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
|
SUBROUTINE MB01UX( SIDE, UPLO, TRANS, M, N, ALPHA, T, LDT, A, LDA,
$ 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 compute one of the matrix products
C
C A : = alpha*op( T ) * A, or A : = alpha*A * op( T ),
C
C where alpha is a scalar, A is an m-by-n matrix, T is a quasi-
C triangular matrix, and op( T ) is one of
C
C op( T ) = T or op( T ) = T', the transpose of T.
C
C ARGUMENTS
C
C Mode Parameters
C
C SIDE CHARACTER*1
C Specifies whether the upper quasi-triangular matrix H
C appears on the left or right in the matrix product as
C follows:
C = 'L': A := alpha*op( T ) * A;
C = 'R': A := alpha*A * op( T ).
C
C UPLO CHARACTER*1.
C Specifies whether the matrix T is an upper or lower
C quasi-triangular matrix as follows:
C = 'U': T is an upper quasi-triangular matrix;
C = 'L': T is a lower quasi-triangular matrix.
C
C TRANS CHARACTER*1
C Specifies the form of op( T ) to be used in the matrix
C multiplication as follows:
C = 'N': op( T ) = T;
C = 'T': op( T ) = T';
C = 'C': op( T ) = T'.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of rows of the matrix A. M >= 0.
C
C N (input) INTEGER
C The number of columns of the matrix A. N >= 0.
C
C ALPHA (input) DOUBLE PRECISION
C The scalar alpha. When alpha is zero then T is not
C referenced and A need not be set before entry.
C
C T (input) DOUBLE PRECISION array, dimension (LDT,k)
C where k is M when SIDE = 'L' and is N when SIDE = 'R'.
C On entry with UPLO = 'U', the leading k-by-k upper
C Hessenberg part of this array must contain the upper
C quasi-triangular matrix T. The elements below the
C subdiagonal are not referenced.
C On entry with UPLO = 'L', the leading k-by-k lower
C Hessenberg part of this array must contain the lower
C quasi-triangular matrix T. The elements above the
C supdiagonal are not referenced.
C
C LDT INTEGER
C The leading dimension of the array T. LDT >= max(1,k),
C where k is M when SIDE = 'L' and is N when SIDE = 'R'.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading M-by-N part of this array must
C contain the matrix A.
C On exit, the leading M-by-N part of this array contains
C the computed product.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= max(1,M).
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0 and ALPHA<>0, DWORK(1) returns the
C optimal value of LDWORK.
C On exit, if INFO = -12, DWORK(1) returns the minimum
C value of LDWORK.
C This array is not referenced when alpha = 0.
C
C LDWORK The length of the array DWORK.
C LDWORK >= 1, if alpha = 0 or MIN(M,N) = 0;
C LDWORK >= 2*(M-1), if SIDE = 'L';
C LDWORK >= 2*(N-1), if SIDE = 'R'.
C For maximal efficiency LDWORK should be at least
C NOFF*N + M - 1, if SIDE = 'L';
C NOFF*M + N - 1, if SIDE = 'R';
C where NOFF is the number of nonzero elements on the
C subdiagonal (if UPLO = 'U') or supdiagonal (if UPLO = 'L')
C of T.
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 METHOD
C
C The technique used in this routine is similiar to the technique
C used in the SLICOT [1] subroutine MB01UW developed by Vasile Sima.
C The required matrix product is computed in two steps. In the first
C step, the triangle of T specified by UPLO is used; in the second
C step, the contribution of the sub-/supdiagonal is added. If the
C workspace can accommodate parts of A, a fast BLAS 3 DTRMM
C operation is used in the first step.
C
C REFERENCES
C
C [1] Benner, P., Mehrmann, V., Sima, V., Van Huffel, S., and
C Varga, A.
C SLICOT - A subroutine library in systems and control theory.
C In: Applied and computational control, signals, and circuits,
C Vol. 1, pp. 499-539, Birkhauser, Boston, 1999.
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, May 2008 (SLICOT version of the HAPACK routine DTRQML).
C
C KEYWORDS
C
C Elementary matrix operations.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER SIDE, TRANS, UPLO
INTEGER INFO, LDA, LDT, LDWORK, M, N
DOUBLE PRECISION ALPHA
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), T(LDT,*)
C .. Local Scalars ..
LOGICAL LSIDE, LTRAN, LUP
CHARACTER ATRAN
INTEGER I, IERR, J, K, NOFF, PDW, PSAV, WRKMIN, WRKOPT,
$ XDIF
DOUBLE PRECISION TEMP
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DLASCL, DLASET, DTRMM, DTRMV,
$ XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, MIN
C
C .. Executable Statements ..
C
C Decode and test the input scalar arguments.
C
INFO = 0
LSIDE = LSAME( SIDE, 'L' )
LUP = LSAME( UPLO, 'U' )
LTRAN = LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' )
IF ( LSIDE ) THEN
K = M
ELSE
K = N
END IF
WRKMIN = 2*( K - 1 )
C
IF ( ( .NOT.LSIDE ).AND.( .NOT.LSAME( SIDE, 'R' ) ) ) THEN
INFO = -1
ELSE IF ( ( .NOT.LUP ).AND.( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
INFO = -2
ELSE IF ( ( .NOT.LTRAN ).AND.( .NOT.LSAME( TRANS, 'N' ) ) ) THEN
INFO = -3
ELSE IF ( M.LT.0 ) THEN
INFO = -4
ELSE IF ( N.LT.0 ) THEN
INFO = -5
ELSE IF ( LDT.LT.MAX( 1, K ) ) THEN
INFO = -8
ELSE IF ( LDA.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF ( LDWORK.LT.0 .OR.
