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
174
175
176
177
178
179
180
|
<HTML>
<HEAD><TITLE>MB03WA - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03WA">MB03WA</A></H2>
<H3>
Swapping two adjacent diagonal blocks in a periodic real Schur canonical form
</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 swap adjacent diagonal blocks A11*B11 and A22*B22 of size
1-by-1 or 2-by-2 in an upper (quasi) triangular matrix product
A*B by an orthogonal equivalence transformation.
(A, B) must be in periodic real Schur canonical form (as returned
by SLICOT Library routine MB03XP), i.e., A is block upper
triangular with 1-by-1 and 2-by-2 diagonal blocks, and B is upper
triangular.
Optionally, the matrices Q and Z of generalized Schur vectors are
updated.
Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)',
Z(in) * B(in) * Q(in)' = Z(out) * B(out) * Q(out)'.
This routine is largely based on the LAPACK routine DTGEX2
developed by Bo Kagstrom and Peter Poromaa.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03WA( WANTQ, WANTZ, N1, N2, A, LDA, B, LDB, Q, LDQ,
$ Z, LDZ, INFO )
C .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER INFO, LDA, LDB, LDQ, LDZ, N1, N2
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
WANTQ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Q as follows:
= .TRUE. : The matrix Q is updated;
= .FALSE.: the matrix Q is not required.
WANTZ LOGICAL
Indicates whether or not the user wishes to accumulate
the matrix Z as follows:
= .TRUE. : The matrix Z is updated;
= .FALSE.: the matrix Z is not required.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N1 (input) INTEGER
The order of the first block A11*B11. N1 = 0, 1 or 2.
N2 (input) INTEGER
The order of the second block A22*B22. N2 = 0, 1 or 2.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix A.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the matrix A of the reordered pair.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N1+N2).
B (input/output) DOUBLE PRECISION array, dimension
(LDB,N1+N2)
On entry, the leading (N1+N2)-by-(N1+N2) part of this
array must contain the matrix B.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the matrix B of the reordered pair.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N1+N2).
Q (input/output) DOUBLE PRECISION array, dimension
(LDQ,N1+N2)
On entry, if WANTQ = .TRUE., the leading
(N1+N2)-by-(N1+N2) part of this array must contain the
orthogonal matrix Q.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the updated matrix Q. Q will be a rotation
matrix for N1=N2=1.
This array is not referenced if WANTQ = .FALSE..
LDQ INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N1+N2.
Z (input/output) DOUBLE PRECISION array, dimension
(LDZ,N1+N2)
On entry, if WANTZ = .TRUE., the leading
(N1+N2)-by-(N1+N2) part of this array must contain the
orthogonal matrix Z.
On exit, the leading (N1+N2)-by-(N1+N2) part of this array
contains the updated matrix Z. Z will be a rotation
matrix for N1=N2=1.
This array is not referenced if WANTZ = .FALSE..
LDZ INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N1+N2.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
= 1: the transformed matrix (A, B) would be
too far from periodic Schur form; the blocks are
not swapped and (A,B) and (Q,Z) are unchanged.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
In the current code both weak and strong stability tests are
performed. The user can omit the strong stability test by changing
the internal logical parameter WANDS to .FALSE.. See ref. [2] for
details.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Kagstrom, B.
A direct method for reordering eigenvalues in the generalized
real Schur form of a regular matrix pair (A,B), in M.S. Moonen
et al (eds.), Linear Algebra for Large Scale and Real-Time
Applications, Kluwer Academic Publ., 1993, pp. 195-218.
[2] Kagstrom, B., and Poromaa, P.
Computing eigenspaces with specified eigenvalues of a regular
matrix pair (A, B) and condition estimation: Theory,
algorithms and software, Numer. Algorithms, 1996, vol. 12,
pp. 369-407.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</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>
|
