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
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
|
SUBROUTINE MB04WD( TRANQ1, TRANQ2, M, N, K, Q1, LDQ1, Q2, LDQ2,
$ CS, TAU, DWORK, LDWORK, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To generate a matrix Q with orthogonal columns (spanning an
C isotropic subspace), which is defined as the first n columns
C of a product of symplectic reflectors and Givens rotators,
C
C Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
C diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
C ....
C diag( H(k),H(k) ) G(k) diag( F(k),F(k) ).
C
C The matrix Q is returned in terms of its first 2*M rows
C
C [ op( Q1 ) op( Q2 ) ]
C Q = [ ].
C [ -op( Q2 ) op( Q1 ) ]
C
C Blocked version of the SLICOT Library routine MB04WU.
C
C ARGUMENTS
C
C Mode Parameters
C
C TRANQ1 CHARACTER*1
C Specifies the form of op( Q1 ) as follows:
C = 'N': op( Q1 ) = Q1;
C = 'T': op( Q1 ) = Q1';
C = 'C': op( Q1 ) = Q1'.
C
C TRANQ2 CHARACTER*1
C Specifies the form of op( Q2 ) as follows:
C = 'N': op( Q2 ) = Q2;
C = 'T': op( Q2 ) = Q2';
C = 'C': op( Q2 ) = Q2'.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of rows of the matrices Q1 and Q2. M >= 0.
C
C N (input) INTEGER
C The number of columns of the matrices Q1 and Q2.
C M >= N >= 0.
C
C K (input) INTEGER
C The number of symplectic Givens rotators whose product
C partly defines the matrix Q. N >= K >= 0.
C
C Q1 (input/output) DOUBLE PRECISION array, dimension
C (LDQ1,N) if TRANQ1 = 'N',
C (LDQ1,M) if TRANQ1 = 'T' or TRANQ1 = 'C'
C On entry with TRANQ1 = 'N', the leading M-by-K part of
C this array must contain in its i-th column the vector
C which defines the elementary reflector F(i).
C On entry with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
C K-by-M part of this array must contain in its i-th row
C the vector which defines the elementary reflector F(i).
C On exit with TRANQ1 = 'N', the leading M-by-N part of this
C array contains the matrix Q1.
C On exit with TRANQ1 = 'T' or TRANQ1 = 'C', the leading
C N-by-M part of this array contains the matrix Q1'.
C
C LDQ1 INTEGER
C The leading dimension of the array Q1.
C LDQ1 >= MAX(1,M), if TRANQ1 = 'N';
C LDQ1 >= MAX(1,N), if TRANQ1 = 'T' or TRANQ1 = 'C'.
C
C Q2 (input/output) DOUBLE PRECISION array, dimension
C (LDQ2,N) if TRANQ2 = 'N',
C (LDQ2,M) if TRANQ2 = 'T' or TRANQ2 = 'C'
C On entry with TRANQ2 = 'N', the leading M-by-K part of
C this array must contain in its i-th column the vector
C which defines the elementary reflector H(i) and, on the
C diagonal, the scalar factor of H(i).
C On entry with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
C K-by-M part of this array must contain in its i-th row the
C vector which defines the elementary reflector H(i) and, on
C the diagonal, the scalar factor of H(i).
C On exit with TRANQ2 = 'N', the leading M-by-N part of this
C array contains the matrix Q2.
C On exit with TRANQ2 = 'T' or TRANQ2 = 'C', the leading
C N-by-M part of this array contains the matrix Q2'.
C
C LDQ2 INTEGER
C The leading dimension of the array Q2.
C LDQ2 >= MAX(1,M), if TRANQ2 = 'N';
C LDQ2 >= MAX(1,N), if TRANQ2 = 'T' or TRANQ2 = 'C'.
C
C CS (input) DOUBLE PRECISION array, dimension (2*K)
C On entry, the first 2*K elements of this array must
C contain the cosines and sines of the symplectic Givens
C rotators G(i).
