update packages

[CreaPhase.git] / octave_packages / m / special-matrix / hilb.m

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+## Copyright (C) 1993-2012 John W. Eaton
+##
+## This file is part of Octave.
+##
+## Octave is free software; you can redistribute it and/or modify it
+## under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or (at
+## your option) any later version.
+##
+## Octave is distributed in the hope that it will be useful, but
+## WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+## General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with Octave; see the file COPYING.  If not, see
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {} hilb (@var{n})
+## Return the Hilbert matrix of order @var{n}.  The @math{i,j} element
+## of a Hilbert matrix is defined as
+## @tex
+## $$
+## H(i, j) = {1 \over (i + j - 1)}
+## $$
+## @end tex
+## @ifnottex
+##
+## @example
+## H(i, j) = 1 / (i + j - 1)
+## @end example
+##
+## @end ifnottex
+##
+## Hilbert matrices are close to being singular which make them difficult to
+## invert with numerical routines.
+## Comparing the condition number of a random matrix 5x5 matrix with that of
+## a Hilbert matrix of order 5 reveals just how difficult the problem is.
+##
+## @example
+## @group
+## cond (rand (5))
+##    @result{} 14.392
+## cond (hilb (5))
+##    @result{} 4.7661e+05
+## @end group
+## @end example
+##
+## @seealso{invhilb}
+## @end deftypefn
+
+## Author: jwe
+
+function retval = hilb (n)
+
+  if (nargin != 1)
+    print_usage ();
+  elseif (! isscalar (n))
+    error ("hilb: N must be a scalar integer");
+  endif
+
+  retval = zeros (n);
+  tmp = 1:n;
+  for i = 1:n
+    retval(i, :) = 1.0 ./ tmp;
+    tmp++;
+  endfor
+
+endfunction
+
+
+%!assert (hilb (2), [1, 1/2; 1/2, 1/3])
+%!assert (hilb (3), [1, 1/2, 1/3; 1/2, 1/3, 1/4; 1/3, 1/4, 1/5])
+
+%!error hilb ()
+%!error hilb (1, 2)
+%!error <N must be a scalar integer> hilb (ones(2))
+