$ ( ALPHA.NE.ZERO .AND. MIN( M, N ).GT.0 .AND.
$ LDWORK.LT.WRKMIN ) ) THEN
DWORK(1) = DBLE( WRKMIN )
INFO = -12
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return.
C
CALL XERBLA( 'MB01UX', -INFO )
RETURN
END IF
C
C Quick return, if possible.
C
IF ( MIN( M, N ).EQ.0 )
$ RETURN
C
IF ( ALPHA.EQ.ZERO ) THEN
C
C Set A to zero and return.
C
CALL DLASET( 'Full', M, N, ZERO, ZERO, A, LDA )
RETURN
END IF
C
C Save and count off-diagonal entries of T.
C
IF ( LUP ) THEN
CALL DCOPY( K-1, T(2,1), LDT+1, DWORK, 1 )
ELSE
CALL DCOPY( K-1, T(1,2), LDT+1, DWORK, 1 )
END IF
NOFF = 0
DO 5 I = 1, K-1
IF ( DWORK(I).NE.ZERO )
$ NOFF = NOFF + 1
5 CONTINUE
C
C Compute optimal workspace.
C
IF ( LSIDE ) THEN
WRKOPT = NOFF*N + M - 1
ELSE
WRKOPT = NOFF*M + N - 1
END IF
PSAV = K
IF ( .NOT.LTRAN ) THEN
XDIF = 0
ELSE
XDIF = 1
END IF
IF ( .NOT.LUP )
$ XDIF = 1 - XDIF
IF ( .NOT.LSIDE )
$ XDIF = 1 - XDIF
C
IF ( LDWORK.GE.WRKOPT ) THEN
C
C Enough workspace for a fast BLAS 3 calculation.
C Save relevant parts of A in the workspace and compute one of
C the matrix products
C A : = alpha*op( triu( T ) ) * A, or
C A : = alpha*A * op( triu( T ) ),
C involving the upper/lower triangle of T.
C
PDW = PSAV
IF ( LSIDE ) THEN
DO 20 J = 1, N
DO 10 I = 1, M-1
IF ( DWORK(I).NE.ZERO ) THEN
DWORK(PDW) = A(I+XDIF,J)
PDW = PDW + 1
END IF
10 CONTINUE
20 CONTINUE
ELSE
DO 30 J = 1, N-1
IF ( DWORK(J).NE.ZERO ) THEN
CALL DCOPY( M, A(1,J+XDIF), 1, DWORK(PDW), 1 )
PDW = PDW + M
END IF
30 CONTINUE
END IF
CALL DTRMM( SIDE, UPLO, TRANS, 'Non-unit', M, N, ALPHA, T,
$ LDT, A, LDA )
C
C Add the contribution of the offdiagonal of T.
C
PDW = PSAV
XDIF = 1 - XDIF
IF( LSIDE ) THEN
DO 50 J = 1, N
DO 40 I = 1, M-1
TEMP = DWORK(I)
IF ( TEMP.NE.ZERO ) THEN
A(I+XDIF,J) = A(I+XDIF,J) + ALPHA * TEMP *
$ DWORK(PDW)
PDW = PDW + 1
END IF
40 CONTINUE
50 CONTINUE
ELSE
DO 60 J = 1, N-1
TEMP = DWORK(J)*ALPHA
IF ( TEMP.NE.ZERO ) THEN
CALL DAXPY( M, TEMP, DWORK(PDW), 1, A(1,J+XDIF), 1 )
PDW = PDW + M
END IF
60 CONTINUE
END IF
ELSE
C
C Use a BLAS 2 calculation.
C
IF ( LSIDE ) THEN
DO 80 J = 1, N
C
C Compute the contribution of the offdiagonal of T to
C the j-th column of the product.
C
DO 70 I = 1, M - 1
DWORK(PSAV+I-1) = DWORK(I)*A(I+XDIF,J)
70 CONTINUE
C
C Multiply the triangle of T by the j-th column of A,
C and add to the above result.
C
CALL DTRMV( UPLO, TRANS, 'Non-unit', M, T, LDT, A(1,J),
$ 1 )
CALL DAXPY( M-1, ONE, DWORK(PSAV), 1, A(2-XDIF,J), 1 )
80 CONTINUE
ELSE
IF ( LTRAN ) THEN
ATRAN = 'N'
ELSE
ATRAN = 'T'
END IF
DO 100 I = 1, M
C
C Compute the contribution of the offdiagonal of T to
C the i-th row of the product.
C
DO 90 J = 1, N - 1
DWORK(PSAV+J-1) = A(I,J+XDIF)*DWORK(J)
90 CONTINUE
C
C Multiply the i-th row of A by the triangle of T,
C and add to the above result.
C
CALL DTRMV( UPLO, ATRAN, 'Non-unit', N, T, LDT, A(I,1),
$ LDA )
CALL DAXPY( N-1, ONE, DWORK(PSAV), 1, A(I,2-XDIF), LDA )
100 CONTINUE
END IF
C
C Scale the result by alpha.
C
IF ( ALPHA.NE.ONE )
$ CALL DLASCL( 'General', 0, 0, ONE, ALPHA, M, N, A, LDA,
$ IERR )
END IF
DWORK(1) = DBLE( MAX( WRKMIN, WRKOPT ) )
RETURN
C *** Last line of MB01UX ***
END
|
