1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
|
## Copyright (C) 1996 John W. Eaton
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2, or (at your option)
## any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, write to the Free
## Software Foundation, 59 Temple Place - Suite 330, Boston, MA
## 02111-1307, USA.
## Usage: x = dlyap (a, b)
##
## Solve a x a' - x + b = 0 (discrete Lyapunov equation) for square
## matrices a and b. If b is not square, then the function attempts
## to solve either
##
## a x a' - x + b b' = 0
##
## or
##
## a' x a - x + b' b = 0
##
## whichever is appropriate. Uses Schur decomposition as in Kitagawa
## (1977).
## Author: A. S. Hodel <scotte@eng.auburn.edu>
## Created: August 1993
## Adapted-By: jwe
function x = dlyap (a, b)
if ((n = is_square (a)) == 0)
warning ("dlyap: a must be square");
endif
if ((m = is_square (b)) == 0)
[n1, m] = size (b);
if (n1 == n)
b = b*b';
m = n1;
else
b = b'*b;
a = a';
endif
endif
if (n != m)
warning ("dlyap: a,b not conformably dimensioned");
endif
## Solve the equation column by column.
[u, s] = schur (a);
b = u'*b*u;
j = n;
while (j > 0)
j1 = j;
## Check for Schur block.
if (j == 1)
blksiz = 1;
elseif (s (j, j-1) != 0)
blksiz = 2;
j = j - 1;
else
blksiz = 1;
endif
Ajj = kron (s (j:j1, j:j1), s) - eye (blksiz*n);
rhs = reshape (b (:, j:j1), blksiz*n, 1);
if (j1 < n)
rhs2 = s*(x (:, (j1+1):n) * s (j:j1, (j1+1):n)');
rhs = rhs + reshape (rhs2, blksiz*n, 1);
endif
v = - Ajj\rhs;
x (:, j) = v (1:n);
if(blksiz == 2)
x (:, j1) = v ((n+1):blksiz*n);
endif
j = j - 1;
endwhile
## Back-transform to original coordinates.
x = u*x*u';
endfunction
|
