@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, Linear Algebra, Matrix Manipulation, Top
@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::           
* Mathematical Constants::      
@end menu

@node Utility Functions, Complex Arithmetic, Arithmetic, Arithmetic
@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.

@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

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

@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

@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

@deftypefn {Mapping Function} {} gcd (@var{x}, @code{...})
Compute the greatest common divisor of the elements of @var{x}, or the
list of all the arguments.  For example, 

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

@noindent
is the same as

@example
gcd ([a1, ..., ak])
@end example

An optional second return value, @var{v}
contains an integer vector such that

@example
g = v(1) * a(k) + ... + v(k) * a(k)
@end example
@end deftypefn

@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

@deftypefn {Mapping Function} {} log (@var{x})
Compute the natural logarithm of @var{x}.  To compute the matrix logarithm, 
see @ref{Linear Algebra}.
@end deftypefn

@deftypefn {Mapping Function} {} log10 (@var{x})
Compute the base-10 logarithm of @var{x}.
@end deftypefn

@deftypefn {Mapping Function} {@var{y} =} 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

@deftypefn {Loadable Function} {} max (@var{x})
For a vector argument, return the maximum value.  For a matrix argument,
return the maximum value from each column, as a row vector.  Thus,

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

@noindent
returns the largest element of @var{x}.

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

@deftypefn {Loadable Function} {} min (@var{x})
Like @code{max}, but return the minimum value.
@end deftypefn

@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

@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

@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

@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

@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

@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

@deftypefn {Mapping Function} {} xor (@var{x}, @var{y})
Return the `exclusive or' of the entries of @var{x} and @var{y}.
For boolean expressions @var{x} and @var{y},
@code{xor (@var{x}, @var{y})} is true if and only if @var{x} or @var{y}
is true, but not if both @var{x} and @var{y} are true.
@end deftypefn

@node Complex Arithmetic, Trigonometry, Utility Functions, 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

@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

@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

@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

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

@deftypefn {Mapping Function} {} real (@var{z})
Return the real part of @var{z}.
@end deftypefn

@node Trigonometry, Sums and Products, Complex Arithmetic, Arithmetic
@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).

@deftypefn {Mapping Function} {} sin (@var{z})
@deftypefnx {Mapping Function} {} cos (@var{z})
@deftypefnx {Mapping Function} {} tan (@var{z})
@deftypefnx {Mapping Function} {} sec (@var{z})
@deftypefnx {Mapping Function} {} csc (@var{z})
@deftypefnx {Mapping Function} {} cot (@var{z})
The ordinary trigonometric functions.
@end deftypefn

@deftypefn {Mapping Function} {} asin (@var{z})
@deftypefnx {Mapping Function} {} acos (@var{z})
@deftypefnx {Mapping Function} {} atan (@var{z})
@deftypefnx {Mapping Function} {} asec (@var{z})
@deftypefnx {Mapping Function} {} acsc (@var{z})
@deftypefnx {Mapping Function} {} acot (@var{z})
The ordinary inverse trigonometric functions.
@end deftypefn

@deftypefn {Mapping Function} {} sinh (@var{z})
@deftypefnx {Mapping Function} {} cosh (@var{z})
@deftypefnx {Mapping Function} {} tanh (@var{z})
@deftypefnx {Mapping Function} {} sech (@var{z})
@deftypefnx {Mapping Function} {} csch (@var{z})
@deftypefnx {Mapping Function} {} coth (@var{z})
Hyperbolic trigonometric functions.
@end deftypefn

@deftypefn {Mapping Function} {} asinh (@var{z})
@deftypefnx {Mapping Function} {} acosh (@var{z})
@deftypefnx {Mapping Function} {} atanh (@var{z})
@deftypefnx {Mapping Function} {} asech (@var{z})
@deftypefnx {Mapping Function} {} acsch (@var{z})
@deftypefnx {Mapping Function} {} acoth (@var{z})
Inverse hyperbolic trigonometric functions.
@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

