Mercurial > octave-antonio
annotate scripts/linear-algebra/qzhess.m @ 19596:0e1f5a750d00
maint: Periodic merge of gui-release to default.
author | John W. Eaton <jwe@octave.org> |
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date | Tue, 20 Jan 2015 10:24:46 -0500 |
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1 ## Copyright (C) 1993-2013 John W. Eaton |
2313 | 2 ## |
3 ## This file is part of Octave. | |
4 ## | |
5 ## Octave is free software; you can redistribute it and/or modify it | |
6 ## under the terms of the GNU General Public License as published by | |
7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
8 ## your option) any later version. | |
2313 | 9 ## |
10 ## Octave is distributed in the hope that it will be useful, but | |
11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
13 ## General Public License for more details. | |
14 ## | |
15 ## You should have received a copy of the GNU General Public License | |
7016 | 16 ## along with Octave; see the file COPYING. If not, see |
17 ## <http://www.gnu.org/licenses/>. | |
245 | 18 |
3372 | 19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {[@var{aa}, @var{bb}, @var{q}, @var{z}] =} qzhess (@var{A}, @var{B}) |
3372 | 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: |
3426 | 26 ## |
3372 | 27 ## @example |
28 ## @group | |
29 ## [aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8]) | |
30 ## @result{} aa = [ -3.02244, -4.41741; 0.92998, 0.69749 ] | |
31 ## @result{} bb = [ -8.60233, -9.99730; 0.00000, -0.23250 ] | |
32 ## @result{} q = [ -0.58124, -0.81373; -0.81373, 0.58124 ] | |
33 ## @result{} z = [ 1, 0; 0, 1 ] | |
34 ## @end group | |
35 ## @end example | |
3426 | 36 ## |
3372 | 37 ## The Hessenberg-triangular decomposition is the first step in |
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38 ## @nospell{Moler and Stewart's} QZ@tie{}decomposition algorithm. |
3426 | 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} |
3372 | 44 ## @end deftypefn |
29 | 45 |
2312 | 46 ## Author: A. S. Hodel <scotte@eng.auburn.edu> |
47 ## Created: August 1993 | |
48 ## Adapted-By: jwe | |
29 | 49 |
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50 function [aa, bb, q, z] = qzhess (A, B) |
72 | 51 |
54 | 52 if (nargin != 2) |
6046 | 53 print_usage (); |
54 | 54 endif |
55 | |
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56 [na, ma] = size (A); |
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57 [nb, mb] = size (B); |
54 | 58 if (na != ma || na != nb || nb != mb) |
59 error ("qzhess: incompatible dimensions"); | |
60 endif | |
61 | |
2303 | 62 ## Reduce to hessenberg-triangular form. |
29 | 63 |
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64 [q, bb] = qr (B); |
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65 aa = q' * A; |
54 | 66 q = q'; |
67 z = eye (na); | |
68 for j = 1:(na-2) | |
69 for i = na:-1:(j+2) | |
70 | |
2303 | 71 ## disp (["zero out aa(", num2str(i), ",", num2str(j), ")"]) |
54 | 72 |
73 rot = givens (aa (i-1, j), aa (i, j)); | |
74 aa ((i-1):i, :) = rot *aa ((i-1):i, :); | |
75 bb ((i-1):i, :) = rot *bb ((i-1):i, :); | |
76 q ((i-1):i, :) = rot *q ((i-1):i, :); | |
77 | |
2303 | 78 ## disp (["now zero out bb(", num2str(i), ",", num2str(i-1), ")"]) |
54 | 79 |
80 rot = givens (bb (i, i), bb (i, i-1))'; | |
81 bb (:, (i-1):i) = bb (:, (i-1):i) * rot'; | |
82 aa (:, (i-1):i) = aa (:, (i-1):i) * rot'; | |
83 z (:, (i-1):i) = z (:, (i-1):i) * rot'; | |
84 | |
85 endfor | |
29 | 86 endfor |
87 | |
54 | 88 bb (2, 1) = 0.0; |
89 for i = 3:na | |
90 bb (i, 1:(i-1)) = zeros (1, i-1); | |
91 aa (i, 1:(i-2)) = zeros (1, i-2); | |
92 endfor | |
93 | |
29 | 94 endfunction |
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95 |
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96 |
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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; |
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132 %! 1 1 0 0 0; |
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133 %! 1 1 1 0 0]; |
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134 %! [aa, bb, q, z] = qzhess (a, b); |
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135 %! assert (inv (q) - q', zeros (5), 2e-8); |
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136 %! assert (inv (z) - z', zeros (5), 2e-8); |
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137 %! assert (q * a * z, aa, 2e-8); |
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138 %! assert (aa .* mask, zeros (5), 2e-8); |
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139 %! assert (q * b * z, bb, 2e-8); |
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141 |
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142 %!error qzhess ([0]) |
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143 %!error qzhess () |
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