|
2847
|
1 ## Copyright (C) 1996, 1997 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 -*- |
|
|
20 ## @deftypefn {Function File} {[@var{aa}, @var{bb}, @var{q}, @var{z}] =} qzhess (@var{a}, @var{b}) |
|
|
21 ## Compute the Hessenberg-triangular decomposition of the matrix pencil |
|
|
22 ## @code{(@var{a}, @var{b})}, returning |
|
3426
|
23 ## @code{@var{aa} = @var{q} * @var{a} * @var{z}}, |
|
3372
|
24 ## @code{@var{bb} = @var{q} * @var{b} * @var{z}}, with @var{q} and @var{z} |
|
|
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 |
|
|
38 ## Moler and Stewart's QZ decomposition algorithm. |
|
3426
|
39 ## |
|
3372
|
40 ## Algorithm taken from Golub and Van Loan, @cite{Matrix Computations, 2nd |
|
|
41 ## edition}. |
|
|
42 ## @end deftypefn |
|
29
|
43 |
|
2312
|
44 ## Author: A. S. Hodel <scotte@eng.auburn.edu> |
|
|
45 ## Created: August 1993 |
|
|
46 ## Adapted-By: jwe |
|
29
|
47 |
|
2312
|
48 function [aa, bb, q, z] = qzhess (a, b) |
|
72
|
49 |
|
54
|
50 if (nargin != 2) |
|
6046
|
51 print_usage (); |
|
54
|
52 endif |
|
|
53 |
|
|
54 [na, ma] = size (a); |
|
|
55 [nb, mb] = size (b); |
|
|
56 if (na != ma || na != nb || nb != mb) |
|
|
57 error ("qzhess: incompatible dimensions"); |
|
|
58 endif |
|
|
59 |
|
2303
|
60 ## Reduce to hessenberg-triangular form. |
|
29
|
61 |
|
54
|
62 [q, bb] = qr (b); |
|
|
63 aa = q' * a; |
|
|
64 q = q'; |
|
|
65 z = eye (na); |
|
|
66 for j = 1:(na-2) |
|
|
67 for i = na:-1:(j+2) |
|
|
68 |
|
2303
|
69 ## disp (["zero out aa(", num2str(i), ",", num2str(j), ")"]) |
|
54
|
70 |
|
|
71 rot = givens (aa (i-1, j), aa (i, j)); |
|
|
72 aa ((i-1):i, :) = rot *aa ((i-1):i, :); |
|
|
73 bb ((i-1):i, :) = rot *bb ((i-1):i, :); |
|
|
74 q ((i-1):i, :) = rot *q ((i-1):i, :); |
|
|
75 |
|
2303
|
76 ## disp (["now zero out bb(", num2str(i), ",", num2str(i-1), ")"]) |
|
54
|
77 |
|
|
78 rot = givens (bb (i, i), bb (i, i-1))'; |
|
|
79 bb (:, (i-1):i) = bb (:, (i-1):i) * rot'; |
|
|
80 aa (:, (i-1):i) = aa (:, (i-1):i) * rot'; |
|
|
81 z (:, (i-1):i) = z (:, (i-1):i) * rot'; |
|
|
82 |
|
|
83 endfor |
|
29
|
84 endfor |
|
|
85 |
|
54
|
86 bb (2, 1) = 0.0; |
|
|
87 for i = 3:na |
|
|
88 bb (i, 1:(i-1)) = zeros (1, i-1); |
|
|
89 aa (i, 1:(i-2)) = zeros (1, i-2); |
|
|
90 endfor |
|
|
91 |
|
29
|
92 endfunction |
