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maint: Periodic merge of gui-release to default.
author John W. Eaton <jwe@octave.org>
date Tue, 20 Jan 2015 10:24:46 -0500
parents 0850b5212619 446c46af4b42
children 4197fc428c7d
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d63878346099 maint: Update copyright notices for release.
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1 ## Copyright (C) 1993-2013 John W. Eaton
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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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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {[@var{aa}, @var{bb}, @var{q}, @var{z}] =} qzhess (@var{A}, @var{B})
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21 ## Compute the Hessenberg-triangular decomposition of the matrix pencil
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22 ## @code{(@var{A}, @var{B})}, returning
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23 ## @code{@var{aa} = @var{q} * @var{A} * @var{z}},
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24 ## @code{@var{bb} = @var{q} * @var{B} * @var{z}}, with @var{q} and @var{z}
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25 ## orthogonal. For example:
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26 ##
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27 ## @example
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28 ## @group
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29 ## [aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8])
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30 ## @result{} aa = [ -3.02244, -4.41741; 0.92998, 0.69749 ]
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31 ## @result{} bb = [ -8.60233, -9.99730; 0.00000, -0.23250 ]
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32 ## @result{} q = [ -0.58124, -0.81373; -0.81373, 0.58124 ]
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33 ## @result{} z = [ 1, 0; 0, 1 ]
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34 ## @end group
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35 ## @end example
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36 ##
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37 ## The Hessenberg-triangular decomposition is the first step in
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38 ## @nospell{Moler and Stewart's} QZ@tie{}decomposition algorithm.
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39 ##
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40 ## Algorithm taken from @nospell{Golub and Van Loan},
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41 ## @cite{Matrix Computations, 2nd edition}.
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42 ##
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43 ## @seealso{lu, chol, hess, qr, qz, schur, svd}
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44 ## @end deftypefn
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46 ## Author: A. S. Hodel <scotte@eng.auburn.edu>
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47 ## Created: August 1993
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48 ## Adapted-By: jwe
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50 function [aa, bb, q, z] = qzhess (A, B)
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52 if (nargin != 2)
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53 print_usage ();
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54 endif
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55
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56 [na, ma] = size (A);
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57 [nb, mb] = size (B);
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58 if (na != ma || na != nb || nb != mb)
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59 error ("qzhess: incompatible dimensions");
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60 endif
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61
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62 ## Reduce to hessenberg-triangular form.
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64 [q, bb] = qr (B);
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65 aa = q' * A;
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66 q = q';
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67 z = eye (na);
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68 for j = 1:(na-2)
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69 for i = na:-1:(j+2)
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71 ## disp (["zero out aa(", num2str(i), ",", num2str(j), ")"])
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73 rot = givens (aa (i-1, j), aa (i, j));
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74 aa ((i-1):i, :) = rot *aa ((i-1):i, :);
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75 bb ((i-1):i, :) = rot *bb ((i-1):i, :);
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76 q ((i-1):i, :) = rot *q ((i-1):i, :);
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78 ## disp (["now zero out bb(", num2str(i), ",", num2str(i-1), ")"])
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79
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80 rot = givens (bb (i, i), bb (i, i-1))';
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81 bb (:, (i-1):i) = bb (:, (i-1):i) * rot';
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82 aa (:, (i-1):i) = aa (:, (i-1):i) * rot';
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83 z (:, (i-1):i) = z (:, (i-1):i) * rot';
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84
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85 endfor
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86 endfor
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87
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88 bb (2, 1) = 0.0;
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89 for i = 3:na
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90 bb (i, 1:(i-1)) = zeros (1, i-1);
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91 aa (i, 1:(i-2)) = zeros (1, i-2);
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92 endfor
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93
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94 endfunction
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97 %!test
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98 %! a = [1 2 1 3;
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99 %! 2 5 3 2;
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100 %! 5 5 1 0;
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101 %! 4 0 3 2];
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102 %! b = [0 4 2 1;
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103 %! 2 3 1 1;
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104 %! 1 0 2 1;
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105 %! 2 5 3 2];
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106 %! mask = [0 0 0 0;
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107 %! 0 0 0 0;
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108 %! 1 0 0 0;
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109 %! 1 1 0 0];
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110 %! [aa, bb, q, z] = qzhess (a, b);
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111 %! assert (inv (q) - q', zeros (4), 2e-8);
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112 %! assert (inv (z) - z', zeros (4), 2e-8);
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113 %! assert (q * a * z, aa, 2e-8);
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114 %! assert (aa .* mask, zeros (4), 2e-8);
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115 %! assert (q * b * z, bb, 2e-8);
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116 %! assert (bb .* mask, zeros (4), 2e-8);
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117
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118 %!test
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119 %! a = [1 2 3 4 5;
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120 %! 3 2 3 1 0;
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121 %! 4 3 2 1 1;
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122 %! 0 1 0 1 0;
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123 %! 3 2 1 0 5];
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124 %! b = [5 0 4 0 1;
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125 %! 1 1 1 2 5;
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126 %! 0 3 2 1 0;
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127 %! 4 3 0 3 5;
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128 %! 2 1 2 1 3];
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129 %! mask = [0 0 0 0 0;
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130 %! 0 0 0 0 0;
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131 %! 1 0 0 0 0;
c5c94b63931f codesprint: linear algebra tests: cross, housh, planerot, qzhess, rref
Roman Belov <romblv@gmail.com>
parents: 11593
diff changeset
132 %! 1 1 0 0 0;
c5c94b63931f codesprint: linear algebra tests: cross, housh, planerot, qzhess, rref
Roman Belov <romblv@gmail.com>
parents: 11593
diff changeset
133 %! 1 1 1 0 0];
14363
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
134 %! [aa, bb, q, z] = qzhess (a, b);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
135 %! assert (inv (q) - q', zeros (5), 2e-8);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
136 %! assert (inv (z) - z', zeros (5), 2e-8);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
137 %! assert (q * a * z, aa, 2e-8);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
138 %! assert (aa .* mask, zeros (5), 2e-8);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
139 %! assert (q * b * z, bb, 2e-8);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
140 %! assert (bb .* mask, zeros (5), 2e-8);
13048
c5c94b63931f codesprint: linear algebra tests: cross, housh, planerot, qzhess, rref
Roman Belov <romblv@gmail.com>
parents: 11593
diff changeset
141
14363
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
142 %!error qzhess ([0])
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14327
diff changeset
143 %!error qzhess ()
13048
c5c94b63931f codesprint: linear algebra tests: cross, housh, planerot, qzhess, rref
Roman Belov <romblv@gmail.com>
parents: 11593
diff changeset
144