C
C TAU (input) DOUBLE PRECISION array, dimension (K)
C On entry, the first K elements of this array must
C contain the scalar factors of the elementary reflectors
C F(i).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal
C value of LDWORK, MAX(M+N,8*N*NB + 15*NB*NB), where NB is
C the optimal block size determined by the function UE01MD.
C On exit, if INFO = -13, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK. LDWORK >= MAX(1,M+N).
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C REFERENCES
C
C [1] Kressner, D.
C Block algorithms for orthogonal symplectic factorizations.
C BIT, 43 (4), pp. 775-790, 2003.
C
C CONTRIBUTORS
C
C D. Kressner, Technical Univ. Berlin, Germany, and
C P. Benner, Technical Univ. Chemnitz, Germany, December 2003.
C
C REVISIONS
C
C V. Sima, June 2008 (SLICOT version of the HAPACK routine DOSGSB).
C
C KEYWORDS
C
C Elementary matrix operations, orthogonal symplectic matrix.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
C .. Scalar Arguments ..
CHARACTER TRANQ1, TRANQ2
INTEGER INFO, K, LDQ1, LDQ2, LDWORK, M, N
C .. Array Arguments ..
DOUBLE PRECISION CS(*), DWORK(*), Q1(LDQ1,*), Q2(LDQ2,*), TAU(*)
C .. Local Scalars ..
LOGICAL LTRQ1, LTRQ2
INTEGER I, IB, IERR, KI, KK, NB, NBMIN, NX, PDRS, PDT,
$ PDW, WRKOPT
C .. External Functions ..
LOGICAL LSAME
INTEGER UE01MD
EXTERNAL LSAME, UE01MD
C .. External Subroutines ..
EXTERNAL MB04QC, MB04QF, MB04WU, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX, MIN, SQRT
C
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
INFO = 0
LTRQ1 = LSAME( TRANQ1, 'T' ) .OR. LSAME( TRANQ1,'C' )
LTRQ2 = LSAME( TRANQ2, 'T' ) .OR. LSAME( TRANQ2,'C' )
NB = UE01MD( 1, 'MB04WD', TRANQ1 // TRANQ2, M, N, K )
C
C Check the scalar input parameters.
C
IF ( .NOT.( LTRQ1 .OR. LSAME( TRANQ1, 'N' ) ) ) THEN
INFO = -1
ELSE IF ( .NOT.( LTRQ2 .OR. LSAME( TRANQ2, 'N' ) ) ) THEN
INFO = -2
ELSE IF ( M.LT.0 ) THEN
INFO = -3
ELSE IF ( N.LT.0 .OR. N.GT.M ) THEN
INFO = -4
ELSE IF ( K.LT.0 .OR. K.GT.N ) THEN
INFO = -5
ELSE IF ( ( LTRQ1 .AND. LDQ1.LT.MAX( 1, N ) ) .OR.
$ ( .NOT.LTRQ1 .AND. LDQ1.LT.MAX( 1, M ) ) ) THEN
INFO = -7
ELSE IF ( ( LTRQ2 .AND. LDQ2.LT.MAX( 1, N ) ) .OR.
$ ( .NOT.LTRQ2 .AND. LDQ2.LT.MAX( 1, M ) ) ) THEN
INFO = -9
ELSE IF ( LDWORK.LT.MAX( 1, M + N ) ) THEN
DWORK(1) = DBLE( MAX( 1, M + N ) )
INFO = -13
END IF
C
C Return if there were illegal values.
C
IF ( INFO.NE.0 ) THEN
CALL XERBLA( 'MB04WD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 ) THEN
DWORK(1) = ONE
RETURN
END IF
C
NBMIN = 2
NX = 0
WRKOPT = M + N
IF( NB.GT.1 .AND. NB.LT.K ) THEN
C
C Determine when to cross over from blocked to unblocked code.
