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

@node Special Matrices, Matrix Manipulation, Input and Output, Top
@chapter Special Matrices

Octave provides a number of functions for creating special matrix forms.
In nearly all cases, it is best to use the built-in functions for this
purpose than to try to use other tricks to achieve the same effect.

@menu
* Special Utility Matrices::    
* Famous Matrices::             
@end menu

@node Special Utility Matrices, Famous Matrices, Special Matrices, Special Matrices
@section Special Utility Matrices

@deftypefn {Built-in Function} {} eye (@var{x})
@deftypefnx {Built-in Function} {} eye (@var{n}, @var{m})
Returns an identity matrix.  If invoked with a single scalar argument,
@code{eye} returns a square matrix with the dimension specified.  If you
supply two scalar arguments, @code{eye} takes them to be the number of
rows and columns.  If given a vector with two elements, @code{eye} uses
the values of the elements as the number of rows and columns,
respecively.  For example,

@example
@group
eye (3)

     @result{}  1  0  0
         0  1  0
         0  0  1
@end group
@end example

The following expressions all produce the same result:

@example
@group
eye (2)
eye (2, 2)
eye (size ([1, 2; 3, 4])
@end group
@end example

For compatibility with @sc{Matlab}, calling @code{eye} with no arguments
is equivalent to calling it with an argument of 1.
@end deftypefn

@deftypefn {Built-in Function} {} ones (@var{x})
@deftypefnx {Built-in Function} {} ones (@var{n}, @var{m})
Returns a matrix whose elements are all 1.  The arguments are handled
the same as the arguments for @code{eye}.

If you need to create a matrix whose values are all the same, you should
use an expression like

@example
val_matrix = val * ones (n, m)
@end example
@end deftypefn

@deftypefn {Built-in Function} {} zeros (@var{x})
@deftypefnx {Built-in Function} {} zeros (@var{n}, @var{m})
Returns a matrix whose elements are all 0.  The arguments are handled
the same as the arguments for @code{eye}.
@end deftypefn

@deftypefn {Built-in Function} {} rand (@var{x})
@deftypefnx {Built-in Function} {} rand (@var{n}, @var{m})
@deftypefnx {Built-in Function} {} rand (@code{"seed"}, @var{x})
Returns a matrix with random elements uniformly distributed on the
interval (0, 1).  The arguments are handled the same as the arguments
for @code{eye}.  In
addition, you can set the seed for the random number generator using the
form

@example
randn ("seed", @var{x})
@end example

@noindent
where @var{x} is a scalar value.  If called as

@example
rand ("seed")
@end example

@noindent
@code{rand} returns the current value of the seed.
@end deftypefn

@deftypefn {Built-in Function} {} randn (@var{x})
@deftypefnx {Built-in Function} {} randn (@var{n}, @var{m})
@deftypefnx {Built-in Function} {} randn (@code{"seed"}, @var{x})
Returns a matrix with normally distributed random elements.  The
arguments are handled the same as the arguments for @code{eye}.  In
addition, you can set the seed for the random number generator using the
form

@example
randn ("seed", @var{x})
@end example

@noindent
where @var{x} is a scalar value.  If called as

@example
randn ("seed")
@end example

@noindent
@code{randn} returns the current value of the seed.
@end deftypefn

The @code{rand} and @code{randn} functions use separate generators.
This ensures that

@example
@group
rand ("seed", 13);
randn ("seed", 13);
u = rand (100, 1);
n = randn (100, 1);
@end group
@end example

@noindent
and

@example
@group
rand ("seed", 13);
randn ("seed", 13);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
  u(i) = rand ();
  n(i) = randn ();
end
@end group
@end example

@noindent
produce equivalent results.

Normally, @code{rand} and @code{randn} obtain their initial
seeds from the system clock, so that the sequence of random numbers is
not the same each time you run Octave.  If you really do need for to
reproduce a sequence of numbers exactly, you can set the seed to a
specific value.

If it is invoked without arguments, @code{rand} and @code{randn} return a
single element of a random sequence.

The @code{rand} and @code{randn} functions use Fortran code from RANLIB,
a library of fortran routines for random number generation, compiled by
Barry W. Brown and James Lovato of the Department of Biomathematics at
The University of Texas, M.D. Anderson Cancer Center, Houston, TX 77030.

@deftypefn {Built-in Function} {} diag (@var{v}, @var{k})
Returns a diagonal matrix with vector @var{v} on diagonal @var{k}.  The
second argument is optional.  If it is positive, the vector is placed on
the @var{k}-th super-diagonal.  If it is negative, it is placed on the
@var{-k}-th sub-diagonal.  The default value of @var{k} is 0, and the
vector is placed on the main diagonal.  For example,

@example
@group
diag ([1, 2, 3], 1)

     @result{}  0  1  0  0
         0  0  2  0
         0  0  0  3
         0  0  0  0
@end group
@end example
@end deftypefn

The functions @code{linspace} and @code{logspace} make it very easy to
create vectors with evenly or logarithmically spaced elements.
@xref{Ranges}.

