dBi(Z)/dZ  exp (-abs (real ((2/3) * Z *sqrt (Z))))
@end example

The function call @code{airy (@var{z})} is equivalent to
@code{airy (0, @var{z})}.

The result is the same size as @var{z}.

If requested, @var{ierr} contains the following status information and
is the same size as the result.

@enumerate 0
@item
Normal return.
@item
Input error, return @code{NaN}.
@item
Overflow, return @code{Inf}.
@item
Loss of significance by argument reduction results in less than half
 of machine accuracy.
@item
Complete loss of significance by argument reduction, return @code{NaN}.
@item
Error---no computation, algorithm termination condition not met,
return @code{NaN}
@end enumerate
@end deftypefn


@anchor{doc-beta}
@deftypefn {Mapping Function} {} beta (@var{a}, @var{b})
Return the Beta function,
@iftex
@tex
$$
 B (a, b) = {\Gamma (a) \Gamma (b) \over \Gamma (a + b)}.
$$
@end tex
@end iftex
@ifinfo

@example
beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
@end example
@end ifinfo
@end deftypefn


@anchor{doc-betainc}
@deftypefn {Mapping Function} {} betainc (@var{x}, @var{a}, @var{b})
Return the incomplete Beta function,
@iftex
@tex
$$
 \beta (a, b, x) = B (a, b)^{-1} \int_0^x t^{(a-z)} (1-t)^{(b-1)} dt.
$$
@end tex
@end iftex
@ifinfo

@smallexample
                                    x
                                   /
betai (a, b, x) = beta (a, b)^(-1) | t^(a-1) (1-t)^(b-1) dt.
                                   /
                                t=0
@end smallexample
@end ifinfo

If x has more than one component, both @var{a} and @var{b} must be
scalars.  If @var{x} is a scalar, @var{a} and @var{b} must be of
compatible dimensions.
@end deftypefn


@anchor{doc-bincoeff}
@deftypefn {Mapping Function} {} bincoeff (@var{n}, @var{k})
Return the binomial coefficient of @var{n} and @var{k}, defined as
@iftex
@tex
$$
 {n \choose k} = {n (n-1) (n-2) \cdots (n-k+1) \over k!}
$$
@end tex
@end iftex
@ifinfo

@example
@group
 /   \
 | n |    n (n-1) (n-2) ... (n-k+1)
 |   |  = -------------------------
 | k |               k!
 \   /
@end group
@end example
@end ifinfo

For example,

@example
@group
bincoeff (5, 2)
@result{} 10
@end group
@end example
@end deftypefn


@anchor{doc-erf}
@deftypefn {Mapping Function} {} erf (@var{z})
Computes the error function,
@iftex
@tex
$$
 {\rm erf} (z) = {2 \over \sqrt{\pi}}\int_0^z e^{-t^2} dt
$$
@end tex
@end iftex
@ifinfo

@smallexample
                         z
                        /
erf (z) = (2/sqrt (pi)) | e^(-t^2) dt
                        /
                     t=0
@end smallexample
@end ifinfo
@end deftypefn
@seealso{erfc and erfinv}


@anchor{doc-erfc}
@deftypefn {Mapping Function} {} erfc (@var{z})
Computes the complementary error function,
@iftex
@tex
$1 - {\rm erf} (z)$.
@end tex
@end iftex
@ifinfo
@code{1 - erf (@var{z})}.
@end ifinfo
@end deftypefn

@seealso{erf and erfinv}


@anchor{doc-erfinv}
@deftypefn {Mapping Function} {} erfinv (@var{z})
Computes the inverse of the error function.
@end deftypefn
@seealso{erf and erfc}


@anchor{doc-gamma}
@deftypefn {Mapping Function} {} gamma (@var{z})
Computes the Gamma function,
@iftex
@tex
$$
 \Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt.
$$
@end tex
@end iftex
@ifinfo

@example
            infinity
            /
gamma (z) = | t^(z-1) exp (-t) dt.
            /
         t=0
@end example
@end ifinfo
@end deftypefn

@seealso{gammai and lgamma}


@anchor{doc-gammainc}
@deftypefn {Mapping Function} {} gammainc (@var{x}, @var{a})
Computes the incomplete gamma function,
@iftex
@tex
$$
 \gamma (x, a) = {\displaystyle\int_0^x e^{-t} t^{a-1} dt \over \Gamma (a)}
$$
@end tex
@end iftex
@ifinfo

@smallexample
                                x
                      1        /
gammainc (x, a) = ---------    | exp (-t) t^(a-1) dt
                  gamma (a)    /
                            t=0
@end smallexample
@end ifinfo

If @var{a} is scalar, then @code{gammainc (@var{x}, @var{a})} is returned
for each element of @var{x} and vice versa.

