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annotate scripts/linear-algebra/ordeig.m @ 30379:363fb10055df stable

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maint: Style check m-files ahead of 7.1 release. * Map.m, integral3.m, logspace.m, quad2d.m, quadgk.m, quadl.m, tsearchn.m, get_first_help_sentence.m, print_usage.m, getframe.m, imformats.m, javaclasspath.m, condest.m, null.m, ordeig.m, inputParser.m, license.m, memory.m, methods.m, __publish_html_output__.m, __publish_latex_output__.m, publish.m, ode15s.m, fminbnd.m, fzero.m, configure_make.m, get_description.m, get_forge_pkg.m, annotation.m, camlookat.m, legend.m, __gnuplot_legend__.m, bar.m, colorbar.m, fill3.m, isosurface.m, plotyy.m, polar.m, __bar__.m, __ezplot__.m, __patch__.m, __pie__.m, __plt__.m, __scatter__.m, smooth3.m, stemleaf.m, __gnuplot_drawnow__.m, print.m, printd.m, __add_default_menu__.m, __gnuplot_draw_axes__.m, __gnuplot_print__.m, __print_parse_opts__.m, struct2hdl.m, profexport.m, profile.m, movfun.m, sprandsym.m, betaincinv.m, factor.m, nchoosek.m, gallery.m, hadamard.m, iqr.m, ranks.m, __run_test_suite__.m, test.m, datevec.m, weboptions.m: Style check m-files ahead of 7.1 release.
author Rik <rik@octave.org>
date Fri, 26 Nov 2021 20:53:22 -0800
parents 7854d5752dd2
children 796f54d4ddbf
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 2018-2021 The Octave Project Developers
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4 ##
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5 ## See the file COPYRIGHT.md in the top-level directory of this
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6 ## distribution or <https://octave.org/copyright/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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11 ## under the terms of the GNU General Public License as published by
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12 ## the Free Software Foundation, either version 3 of the License, or
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13 ## (at your option) any later version.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 ## GNU General Public License for more details.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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21 ## along with Octave; see the file COPYING. If not, see
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22 ## <https://www.gnu.org/licenses/>.
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23 ##
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24 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {@var{lambda} =} ordeig (@var{A})
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28 ## @deftypefnx {} {@var{lambda} =} ordeig (@var{A}, @var{B})
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29 ## Return the eigenvalues of quasi-triangular matrices in their order of
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30 ## appearance in the matrix @var{A}.
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31 ##
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32 ## The quasi-triangular matrix @var{A} is usually the result of a Schur
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33 ## factorization. If called with a second input @var{B} then the generalized
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34 ## eigenvalues of the pair @var{A}, @var{B} are returned in the order of
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35 ## appearance of the matrix @code{@var{A}-@var{lambda}*@var{B}}. The pair
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36 ## @var{A}, @var{B} is usually the result of a QZ decomposition.
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37 ##
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38 ## @seealso{ordschur, ordqz, eig, schur, qz}
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39 ## @end deftypefn
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40
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41 function lambda = ordeig (A, B)
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42
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43 if (nargin < 1)
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44 print_usage ();
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45 endif
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46
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47 if (! isnumeric (A) || ! issquare (A))
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48 error ("ordeig: A must be a square matrix");
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49 endif
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50
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51 n = length (A);
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52
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53 if (nargin == 1)
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54 B = eye (n);
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55 if (isreal (A))
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56 if (! is_quasitri (A))
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57 error ("ordeig: A must be quasi-triangular (i.e., upper block triangular with 1x1 or 2x2 blocks on the diagonal)");
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58 endif
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59 else
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60 if (! istriu (A))
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61 error ("ordeig: A must be upper-triangular when it is complex");
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62 endif
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63 endif
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64 else
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65 if (! isnumeric (B) || ! issquare (B))
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66 error ("ordeig: B must be a square matrix");
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67 elseif (length (B) != n)
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68 error ("ordeig: A and B must be the same size");
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69 endif
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70 if (isreal (A) && isreal (B))
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71 if (! is_quasitri (A) || ! is_quasitri (B))
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72 error ("ordeig: A and B must be quasi-triangular (i.e., upper block triangular with 1x1 or 2x2 blocks on the diagonal)");
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73 endif
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74 else
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75 if (! istriu (A) || ! istriu (B))
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76 error ("ordeig: A and B must both be upper-triangular if either is complex");
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77 endif
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78 endif
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79 endif
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80
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81 ## Start of algorithm
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82 lambda = zeros (n, 1);
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83
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84 i = 1;
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85 while (i <= n)
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86 if (i == n || (A(i+1,i) == 0 && B(i+1,i) == 0))
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87 lambda(i) = A(i,i) / B(i,i);
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88 else
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89 a = B(i,i) * B(i+1,i+1);
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90 b = - (A(i,i) * B(i+1,i+1) + A(i+1,i+1) * B(i,i));
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91 c = A(i,i) * A(i+1,i+1) - ...
