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
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
|
SUBROUTINE SB04RX( RC, UL, M, A, LDA, LAMBD1, LAMBD2, LAMBD3,
$ LAMBD4, D, TOL, IWORK, DWORK, LDDWOR, INFO )
C
C RELEASE 4.0, WGS COPYRIGHT 2000.
C
C PURPOSE
C
C To solve a system of equations in quasi-Hessenberg form
C (Hessenberg form plus two consecutive offdiagonals) with two
C right-hand sides.
C
C ARGUMENTS
C
C Mode Parameters
C
C RC CHARACTER*1
C Indicates processing by columns or rows, as follows:
C = 'R': Row transformations are applied;
C = 'C': Column transformations are applied.
C
C UL CHARACTER*1
C Indicates whether A is upper or lower Hessenberg matrix,
C as follows:
C = 'U': A is upper Hessenberg;
C = 'L': A is lower Hessenberg.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The order of the matrix A. M >= 0.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,M)
C The leading M-by-M part of this array must contain a
C matrix A in Hessenberg form.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,M).
C
C LAMBD1, (input) DOUBLE PRECISION
C LAMBD2, These variables must contain the 2-by-2 block to be
C LAMBD3, multiplied to the elements of A.
C LAMBD4
C
C D (input/output) DOUBLE PRECISION array, dimension (2*M)
C On entry, this array must contain the two right-hand
C side vectors of the quasi-Hessenberg system, stored
C row-wise.
C On exit, if INFO = 0, this array contains the two solution
C vectors of the quasi-Hessenberg system, stored row-wise.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used to test for near singularity of
C the triangular factor R of the quasi-Hessenberg matrix.
C A matrix whose estimated condition number is less
C than 1/TOL is considered to be nonsingular.
C
C Workspace
C
C IWORK INTEGER array, dimension (2*M)
C
C DWORK DOUBLE PRECISION array, dimension (LDDWOR,2*M+3)
C The leading 2*M-by-2*M part of this array is used for
C computing the triangular factor of the QR decomposition
C of the quasi-Hessenberg matrix. The remaining 6*M elements
C are used as workspace for the computation of the
C reciprocal condition estimate.
C
C LDDWOR INTEGER
C The leading dimension of array DWORK.
C LDDWOR >= MAX(1,2*M).
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C = 1: if the quasi-Hessenberg matrix is (numerically)
C singular. That is, its estimated reciprocal
C condition number is less than or equal to TOL.
C
C NUMERICAL ASPECTS
C
C None.
C
C CONTRIBUTORS
C
C D. Sima, University of Bucharest, May 2000.
C
C REVISIONS
C
C -
C
C Note that RC, UL, M, LDA, and LDDWOR must be such that the value
C of the LOGICAL variable OK in the following statement is true.
C
C OK = ( ( UL.EQ.'U' ) .OR. ( UL.EQ.'u' ) .OR.
C ( UL.EQ.'L' ) .OR. ( UL.EQ.'l' ) )
C .AND.
C ( ( RC.EQ.'R' ) .OR. ( RC.EQ.'r' ) .OR.
C ( RC.EQ.'C' ) .OR. ( RC.EQ.'c' ) )
C .AND.
C ( M.GE.0 )
C .AND.
C ( LDA.GE.MAX( 1, M ) )
C .AND.
C ( LDDWOR.GE.MAX( 1, 2*M ) )
C
C These conditions are not checked by the routine.
C
C KEYWORDS
C
C Hessenberg form, orthogonal transformation, real Schur form,
C Sylvester equation.
C
C ******************************************************************
C
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBD1, LAMBD2, LAMBD3, LAMBD4, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)
C .. Local Scalars ..
CHARACTER TRANS
INTEGER J, J1, J2, M2, MJ, ML
DOUBLE PRECISION C, R, RCOND, S
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DLARTG, DLASET, DROT, DSCAL, DTRCON,
$ DTRSV
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN, MOD
C .. Executable Statements ..
C
INFO = 0
C
C For speed, no tests on the input scalar arguments are made.
C Quick return if possible.