C
NX = MAX( 0, UE01MD( 3, 'MB04WD', TRANQ1 // TRANQ2, M, N, K ) )
IF ( NX.LT.K ) THEN
C
C Determine if workspace is large enough for blocked code.
C
WRKOPT = MAX( WRKOPT, 8*N*NB + 15*NB*NB )
IF( LDWORK.LT.WRKOPT ) THEN
C
C Not enough workspace to use optimal NB: reduce NB and
C determine the minimum value of NB.
C
NB = INT( ( SQRT( DBLE( 16*N*N + 15*LDWORK ) )
$ - DBLE( 4*N ) ) / 15.0D0 )
NBMIN = MAX( 2, UE01MD( 2, 'MB04WD', TRANQ1 // TRANQ2, M,
$ N, K ) )
END IF
END IF
END IF
C
IF ( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
C
C Use blocked code after the last block.
C The first kk columns are handled by the block method.
C
KI = ( ( K-NX-1 ) / NB )*NB
KK = MIN( K, KI+NB )
ELSE
KK = 0
END IF
C
C Use unblocked code for the last or only block.
C
IF ( KK.LT.N )
$ CALL MB04WU( TRANQ1, TRANQ2, M-KK, N-KK, K-KK, Q1(KK+1,KK+1),
$ LDQ1, Q2(KK+1,KK+1), LDQ2, CS(2*KK+1), TAU(KK+1),
$ DWORK, LDWORK, IERR )
C
C Blocked code.
C
IF ( KK.GT.0 ) THEN
PDRS = 1
PDT = PDRS + 6*NB*NB
PDW = PDT + 9*NB*NB
IF ( LTRQ1.AND.LTRQ2 ) THEN
DO 10 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF ( I+IB.LE.N ) THEN
C
C Form the triangular factors of the symplectic block
C reflector SH.
C
CALL MB04QF( 'Forward', 'Rowwise', 'Rowwise', M-I+1,
$ IB, Q1(I,I), LDQ1, Q2(I,I), LDQ2,
$ CS(2*I-1), TAU(I), DWORK(PDRS), NB,
$ DWORK(PDT), NB, DWORK(PDW) )
C
C Apply SH to Q1(i+ib:n,i:m) and Q2(i+ib:n,i:m) from
C the right.
C
CALL MB04QC( 'Zero Structure', 'Transpose',
$ 'Transpose', 'No Transpose', 'Forward',
$ 'Rowwise', 'Rowwise', M-I+1, N-I-IB+1,
$ IB, Q1(I,I), LDQ1, Q2(I,I), LDQ2,
$ DWORK(PDRS), NB, DWORK(PDT), NB,
$ Q2(I+IB,I), LDQ2, Q1(I+IB,I), LDQ1,
$ DWORK(PDW) )
END IF
C
C Apply SH to columns i:m of the current block.
C
CALL MB04WU( 'Transpose', 'Transpose', M-I+1, IB, IB,
$ Q1(I,I), LDQ1, Q2(I,I), LDQ2, CS(2*I-1),
$ TAU(I), DWORK, LDWORK, IERR )
10 CONTINUE
C
ELSE IF ( LTRQ1 ) THEN
DO 20 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF ( I+IB.LE.N ) THEN
C
C Form the triangular factors of the symplectic block
C reflector SH.
C
CALL MB04QF( 'Forward', 'Rowwise', 'Columnwise',
$ M-I+1, IB, Q1(I,I), LDQ1, Q2(I,I), LDQ2,
$ CS(2*I-1), TAU(I), DWORK(PDRS), NB,
$ DWORK(PDT), NB, DWORK(PDW) )
C
C Apply SH to Q1(i+ib:n,i:m) from the right and to
C Q2(i:m,i+ib:n) from the left.
