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
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
|
<HTML>
<HEAD><TITLE>MB03RY - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03RY">MB03RY</A></H2>
<H3>
Solution of a Sylvester equation -AX + XB = C, with A and B in real Schur form, aborting the computations when the norm of X is too large
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To solve the Sylvester equation -AX + XB = C, where A and B are
M-by-M and N-by-N matrices, respectively, in real Schur form.
This routine is intended to be called only by SLICOT Library
routine MB03RD. For efficiency purposes, the computations are
aborted when the infinity norm of an elementary submatrix of X is
greater than a given value PMAX.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03RY( M, N, PMAX, A, LDA, B, LDB, C, LDC, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, M, N
DOUBLE PRECISION PMAX
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
M (input) INTEGER
The order of the matrix A and the number of rows of the
matrices C and X. M >= 0.
N (input) INTEGER
The order of the matrix B and the number of columns of the
matrices C and X. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the infinity norm of an elementary
submatrix of X (see METHOD).
A (input) DOUBLE PRECISION array, dimension (LDA,M)
The leading M-by-M part of this array must contain the
matrix A of the Sylvester equation, in real Schur form.
The elements below the real Schur form are not referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,M).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The leading N-by-N part of this array must contain the
matrix B of the Sylvester equation, in real Schur form.
The elements below the real Schur form are not referenced.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading M-by-N part of this array must
contain the matrix C of the Sylvester equation.
On exit, if INFO = 0, the leading M-by-N part of this
array contains the solution matrix X of the Sylvester
equation, and each elementary submatrix of X (see METHOD)
has the infinity norm less than or equal to PMAX.
On exit, if INFO = 1, the solution matrix X has not been
computed completely, because an elementary submatrix of X
had the infinity norm greater than PMAX. Part of the
matrix C has possibly been overwritten with the
corresponding part of X.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,M).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
= 1: an elementary submatrix of X had the infinity norm
greater than the given value PMAX.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine uses an adaptation of the standard method for solving
Sylvester equations [1], which controls the magnitude of the
individual elements of the computed solution [2]. The equation
-AX + XB = C can be rewritten as
p l-1
-A X + X B = C + sum A X - sum X B
kk kl kl ll kl i=k+1 ki il j=1 kj jl
for l = 1:q, and k = p:-1:1, where A , B , C , and X , are
kk ll kl kl
block submatrices defined by the partitioning induced by the Schur
form of A and B, and p and q are the numbers of the diagonal
blocks of A and B, respectively. So, the elementary submatrices of
X are found block column by block column, starting from the
bottom. If any such elementary submatrix has the infinity norm
greater than the given value PMAX, the calculations are ended.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 2 2
The algorithm requires 0(M N + MN ) operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
Let
( A C ) ( I X )
M = ( ), Y = ( ).
( 0 B ) ( 0 I )
Then
-1 ( A 0 )
Y M Y = ( ),
( 0 B )
hence Y is an non-orthogonal transformation matrix which performs
the reduction of M to a block-diagonal form. Bounding a norm of
X is equivalent to setting an upper bound to the condition number
of the transformation matrix Y.
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
<HR>
<A HREF=support.html><B>Return to Supporting Routines index</B></A></BODY>
</HTML>
|