@deftypefn {Function File} {} linspace (@var{base}, @var{limit}, @var{n})
creates a row vector with @var{n} (@var{n} greater than 1) linearly
spaced elements between @var{base} and @var{limit}.  The @var{base} and
@var{limit} are always included in the range.  If @var{base} is greater
than @var{limit}, the elements are stored in decreasing order.  If the
number of points is not specified, a value of 100 is used.

The @code{linspace} function always returns a row vector, regardless of
the value of @code{prefer_column_vectors}.
@end deftypefn

@deftypefn {Function File} {} logspace (@var{base}, @var{limit}, @var{n})
Similar to @code{linspace} except that the values are logarithmically
spaced.

If @var{limit} is equal to
@iftex
@tex
$\pi$,
@end tex
@end iftex
@ifinfo
pi,
@end ifinfo
the points are between
@iftex
@tex
$10^{base}$ and $\pi$,
@end tex
@end iftex
@ifinfo
10^base and pi,
@end ifinfo
@emph{not}
@iftex
@tex
$10^{base}$ and $10^{\pi}$,
@end tex
@end iftex
@ifinfo
10^base and 10^pi,
@end ifinfo
in order to  be compatible with the corresponding @sc{Matlab} function.
@end deftypefn

@node Famous Matrices,  , Special Utility Matrices, Special Matrices
@section Famous Matrices

The following functions return famous matrix forms.

@deftypefn {Function File} {} hadamard (@var{k})
Return the Hadamard matrix of order n = 2^k.
@end deftypefn

@deftypefn {Function File} {} hankel (@var{c}, @var{r})
Return the Hankel matrix constructed given the first column @var{c}, and
(optionally) the last row @var{r}.  If the last element of @var{c} is
not the same as the first element of @var{r}, the last element of
@var{c} is used.  If the second argument is omitted, the last row is
taken to be the same as the first column.

A Hankel matrix formed from an m-vector @var{c}, and an n-vector
@var{r}, has the elements
@iftex
@tex
$$
H (i, j) = \cases{c_{i+j-1},&$i+j-1\le m$;\cr r_{i+j-m},&otherwise.\cr}
$$
@end tex
@end iftex
@ifinfo

@example
@group
H (i, j) = c (i+j-1),  i+j-1 <= m;
H (i, j) = r (i+j-m),  otherwise
@end group
@end example
@end ifinfo
@end deftypefn

@deftypefn {Function File} {} hilb (@var{n})
Return the Hilbert matrix of order @var{n}.  The
@iftex
@tex
$i,\,j$
@end tex
@end iftex
@ifinfo
i, j
@end ifinfo
element of a Hilbert matrix is defined as
@iftex
@tex
$$
H (i, j) = {1 \over (i + j - 1)}
$$
@end tex
@end iftex
@ifinfo

@example
H (i, j) = 1 / (i + j - 1)
@end example
@end ifinfo
@end deftypefn

@deftypefn {Function File} {} invhilb (@var{n})
Return the inverse of a Hilbert matrix of order @var{n}.  This is exact.
Compare with the numerical calculation of @code{inverse (hilb (n))},
which suffers from the ill-conditioning of the Hilbert matrix, and the
finite precision of your computer's floating point arithmetic.
@end deftypefn

@deftypefn {Function File} {} toeplitz (@var{c}, @var{r})
Return the Toeplitz matrix constructed given the first column @var{c},
and (optionally) the first row @var{r}.  If the first element of @var{c}
is not the same as the first element of @var{r}, the first element of
@var{c} is used.  If the second argument is omitted, the first row is
taken to be the same as the first column.

A square Toeplitz matrix has the form
@iftex
@tex
$$
\left[\matrix{c_0    & r_1     & r_2      & \ldots & r_n\cr
              c_1    & c_0     & r_1      &        & c_{n-1}\cr
              c_2    & c_1     & c_0      &        & c_{n-2}\cr
              \vdots &         &          &        & \vdots\cr
              c_n    & c_{n-1} & c_{n-2} & \ldots & c_0}\right].
$$
@end tex
@end iftex
@ifinfo

@example
@group
c(0)  r(1)   r(2)  ...  r(n)
c(1)  c(0)   r(1)      r(n-1)
c(2)  c(1)   c(0)      r(n-2)
 .                       .
 .                       .
 .                       .

c(n) c(n-1) c(n-2) ...  c(0)
@end group
@end example
@end ifinfo
@end deftypefn

@deftypefn {Function File} {} vander (@var{c})
Return the Vandermonde matrix whose next to last column is @var{c}.

A Vandermonde matrix has the form
@iftex
@tex
$$
\left[\matrix{c_0^n  & \ldots & c_0^2  & c_0    & 1\cr
              c_1^n  & \ldots & c_1^2  & c_1    & 1\cr
              \vdots &        & \vdots & \vdots & \vdots\cr
              c_n^n  & \ldots & c_n^2  & c_n    & 1}\right].
$$
@end tex
@end iftex
@ifinfo

@example
@group
c(0)^n ... c(0)^2  c(0)  1
c(1)^n ... c(1)^2  c(1)  1
 .           .      .    .
 .           .      .    .
 .           .      .    .
                 
c(n)^n ... c(n)^2  c(n)  1
@end group
@end example
@end ifinfo
@end deftypefn