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Initial commit into CVS.
author whyly
date Tue, 17 Oct 2006 22:18:07 +0000
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1 ## Copyright (C) 2006 Arno Onken
2 ##
3 ## This program is free software; you can redistribute it and/or modify
4 ## it under the terms of the GNU General Public License as published by
5 ## the Free Software Foundation; either version 2 of the License, or
6 ## (at your option) any later version.
7 ##
8 ## This program is distributed in the hope that it will be useful,
9 ## but WITHOUT ANY WARRANTY; without even the implied warranty of
10 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
11 ## GNU General Public License for more details.
12 ##
13 ## You should have received a copy of the GNU General Public License
14 ## along with this program; if not, write to the Free Software
15 ## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
16
17 ## -*- texinfo -*-
18 ## @deftypefn {Function File} {@var{c} =} condeig (@var{a})
19 ## @deftypefnx {Function File} {[@var{v}, @var{lambda}, @var{c}] =} condeig (@var{a})
20 ## Computes condition numbers for the eigenvalues of a matrix. The
21 ## condition numbers are the reciprocals of the cosines of the angles
22 ## between the left and right eigenvectors.
23 ##
24 ## Arguments are
25 ##
26 ## @itemize @bullet
27 ## @item
28 ## @var{a} must be a square numeric matrix.
29 ## @end itemize
30 ##
31 ## Return values are
32 ##
33 ## @itemize @bullet
34 ## @item
35 ## @var{c} is a vector of condition numbers for the eigenvalue of
36 ## @var{a}.
37 ##
38 ## @item
39 ## @var{v} is the matrix of right eigenvectors of @var{a}. The result is
40 ## the same as for @code{[v, lambda] = eig (a)}.
41 ##
42 ## @item
43 ## @var{lambda} is the diagonal matrix of eigenvalues of @var{a}. The
44 ## result is the same as for @code{[v, lambda] = eig (a)}.
45 ## @end itemize
46 ##
47 ## Example:
48 ##
49 ## @example
50 ## @group
51 ## a = [1, 2; 3, 4];
52 ## c = condeig (a)
53 ## @result{} [1.0150; 1.0150]
54 ## @end group
55 ## @end example
56 ## @end deftypefn
57
58 ## Author: Arno Onken <whyly@gmx.net>
59 ## Description: Condition numbers for eigenvalues
60
61 function [v, lambda, c] = condeig (a)
62
63 # Check arguments
64 if (nargin != 1 || nargout > 3)
65 usage ("[v, lambda, c] = condeig (a)");
66 endif
67
68 if (! isempty (a) && ! ismatrix (a))
69 error ("condeig: a must be a numeric matrix");
70 endif
71
72 if (columns (a) != rows (a))
73 error ("condeig: a must be a square matrix");
74 endif
75
76 # Right eigenvectors
77 [v, lambda] = eig (a);
78
79 if (isempty (a))
80 c = lambda;
81 else
82 # Corresponding left eigenvectors
83 vl = inv (v)';
84 # Normalize vectors
85 vl = vl ./ repmat (sqrt (sum (abs (vl .^ 2))), rows (vl), 1);
86
87 # Condition numbers
88 # cos (angle) = (norm (v1) * norm (v2)) / dot (v1, v2)
89 # Norm of the eigenvectors is 1 => norm (v1) * norm (v2) = 1
90 c = abs (1 ./ dot (vl, v)');
91 endif
92
93 if (nargout == 0 || nargout == 1)
94 v = c;
95 endif
96
97 endfunction
98
99 %!test
100 %! a = [1, 2; 3, 4];
101 %! c = condeig (a);
102 %! expected_c = [1.0150; 1.0150];
103 %! assert (c, expected_c, 0.001);
104
105 %!test
106 %! a = [1, 3; 5, 8];
107 %! [v, lambda, c] = condeig (a);
108 %! [expected_v, expected_lambda] = eig (a);
109 %! expected_c = [1.0182; 1.0182];
110 %! assert (v, expected_v, 0.001);
111 %! assert (lambda, expected_lambda, 0.001);
112 %! assert (c, expected_c, 0.001);