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
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
|
SUBROUTINE SB03MV( LTRAN, LUPPER, T, LDT, B, LDB, SCALE, X, LDX,
$ XNORM, INFO )
C
C RELEASE 4.0, WGS COPYRIGHT 1999.
C
C PURPOSE
C
C To solve for the 2-by-2 symmetric matrix X in
C
C op(T)'*X*op(T) - X = SCALE*B,
C
C where T is 2-by-2, B is symmetric 2-by-2, and op(T) = T or T',
C where T' denotes the transpose of T.
C
C ARGUMENTS
C
C Mode Parameters
C
C LTRAN LOGICAL
C Specifies the form of op(T) to be used, as follows:
C = .FALSE.: op(T) = T,
C = .TRUE. : op(T) = T'.
C
C LUPPER LOGICAL
C Specifies which triangle of the matrix B is used, and
C which triangle of the matrix X is computed, as follows:
C = .TRUE. : The upper triangular part;
C = .FALSE.: The lower triangular part.
C
C Input/Output Parameters
C
C T (input) DOUBLE PRECISION array, dimension (LDT,2)
C The leading 2-by-2 part of this array must contain the
C matrix T.
C
C LDT INTEGER
C The leading dimension of array T. LDT >= 2.
C
C B (input) DOUBLE PRECISION array, dimension (LDB,2)
C On entry with LUPPER = .TRUE., the leading 2-by-2 upper
C triangular part of this array must contain the upper
C triangular part of the symmetric matrix B and the strictly
C lower triangular part of B is not referenced.
C On entry with LUPPER = .FALSE., the leading 2-by-2 lower
C triangular part of this array must contain the lower
C triangular part of the symmetric matrix B and the strictly
C upper triangular part of B is not referenced.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= 2.
C
C SCALE (output) DOUBLE PRECISION
C The scale factor. SCALE is chosen less than or equal to 1
C to prevent the solution overflowing.
C
C X (output) DOUBLE PRECISION array, dimension (LDX,2)
C On exit with LUPPER = .TRUE., the leading 2-by-2 upper
C triangular part of this array contains the upper
C triangular part of the symmetric solution matrix X and the
C strictly lower triangular part of X is not referenced.
C On exit with LUPPER = .FALSE., the leading 2-by-2 lower
C triangular part of this array contains the lower
C triangular part of the symmetric solution matrix X and the
C strictly upper triangular part of X is not referenced.
C Note that X may be identified with B in the calling
C statement.
C
C LDX INTEGER
C The leading dimension of array X. LDX >= 2.
C
C XNORM (output) DOUBLE PRECISION
C The infinity-norm of the solution.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C = 1: if T has almost reciprocal eigenvalues, so T
C is perturbed to get a nonsingular equation.
C
C NOTE: In the interests of speed, this routine does not
C check the inputs for errors.
C
C METHOD
C
C The equivalent linear algebraic system of equations is formed and
C solved using Gaussian elimination with complete pivoting.
C
C REFERENCES
C
C [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
C Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
C Ostrouchov, S., and Sorensen, D.
C LAPACK Users' Guide: Second Edition.
C SIAM, Philadelphia, 1995.
C
C NUMERICAL ASPECTS
C
C The algorithm is stable and reliable, since Gaussian elimination
C with complete pivoting is used.
C
C CONTRIBUTOR
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, May 1997.
C Based on DLALD2 by P. Petkov, Tech. University of Sofia, September
C 1993.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Discrete-time system, Lyapunov equation, matrix algebra.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO, FOUR
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ FOUR = 4.0D+0 )
C ..
C .. Scalar Arguments ..
LOGICAL LTRAN, LUPPER
INTEGER INFO, LDB, LDT, LDX
DOUBLE PRECISION SCALE, XNORM
C ..
C .. Array Arguments ..
DOUBLE PRECISION B( LDB, * ), T( LDT, * ), X( LDX, * )
C ..
C .. Local Scalars ..
INTEGER I, IP, IPSV, J, JP, JPSV, K
DOUBLE PRECISION EPS, SMIN, SMLNUM, TEMP, XMAX
C ..
C .. Local Arrays ..
INTEGER JPIV( 3 )
DOUBLE PRECISION BTMP( 3 ), T9( 3, 3 ), TMP( 3 )
C ..
C .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
C ..
C .. External Subroutines ..
EXTERNAL DSWAP
C ..
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX
C ..
C .. Executable Statements ..
C
C Do not check the input parameters for errors.
C
INFO = 0
C
C Set constants to control overflow.
C
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
C
C Solve equivalent 3-by-3 system using complete pivoting.
C Set pivots less than SMIN to SMIN.