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92 (A(i,i+1) - B(i,i+1)) * (A(i+1,i) - B(i+1,i));
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93 if (b > 0)
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94 lambda(i) = 2*c / (-b - sqrt (b^2 - 4*a*c));
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95 i += 1;
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96 lambda(i) = (-b - sqrt (b^2 - 4*a*c)) / 2 / a;
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97 else
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98 lambda(i) = (-b + sqrt (b^2 - 4*a*c)) / 2 / a;
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99 i += 1;
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100 lambda(i) = 2*c / (-b + sqrt (b^2 - 4*a*c));
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101 endif
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102 endif
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103 i += 1;
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104 endwhile
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105
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106 endfunction
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107
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108 ## Check whether a matrix is quasi-triangular
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109 function retval = is_quasitri (A)
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110
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111 if (length (A) <= 2)
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112 retval = true;
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113 else
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114 v = diag (A, -1) != 0;
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115 retval = (all (tril (A, -2)(:) == 0) && all (v(1:end-1) + v(2:end) < 2));
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116 endif
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117
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118 endfunction
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119
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120
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121 %!test
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122 %! A = toeplitz ([0, 1, 0, 0], [0, -1, 0, 0]);
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123 %! T = schur (A);
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124 %! lambda = ordeig (T);
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125 %! assert (lambda, [1.61803i; -1.61803i; 0.61803i; -0.61803i], 1e-4);
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126
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127 %!test
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128 %! A = toeplitz ([0, 1, 0, 0], [0, -1, 0, 0]);
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129 %! B = toeplitz ([0, 0, 0, 1], [0, -1, 0, 2]);
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130 %! [AA, BB] = qz (A, B);
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131 %! assert (isreal (AA) && isreal (BB));
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132 %! lambda = ordeig (AA, BB);
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133 %! assert (lambda, [0.5+0.86603i; 0.5-0.86603i; i; -i], 1e-4);
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134 %! [AA, BB] = qz (complex (A), complex (B));
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135 %! assert (iscomplex (AA) && iscomplex (BB));
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136 %! lambda = ordeig (AA, BB);
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137 %! assert (lambda, diag (AA) ./ diag (BB));
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138
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139 ## Check trivial 1x1 case
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140 %!test <*55779>
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141 %! lambda = ordeig ([6], [2]);
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142 %! assert (lambda, 3);
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143
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144 ## Test input validation
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145 %!error <Invalid call> ordeig ()
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146 %!error <A must be a square matrix> ordeig ('a')
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147 %!error <A must be a square matrix> ordeig ([1, 2, 3])
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148 %!error <A must be quasi-triangular> ordeig (magic (3))
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149 %!error <A must be upper-triangular> ordeig ([1, 0; i, 1])
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150 %!error <B must be a square matrix> ordeig (1, 'a')
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151 %!error <B must be a square matrix> ordeig (1, [1, 2])
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152 %!error <A and B must be the same size> ordeig (1, ones (2,2))
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153 %!error <A and B must be quasi-triangular>
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154 %! ordeig (triu (magic (3)), magic (3))
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155 %!error <A and B must both be upper-triangular>
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156 %! ordeig ([1, 1; 0, 1], [1, 0; i, 1])