C
IF ( M.EQ.0 )
$ RETURN
C
M2 = M*2
IF ( LSAME( UL, 'U' ) ) THEN
C
DO 20 J = 1, M
J2 = J*2
ML = MIN( M, J + 1 )
CALL DLASET( 'Full', M2, 2, ZERO, ZERO, DWORK(1,J2-1),
$ LDDWOR )
CALL DCOPY( ML, A(1,J), 1, DWORK(1,J2-1), 2 )
CALL DSCAL( ML, LAMBD1, DWORK(1,J2-1), 2 )
CALL DCOPY( ML, A(1,J), 1, DWORK(2,J2-1), 2 )
CALL DSCAL( ML, LAMBD3, DWORK(2,J2-1), 2 )
CALL DCOPY( ML, A(1,J), 1, DWORK(1,J2), 2 )
CALL DSCAL( ML, LAMBD2, DWORK(1,J2), 2 )
CALL DCOPY( ML, A(1,J), 1, DWORK(2,J2), 2 )
CALL DSCAL( ML, LAMBD4, DWORK(2,J2), 2 )
C
DWORK(J2-1,J2-1) = DWORK(J2-1,J2-1) + ONE
DWORK(J2,J2) = DWORK(J2,J2) + ONE
20 CONTINUE
C
IF ( LSAME( RC, 'R' ) ) THEN
TRANS = 'N'
C
C A is an upper Hessenberg matrix, row transformations.
C
DO 40 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(J+3,J).NE.ZERO ) THEN
CALL DLARTG( DWORK(J+2,J), DWORK(J+3,J), C, S, R )
DWORK(J+2,J) = R
DWORK(J+3,J) = ZERO
CALL DROT( MJ, DWORK(J+2,J+1), LDDWOR,
$ DWORK(J+3,J+1), LDDWOR, C, S )
CALL DROT( 1, D(J+2), 1, D(J+3), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(J+2,J).NE.ZERO ) THEN
CALL DLARTG( DWORK(J+1,J), DWORK(J+2,J), C, S, R )
DWORK(J+1,J) = R
DWORK(J+2,J) = ZERO
CALL DROT( MJ, DWORK(J+1,J+1), LDDWOR,
$ DWORK(J+2,J+1), LDDWOR, C, S )
CALL DROT( 1, D(J+1), 1, D(J+2), 1, C, S )
END IF
END IF
IF ( DWORK(J+1,J).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J), DWORK(J+1,J), C, S, R )
DWORK(J,J) = R
DWORK(J+1,J) = ZERO
CALL DROT( MJ, DWORK(J,J+1), LDDWOR, DWORK(J+1,J+1),
$ LDDWOR, C, S )
CALL DROT( 1, D(J), 1, D(J+1), 1, C, S )
END IF
40 CONTINUE
C
ELSE
TRANS = 'T'
C
C A is an upper Hessenberg matrix, column transformations.
C
DO 60 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(MJ+1,MJ-2).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ-1), DWORK(MJ+1,MJ-2), C,
$ S, R )
DWORK(MJ+1,MJ-1) = R
DWORK(MJ+1,MJ-2) = ZERO
CALL DROT( MJ, DWORK(1,MJ-1), 1, DWORK(1,MJ-2), 1,
$ C, S )
CALL DROT( 1, D(MJ-1), 1, D(MJ-2), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(MJ+1,MJ-1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ), DWORK(MJ+1,MJ-1), C,
$ S, R )
DWORK(MJ+1,MJ) = R
DWORK(MJ+1,MJ-1) = ZERO
CALL DROT( MJ, DWORK(1,MJ), 1, DWORK(1,MJ-1), 1, C,
$ S )
CALL DROT( 1, D(MJ), 1, D(MJ-1), 1, C, S )
END IF
END IF
IF ( DWORK(MJ+1,MJ).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ+1,MJ), C, S,
$ R )
DWORK(MJ+1,MJ+1) = R
DWORK(MJ+1,MJ) = ZERO
CALL DROT( MJ, DWORK(1,MJ+1), 1, DWORK(1,MJ), 1, C,
$ S )
CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S )
END IF
60 CONTINUE
C
END IF
ELSE
C
DO 80 J = 1, M
J2 = J*2
J1 = MAX( J - 1, 1 )
ML = MIN( M - J + 2, M )
CALL DLASET( 'Full', M2, 2, ZERO, ZERO, DWORK(1,J2-1),
$ LDDWOR )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2-1,J2-1), 2 )
CALL DSCAL( ML, LAMBD1, DWORK(J1*2-1,J2-1), 2 )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2,J2-1), 2 )
CALL DSCAL( ML, LAMBD3, DWORK(J1*2,J2-1), 2 )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2-1,J2), 2 )
CALL DSCAL( ML, LAMBD2, DWORK(J1*2-1,J2), 2 )
CALL DCOPY( ML, A(J1,J), 1, DWORK(J1*2,J2), 2 )
CALL DSCAL( ML, LAMBD4, DWORK(J1*2,J2), 2 )
C
DWORK(J2-1,J2-1) = DWORK(J2-1,J2-1) + ONE
DWORK(J2,J2) = DWORK(J2,J2) + ONE
80 CONTINUE
C
IF ( LSAME( RC, 'R' ) ) THEN
TRANS = 'N'
C
C A is a lower Hessenberg matrix, row transformations.
