Add a useful package (from Source forge) for octave

[CreaPhase.git] / octave_packages / linear-algebra-2.2.0 / condeig.m

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+## Copyright (C) 2006, 2007 Arno Onken <asnelt@asnelt.org>
+##
+## This program 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.
+##
+## This program 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
+## this program; if not, see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{c} =} condeig (@var{a})
+## @deftypefnx {Function File} {[@var{v}, @var{lambda}, @var{c}] =} condeig (@var{a})
+## Compute condition numbers of the eigenvalues of a matrix. The
+## condition numbers are the reciprocals of the cosines of the angles
+## between the left and right eigenvectors.
+##
+## @subheading Arguments
+##
+## @itemize @bullet
+## @item
+## @var{a} must be a square numeric matrix.
+## @end itemize
+##
+## @subheading Return values
+##
+## @itemize @bullet
+## @item
+## @var{c} is a vector of condition numbers of the eigenvalue of
+## @var{a}.
+##
+## @item
+## @var{v} is the matrix of right eigenvectors of @var{a}. The result is
+## the same as for @code{[v, lambda] = eig (a)}.
+##
+## @item
+## @var{lambda} is the diagonal matrix of eigenvalues of @var{a}. The
+## result is the same as for @code{[v, lambda] = eig (a)}.
+## @end itemize
+##
+## @subheading Example
+##
+## @example
+## @group
+## a = [1, 2; 3, 4];
+## c = condeig (a)
+## @result{} [1.0150; 1.0150]
+## @end group
+## @end example
+## @end deftypefn
+
+function [v, lambda, c] = condeig (a)
+
+  # Check arguments
+  if (nargin != 1 || nargout > 3)
+    print_usage ();
+  endif
+
+  if (! isempty (a) && ! ismatrix (a))
+    error ("condeig: a must be a numeric matrix");
+  endif
+
+  if (columns (a) != rows (a))
+    error ("condeig: a must be a square matrix");
+  endif
+
+  if (issparse (a) && (nargout == 0 || nargout == 1) && exist ("svds", "file"))
+    ## Try to use svds to calculate the condition as it will typically be much 
+    ## faster than calling eig as only the smallest and largest eigenvalue are
+    ## calculated.
+    try
+      s0 = svds (a, 1, 0);
+      v = svds (a, 1) / s0;
+    catch
+      ## Caught an error as there is a singular value exactly at Zero!!
+      v = Inf;
+    end_try_catch
+    return;
+  endif
+
+  # Right eigenvectors
+  [v, lambda] = eig (a);
+
+  if (isempty (a))
+    c = lambda;
+  else
+    # Corresponding left eigenvectors
+    vl = inv (v)';
+    # Normalize vectors
+    vl = vl ./ repmat (sqrt (sum (abs (vl .^ 2))), rows (vl), 1);
+
+    # Condition numbers
+    # cos (angle) = (norm (v1) * norm (v2)) / dot (v1, v2)
+    # Norm of the eigenvectors is 1 => norm (v1) * norm (v2) = 1
+    c = abs (1 ./ dot (vl, v)');
+  endif
+
+  if (nargout == 0 || nargout == 1)
+    v = c;
+  endif
+
+endfunction
+
+%!test
+%! a = [1, 2; 3, 4];
+%! c = condeig (a);
+%! expected_c = [1.0150; 1.0150];
+%! assert (c, expected_c, 0.001);
+
+%!test
+%! a = [1, 3; 5, 8];
+%! [v, lambda, c] = condeig (a);
+%! [expected_v, expected_lambda] = eig (a);
+%! expected_c = [1.0182; 1.0182];
+%! assert (v, expected_v, 0.001);
+%! assert (lambda, expected_lambda, 0.001);
+%! assert (c, expected_c, 0.001);