@deftypefn {Mapping Function} {} atan2 (@var{y}, @var{x})
Return the arctangent of @var{y}/@var{x}.  The signs of the arguments
are used to determine the quadrant of the result, which is in the range
@iftex
@tex
$\pi$ to $-\pi$.
@end tex
@end iftex
@ifinfo
@code{pi} to -@code{pi}.
@end ifinfo
@end deftypefn

@node Sums and Products, Special Functions, Trigonometry, Arithmetic
@section Sums and Products

@deftypefn {Built-in Function} {} sum (@var{x})
For a vector argument, return the sum of all the elements.  For a matrix
argument, return the sum of the elements in each column, as a row
vector.  The sum of an empty matrix is 0 if it has no columns, or a
vector of zeros if it has no rows (@pxref{Empty Matrices}).
@end deftypefn

@deftypefn {Built-in Function} {} prod (@var{x})
For a vector argument, return the product of all the elements.  For a
matrix argument, return the product of the elements in each column, as a
row vector.  The product of an empty matrix is 1 if it has no columns,
or a vector of ones if it has no rows (@pxref{Empty Matrices}).
@end deftypefn

@deftypefn {Built-in Function} {} cumsum (@var{x})
Return the cumulative sum of each column of @var{x}.  For example,

@example
@group
cumsum ([1, 2; 3, 4])
     @result{}  1  2
         4  6
@end group
@end example
@end deftypefn

@deftypefn {Built-in Function} {} cumprod (@var{x})
Return the cumulative product of each column of @var{x}.  For example,

@example
@group
cumprod ([1, 2; 3, 4])
     @result{}  1  2
         3  8
@end group
@end example
@end deftypefn

@deftypefn {Built-in Function} {} sumsq (@var{x})
For a vector argument, return the sum of the squares of all the
elements.  For a matrix argument, return the sum of the squares of the
elements in each column, as a row vector.
@end deftypefn

@node Special Functions, Mathematical Constants, Sums and Products, Arithmetic
@section Special Functions

@deftypefn {Mapping Function} {} besseli (@var{alpha}, @var{x})
@deftypefnx {Mapping Function} {} besselj (@var{alpha}, @var{x})
@deftypefnx {Mapping Function} {} besselk (@var{alpha}, @var{x})
@deftypefnx {Mapping Function} {} bessely (@var{alpha}, @var{x})
Compute Bessel functions of the following types:

@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.
@end table

The second argument, @var{x}, must be a real matrix, vector, or scalar.

The first argument, @var{alpha}, must be greater than or equal to zero.
If @var{alpha} is a range, it must have an increment equal to one.

If @var{alpha} is a scalar, the result is the same size as @var{x}.

If @var{alpha} is a range, @var{x} must be a vector or scalar, and the
result is a matrix with @code{length(@var{x})} rows and
@code{length(@var{alpha})} columns.
@end deftypefn

@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

@deftypefn {Mapping Function} {} betai (@var{a}, @var{b}, @var{x})
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

@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

@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

@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

@deftypefn {Mapping Function} {} erfinv (@var{z})
Computes the inverse of the error function,
@end deftypefn

@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

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

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

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

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

@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

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

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

@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
 @var{K}(@var{m},@var{n}) * vec (@var{A}) = vec (@var{A}')
@end ifinfo
 for all
@iftex
@tex
 $m\times n$
@end tex
@end iftex
@ifinfo
 @var{m} by @var{n}
@end ifinfo
 matrices
@iftex
@tex
 $A$.
@end tex
@end iftex
@ifinfo
 @var{A}.
@end ifinfo

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

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

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

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

@node Mathematical Constants,  , Special Functions, Arithmetic
@section Mathematical Constants

@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
The @code{I} and @code{J} forms are true constants, and cannot be
modified.  The @code{i} and @code{j} forms are like ordinary variables,
and may be used for other purposes.  However, unlike other variables,
they once again assume their special predefined values if they are
cleared @xref{Status of Variables}.
@end defvr

@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

@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

@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

@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

@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

@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

@defvr {Built-in Variable} realmin
The smallest 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