C
DO 100 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(MJ-2,MJ+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ-1,MJ+1), DWORK(MJ-2,MJ+1), C,
$ S, R )
DWORK(MJ-1,MJ+1) = R
DWORK(MJ-2,MJ+1) = ZERO
CALL DROT( MJ, DWORK(MJ-1,1), LDDWOR,
$ DWORK(MJ-2,1), LDDWOR, C, S )
CALL DROT( 1, D(MJ-1), 1, D(MJ-2), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(MJ-1,MJ+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ,MJ+1), DWORK(MJ-1,MJ+1), C,
$ S, R )
DWORK(MJ,MJ+1) = R
DWORK(MJ-1,MJ+1) = ZERO
CALL DROT( MJ, DWORK(MJ,1), LDDWOR, DWORK(MJ-1,1),
$ LDDWOR, C, S )
CALL DROT( 1, D(MJ), 1, D(MJ-1), 1, C, S )
END IF
END IF
IF ( DWORK(MJ,MJ+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ,MJ+1), C, S,
$ R )
DWORK(MJ+1,MJ+1) = R
DWORK(MJ,MJ+1) = ZERO
CALL DROT( MJ, DWORK(MJ+1,1), LDDWOR, DWORK(MJ,1),
$ LDDWOR, C, S)
CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S )
END IF
100 CONTINUE
C
ELSE
TRANS = 'T'
C
C A is a lower Hessenberg matrix, column transformations.
C
DO 120 J = 1, M2 - 1
MJ = M2 - J
IF ( MOD(J,2).EQ.1 .AND. J.LT.M2-2 ) THEN
IF ( DWORK(J,J+3).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J+2), DWORK(J,J+3), C, S, R )
DWORK(J,J+2) = R
DWORK(J,J+3) = ZERO
CALL DROT( MJ, DWORK(J+1,J+2), 1, DWORK(J+1,J+3),
$ 1, C, S )
CALL DROT( 1, D(J+2), 1, D(J+3), 1, C, S )
END IF
END IF
IF ( J.LT.M2-1 ) THEN
IF ( DWORK(J,J+2).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J+1), DWORK(J,J+2), C, S, R )
DWORK(J,J+1) = R
DWORK(J,J+2) = ZERO
CALL DROT( MJ, DWORK(J+1,J+1), 1, DWORK(J+1,J+2),
$ 1, C, S )
CALL DROT( 1, D(J+1), 1, D(J+2), 1, C, S )
END IF
END IF
IF ( DWORK(J,J+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J), DWORK(J,J+1), C, S, R )
DWORK(J,J) = R
DWORK(J,J+1) = ZERO
CALL DROT( MJ, DWORK(J+1,J), 1, DWORK(J+1,J+1), 1, C,
$ S )
CALL DROT( 1, D(J), 1, D(J+1), 1, C, S )
END IF
120 CONTINUE
C
END IF
END IF
C
CALL DTRCON( '1-norm', UL, 'Non-unit', M2, DWORK, LDDWOR, RCOND,
$ DWORK(1,M2+1), IWORK, INFO )
IF ( RCOND.LE.TOL ) THEN
INFO = 1
ELSE
CALL DTRSV( UL, TRANS, 'Non-unit', M2, DWORK, LDDWOR, D, 1 )
END IF
C
RETURN
C *** Last line of SB04RX ***
END
|