If neither @var{x} nor @var{a} is scalar, the sizes of @var{x} and
@var{a} must agree, and @var{gammainc} is applied element-by-element.
@end deftypefn
@seealso{gamma and lgamma}


@anchor{doc-lgamma}
@deftypefn {Mapping Function} {} lgamma (@var{a}, @var{x})
@deftypefnx {Mapping Function} {} gammaln (@var{a}, @var{x})
Return the natural logarithm of the gamma function.
@end deftypefn
@seealso{gamma and gammai}


@anchor{doc-cross}
@deftypefn {Function File} {} cross (@var{x}, @var{y}, @var{dim})
Computes the vector cross product of the two 3-dimensional vectors
@var{x} and @var{y}.

@example
@group
cross ([1,1,0], [0,1,1])
@result{} [ 1; -1; 1 ]
@end group
@end example

If @var{x} and @var{y} are matrices, the cross product is applied 
along the first dimension with 3 elements. The optional argument 
@var{dim} is used to force the cross product to be calculated along
the dimension defiend by @var{dim}.
@end deftypefn


@anchor{doc-commutation_matrix}
@deftypefn {Function File} {} commutation_matrix (@var{m}, @var{n})
Return the commutation matrix
@iftex
@tex
 $K_{m,n}$
@end tex
@end iftex
@ifinfo
 K(m,n)
@end ifinfo
 which is the unique
@iftex
@tex
 $m n \times m n$
@end tex
@end iftex
@ifinfo
@var{m}*@var{n} by @var{m}*@var{n}
@end ifinfo
 matrix such that
@iftex
@tex
 $K_{m,n} \cdot {\rm vec} (A) = {\rm vec} (A^T)$
@end tex
@end iftex
@ifinfo
@math{K(m,n) * vec(A) = vec(A')}
@end ifinfo
 for all
@iftex
@tex
 $m\times n$
@end tex
@end iftex
@ifinfo
@math{m} by @math{n}
@end ifinfo
 matrices
@iftex
@tex
 $A$.
@end tex
@end iftex
@ifinfo
@math{A}.
@end ifinfo

If only one argument @var{m} is given,
@iftex
@tex
 $K_{m,m}$
@end tex
@end iftex
@ifinfo
@math{K(m,m)}
@end ifinfo
 is returned.

See Magnus and Neudecker (1988), Matrix differential calculus with
applications in statistics and econometrics.
@end deftypefn


@anchor{doc-duplication_matrix}
@deftypefn {Function File} {} duplication_matrix (@var{n})
Return the duplication matrix
@iftex
@tex
 $D_n$
@end tex
@end iftex
@ifinfo
@math{Dn}
@end ifinfo
 which is the unique
@iftex
@tex
 $n^2 \times n(n+1)/2$
@end tex
@end iftex
@ifinfo
@math{n^2} by @math{n*(n+1)/2}
@end ifinfo
 matrix such that
@iftex
@tex
 $D_n * {\rm vech} (A) = {\rm vec} (A)$
@end tex
@end iftex
@ifinfo
@math{Dn vech (A) = vec (A)}
@end ifinfo
 for all symmetric
@iftex
@tex
 $n \times n$
@end tex
@end iftex
@ifinfo
@math{n} by @math{n}
@end ifinfo
 matrices
@iftex
@tex
 $A$.
@end tex
@end iftex
@ifinfo
@math{A}.
@end ifinfo

See Magnus and Neudecker (1988), Matrix differential calculus with
applications in statistics and econometrics.
@end deftypefn


@node Coordinate Transformations
@section Coordinate Transformations

@anchor{doc-cart2pol}
@deftypefn {Function File} {} [@var{theta}, @var{r}] = cart2pol (@var{x}, @var{y})
@deftypefnx {Function File} {} [@var{theta}, @var{r}, @var{z}] = cart2pol (@var{x}, @var{y}, @var{z})
Transform cartesian to polar or cylindrical coordinates.
@var{x}, @var{y} (and @var{z}) must be of same shape.
@var{theta} describes the angle relative to the x - axis.
@var{r} is the distance to the z - axis (0, 0, z).
@end deftypefn
@seealso{pol2cart, cart2sph, sph2cart}


@anchor{doc-pol2cart}
@deftypefn {Function File} {} [@var{x}, @var{y}] = pol2cart (@var{theta}, @var{r})
@deftypefnx {Function File} {} [@var{x}, @var{y}, @var{z}] = pol2cart (@var{theta}, @var{r}, @var{z})
Transform polar or cylindrical to cartesian coordinates.
@var{theta}, @var{r} (and @var{z}) must be of same shape.
@var{theta} describes the angle relative to the x - axis.
@var{r} is the distance to the z - axis (0, 0, z).
@end deftypefn
@seealso{cart2pol, cart2sph, sph2cart}