C
SMIN = MAX( ABS( T( 1, 1 ) ), ABS( T( 1, 2 ) ),
$ ABS( T( 2, 1 ) ), ABS( T( 2, 2 ) ) )
SMIN = MAX( EPS*SMIN, SMLNUM )
T9( 1, 1 ) = T( 1, 1 )*T( 1, 1 ) - ONE
T9( 2, 2 ) = T( 1, 1 )*T( 2, 2 ) + T( 1, 2 )*T( 2, 1 ) - ONE
T9( 3, 3 ) = T( 2, 2 )*T( 2, 2 ) - ONE
IF( LTRAN ) THEN
T9( 1, 2 ) = T( 1, 1 )*T( 1, 2 ) + T( 1, 1 )*T( 1, 2 )
T9( 1, 3 ) = T( 1, 2 )*T( 1, 2 )
T9( 2, 1 ) = T( 1, 1 )*T( 2, 1 )
T9( 2, 3 ) = T( 1, 2 )*T( 2, 2 )
T9( 3, 1 ) = T( 2, 1 )*T( 2, 1 )
T9( 3, 2 ) = T( 2, 1 )*T( 2, 2 ) + T( 2, 1 )*T( 2, 2 )
ELSE
T9( 1, 2 ) = T( 1, 1 )*T( 2, 1 ) + T( 1, 1 )*T( 2, 1 )
T9( 1, 3 ) = T( 2, 1 )*T( 2, 1 )
T9( 2, 1 ) = T( 1, 1 )*T( 1, 2 )
T9( 2, 3 ) = T( 2, 1 )*T( 2, 2 )
T9( 3, 1 ) = T( 1, 2 )*T( 1, 2 )
T9( 3, 2 ) = T( 1, 2 )*T( 2, 2 ) + T( 1, 2 )*T( 2, 2 )
END IF
BTMP( 1 ) = B( 1, 1 )
IF ( LUPPER ) THEN
BTMP( 2 ) = B( 1, 2 )
ELSE
BTMP( 2 ) = B( 2, 1 )
END IF
BTMP( 3 ) = B( 2, 2 )
C
C Perform elimination.
C
DO 50 I = 1, 2
XMAX = ZERO
C
DO 20 IP = I, 3
C
DO 10 JP = I, 3
IF( ABS( T9( IP, JP ) ).GE.XMAX ) THEN
XMAX = ABS( T9( IP, JP ) )
IPSV = IP
JPSV = JP
END IF
10 CONTINUE
C
20 CONTINUE
C
IF( IPSV.NE.I ) THEN
CALL DSWAP( 3, T9( IPSV, 1 ), 3, T9( I, 1 ), 3 )
TEMP = BTMP( I )
BTMP( I ) = BTMP( IPSV )
BTMP( IPSV ) = TEMP
END IF
IF( JPSV.NE.I )
$ CALL DSWAP( 3, T9( 1, JPSV ), 1, T9( 1, I ), 1 )
JPIV( I ) = JPSV
IF( ABS( T9( I, I ) ).LT.SMIN ) THEN
INFO = 1
T9( I, I ) = SMIN
END IF
C
DO 40 J = I + 1, 3
T9( J, I ) = T9( J, I ) / T9( I, I )
BTMP( J ) = BTMP( J ) - T9( J, I )*BTMP( I )
C
DO 30 K = I + 1, 3
T9( J, K ) = T9( J, K ) - T9( J, I )*T9( I, K )
30 CONTINUE
C
40 CONTINUE
C
50 CONTINUE
C
IF( ABS( T9( 3, 3 ) ).LT.SMIN )
$ T9( 3, 3 ) = SMIN
SCALE = ONE
IF( ( FOUR*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T9( 1, 1 ) ) .OR.
$ ( FOUR*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T9( 2, 2 ) ) .OR.
$ ( FOUR*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T9( 3, 3 ) ) ) THEN
SCALE = ( ONE / FOUR ) / MAX( ABS( BTMP( 1 ) ),
$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ) )
BTMP( 1 ) = BTMP( 1 )*SCALE
BTMP( 2 ) = BTMP( 2 )*SCALE
BTMP( 3 ) = BTMP( 3 )*SCALE
END IF
C
DO 70 I = 1, 3
K = 4 - I
TEMP = ONE / T9( K, K )
TMP( K ) = BTMP( K )*TEMP
C
DO 60 J = K + 1, 3
TMP( K ) = TMP( K ) - ( TEMP*T9( K, J ) )*TMP( J )
60 CONTINUE
C
70 CONTINUE
C
DO 80 I = 1, 2
IF( JPIV( 3-I ).NE.3-I ) THEN
TEMP = TMP( 3-I )
TMP( 3-I ) = TMP( JPIV( 3-I ) )
TMP( JPIV( 3-I ) ) = TEMP
END IF
80 CONTINUE
C
X( 1, 1 ) = TMP( 1 )
IF ( LUPPER ) THEN
X( 1, 2 ) = TMP( 2 )
ELSE
X( 2, 1 ) = TMP( 2 )
END IF
X( 2, 2 ) = TMP( 3 )
XNORM = MAX( ABS( TMP( 1 ) ) + ABS( TMP( 2 ) ),
$ ABS( TMP( 2 ) ) + ABS( TMP( 3 ) ) )
C
RETURN
C *** Last line of SB03MV ***
END
|