C
CALL MB04QC( 'Zero Structure', 'No Transpose',
$ 'Transpose', 'No Transpose',
$ 'Forward', 'Rowwise', 'Columnwise',
$ M-I+1, N-I-IB+1, IB, Q1(I,I), LDQ1,
$ Q2(I,I), LDQ2, DWORK(PDRS), NB,
$ DWORK(PDT), NB, Q2(I,I+IB), LDQ2,
$ Q1(I+IB,I), LDQ1, DWORK(PDW) )
END IF
C
C Apply SH to columns/rows i:m of the current block.
C
CALL MB04WU( 'Transpose', 'No Transpose', M-I+1, IB, IB,
$ Q1(I,I), LDQ1, Q2(I,I), LDQ2, CS(2*I-1),
$ TAU(I), DWORK, LDWORK, IERR )
20 CONTINUE
C
ELSE IF ( LTRQ2 ) THEN
DO 30 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF ( I+IB.LE.N ) THEN
C
C Form the triangular factors of the symplectic block
C reflector SH.
C
CALL MB04QF( 'Forward', 'Columnwise', 'Rowwise',
$ M-I+1, IB, Q1(I,I), LDQ1, Q2(I,I), LDQ2,
$ CS(2*I-1), TAU(I), DWORK(PDRS), NB,
$ DWORK(PDT), NB, DWORK(PDW) )
C
C Apply SH to Q1(i:m,i+ib:n) from the left and to
C Q2(i+ib:n,i:m) from the right.
C
CALL MB04QC( 'Zero Structure', 'Transpose',
$ 'No Transpose', 'No Transpose', 'Forward',
$ 'Columnwise', 'Rowwise', M-I+1, N-I-IB+1,
$ IB, Q1(I,I), LDQ1, Q2(I,I), LDQ2,
$ DWORK(PDRS), NB, DWORK(PDT), NB,
$ Q2(I+IB,I), LDQ2, Q1(I,I+IB), LDQ1,
$ DWORK(PDW) )
END IF
C
C Apply SH to columns/rows i:m of the current block.
C
CALL MB04WU( 'No Transpose', 'Transpose', M-I+1, IB, IB,
$ Q1(I,I), LDQ1, Q2(I,I), LDQ2, CS(2*I-1),
$ TAU(I), DWORK, LDWORK, IERR )
30 CONTINUE
C
ELSE
DO 40 I = KI + 1, 1, -NB
IB = MIN( NB, K-I+1 )
IF ( I+IB.LE.N ) THEN
C
C Form the triangular factors of the symplectic block
C reflector SH.
C
CALL MB04QF( 'Forward', 'Columnwise', 'Columnwise',
$ M-I+1, IB, Q1(I,I), LDQ1, Q2(I,I), LDQ2,
$ CS(2*I-1), TAU(I), DWORK(PDRS), NB,
$ DWORK(PDT), NB, DWORK(PDW) )
C
C Apply SH to Q1(i:m,i+ib:n) and Q2(i:m,i+ib:n) from
C the left.
C
CALL MB04QC( 'Zero Structure', 'No Transpose',
$ 'No Transpose', 'No Transpose',
$ 'Forward', 'Columnwise', 'Columnwise',
$ M-I+1, N-I-IB+1, IB, Q1(I,I), LDQ1,
$ Q2(I,I), LDQ2, DWORK(PDRS), NB,
$ DWORK(PDT), NB, Q2(I,I+IB), LDQ2,
$ Q1(I,I+IB), LDQ1, DWORK(PDW) )
END IF
C
C Apply SH to rows i:m of the current block.
C
CALL MB04WU( 'No Transpose', 'No Transpose', M-I+1, IB,
$ IB, Q1(I,I), LDQ1, Q2(I,I), LDQ2, CS(2*I-1),
$ TAU(I), DWORK, LDWORK, IERR )
40 CONTINUE
END IF
END IF
C
DWORK(1) = DBLE( WRKOPT )
C
RETURN
C *** Last line of MB04WD ***
END
|