@anchor{doc-cart2sph}
@deftypefn {Function File} {} [@var{theta}, @var{phi}, @var{r}] = cart2sph (@var{x}, @var{y}, @var{z})
Transform cartesian to spherical coordinates.
@var{x}, @var{y} and @var{z} must be of same shape.
@var{theta} describes the angle relative to the x - axis.
@var{phi} is the angle relative to the xy - plane.
@var{r} is the distance to the origin (0, 0, 0).
@end deftypefn
@seealso{pol2cart, cart2pol, sph2cart}


@anchor{doc-sph2cart}
@deftypefn {Function File} {} [@var{x}, @var{y}, @var{z}] = sph2cart (@var{theta}, @var{phi}, @var{r})
Transform spherical to cartesian coordinates.
@var{x}, @var{y} and @var{z} must be of same shape.
@var{theta} describes the angle relative to the x-axis.
@var{phi} is the angle relative to the xy-plane.
@var{r} is the distance to the origin (0, 0, 0).
@end deftypefn
@seealso{pol2cart, cart2pol, cart2sph}


@node Mathematical Constants
@section Mathematical Constants

@anchor{doc-I}
@defvr {Built-in Variable} I
@defvrx {Built-in Variable} J
@defvrx {Built-in Variable} i
@defvrx {Built-in Variable} j
A pure imaginary number, defined as
@iftex
@tex
  $\sqrt{-1}$.
@end tex
@end iftex
@ifinfo
  @code{sqrt (-1)}.
@end ifinfo
These built-in variables behave like functions so you can use the names
for other purposes.  If you use them as variables and assign values to
them and then clear them, they once again assume their special predefined
values @xref{Status of Variables}.
@end defvr


@anchor{doc-Inf}
@defvr {Built-in Variable} Inf
@defvrx {Built-in Variable} inf
Infinity.  This is the result of an operation like 1/0, or an operation
that results in a floating point overflow.
@end defvr


@anchor{doc-NaN}
@defvr {Built-in Variable} NaN
@defvrx {Built-in Variable} nan
Not a number.  This is the result of an operation like
@iftex
@tex
$0/0$, or $\infty - \infty$,
@end tex
@end iftex
@ifinfo
0/0, or @samp{Inf - Inf},
@end ifinfo
or any operation with a NaN.

Note that NaN always compares not equal to NaN.  This behavior is
specified by the IEEE standard for floating point arithmetic.  To
find NaN values, you must use the @code{isnan} function.
@end defvr


@anchor{doc-pi}
@defvr {Built-in Variable} pi
The ratio of the circumference of a circle to its diameter.
Internally, @code{pi} is computed as @samp{4.0 * atan (1.0)}.
@end defvr


@anchor{doc-e}
@defvr {Built-in Variable} e
The base of natural logarithms.  The constant
@iftex
@tex
 $e$
@end tex
@end iftex
@ifinfo
 @var{e}
@end ifinfo
 satisfies the equation
@iftex
@tex
 $\log (e) = 1$.
@end tex
@end iftex
@ifinfo
 @code{log} (@var{e}) = 1.
@end ifinfo
@end defvr


@anchor{doc-eps}
@defvr {Built-in Variable} eps
The machine precision.  More precisely, @code{eps} is the largest
relative spacing between any two adjacent numbers in the machine's
floating point system.  This number is obviously system-dependent.  On
machines that support 64 bit IEEE floating point arithmetic, @code{eps}
is approximately
@ifinfo
 2.2204e-16.
@end ifinfo
@iftex
@tex
 $2.2204\times10^{-16}$.
@end tex
@end iftex
@end defvr


@anchor{doc-realmax}
@defvr {Built-in Variable} realmax
The largest floating point number that is representable.  The actual
value is system-dependent.  On machines that support 64-bit IEEE
floating point arithmetic, @code{realmax} is approximately
@ifinfo
 1.7977e+308
@end ifinfo
@iftex
@tex
 $1.7977\times10^{308}$.
@end tex
@end iftex
@end defvr


@anchor{doc-realmin}
@defvr {Built-in Variable} realmin
The smallest normalized floating point number that is representable.
The actual value is system-dependent.  On machines that support
64-bit IEEE floating point arithmetic, @code{realmin} is approximately
@ifinfo
 2.2251e-308
@end ifinfo
@iftex
@tex
 $2.2251\times10^{-308}$.
@end tex
@end iftex
@end defvr

