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* condeig.m: fix Texinfo markup typo in condeig doc string (bug #48672)
author Mike Miller <mtmiller@octave.org>
date Mon, 01 Aug 2016 09:52:51 -0700
parents e859caa53399
children bac0d6f07a3e
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rev   line source
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1 ## Copyright (C) 2006, 2007 Arno Onken <asnelt@asnelt.org>
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {} {@var{c} =} condeig (@var{a})
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21 ## @deftypefnx {} {[@var{v}, @var{lambda}, @var{c}] =} condeig (@var{a})
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22 ## Compute condition numbers of a matrix with respect to eigenvalues.
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23 ##
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24 ## The condition numbers are the reciprocals of the cosines of the angles
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25 ## between the left and right eigenvectors; Large values indicate that the
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26 ## matrix has multiple distinct eigenvalues.
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27 ##
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28 ## The input @var{a} must be a square numeric matrix.
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29 ##
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30 ## The outputs are:
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31 ##
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32 ## @itemize @bullet
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33 ## @item
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34 ## @var{c} is a vector of condition numbers for the eigenvalues of
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35 ## @var{a}.
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36 ##
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37 ## @item
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38 ## @var{v} is the matrix of right eigenvectors of @var{a}. The result is
22195
93ed9396f2c3 * condeig.m: fix Texinfo markup typo in condeig doc string (bug #48672)
Mike Miller <mtmiller@octave.org>
parents: 22129
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39 ## equivalent to calling @code{[@var{v}, @var{lambda}] = eig (@var{a})}.
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40 ##
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41 ## @item
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42 ## @var{lambda} is the diagonal matrix of eigenvalues of @var{a}. The
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43 ## result is equivalent to calling
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44 ## @code{[@var{v}, @var{lambda}] = eig (@var{a})}.
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45 ## @end itemize
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46 ##
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47 ## Example
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48 ##
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49 ## @example
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50 ## @group
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51 ## a = [1, 2; 3, 4];
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52 ## c = condeig (a)
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53 ## @result{} [1.0150; 1.0150]
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54 ## @end group
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55 ## @end example
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56 ## @seealso{eig, cond, balance}
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57 ## @end deftypefn
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58
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59 function [v, lambda, c] = condeig (a)
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60
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61 if (nargin != 1)
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62 print_usage ();
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63 endif
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64
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65 if (! (isnumeric (a) && issquare (a)))
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66 error ("condeig: A must be a square numeric matrix");
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67 endif
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68
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69 if (issparse (a) && nargout <= 1)
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70 ## Try to use svds to calculate the condition number as it will typically
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71 ## be much faster than calling eig as only the smallest and largest
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72 ## eigenvalue are calculated.
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73
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74 ## FIXME: This calculates one condition number for the entire matrix.
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75 ## In the full case, separate condition numbers are calculated for every
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76 ## eigenvalue.
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77 try
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78 s0 = svds (a, 1, 0); # min eigenvalue
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79 v = svds (a, 1) / s0; # max eigenvalue
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80 catch
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81 ## Caught an error as there is a singular value exactly at zero!!
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82 v = Inf;
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83 end_try_catch
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84 return;
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85 endif
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86
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87 ## Right eigenvectors
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88 [v, lambda] = eig (a);
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89
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90 if (isempty (a))
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91 c = [];
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92 else
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93 ## Corresponding left eigenvectors
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94 vl = inv (v)';
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95 ## Normalize vectors
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96 vl ./= repmat (sqrt (sum (abs (vl .^ 2))), rows (vl), 1);
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97
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98 ## Condition numbers
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99 ## Definition: cos (angle) = (norm (v1) * norm (v2)) / dot (v1, v2)
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100 ## Eigenvectors have been normalized so `norm (v1) * norm (v2)' = 1
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101 c = abs (1 ./ dot (vl, v)');
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102 endif
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103
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104 if (nargout <= 1)
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105 v = c;
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106 endif
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107
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108 endfunction
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109
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110
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111 %!test
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112 %! a = [1, 2; 3, 4];
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113 %! c = condeig (a);
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114 %! expected_c = [1.0150; 1.0150];
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115 %! assert (c, expected_c, 0.001);
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116
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117 %!test
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118 %! a = [1, 3; 5, 8];
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119 %! [v, lambda, c] = condeig (a);
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120 %! [expected_v, expected_lambda] = eig (a);
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121 %! expected_c = [1.0182; 1.0182];
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122 %! assert (v, expected_v, 0.001);
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123 %! assert (lambda, expected_lambda, 0.001);
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124 %! assert (c, expected_c, 0.001);
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125
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126 # Test empty input
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127 %!assert (condeig ([]), [])
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128
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129 ## Test input validation
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130 %!error condeig ()
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131 %!error condeig (1,2)
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132 %!error <A must be a square numeric matrix> condeig ({1})
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133 %!error <A must be a square numeric matrix> condeig (ones (3,2))
e859caa53399 condeig.m: New function imported from Octave-Forge linear-algebra package.
Rik <rik@octave.org>
parents:
diff changeset
134 %!error <A must be a square numeric matrix> condeig (ones (2,2,2))
e859caa53399 condeig.m: New function imported from Octave-Forge linear-algebra package.
Rik <rik@octave.org>
parents:
diff changeset
135