1 SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
2 $ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
3 $ Q, LDQ, WORK, NCYCLE, INFO )
4 *
5 * -- LAPACK routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2009 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBQ, JOBU, JOBV
12 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
13 $ NCYCLE, P
14 DOUBLE PRECISION TOLA, TOLB
15 * ..
16 * .. Array Arguments ..
17 DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
18 $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
19 $ V( LDV, * ), WORK( * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * DTGSJA computes the generalized singular value decomposition (GSVD)
26 * of two real upper triangular (or trapezoidal) matrices A and B.
27 *
28 * On entry, it is assumed that matrices A and B have the following
29 * forms, which may be obtained by the preprocessing subroutine DGGSVP
30 * from a general M-by-N matrix A and P-by-N matrix B:
31 *
32 * N-K-L K L
33 * A = K ( 0 A12 A13 ) if M-K-L >= 0;
34 * L ( 0 0 A23 )
35 * M-K-L ( 0 0 0 )
36 *
37 * N-K-L K L
38 * A = K ( 0 A12 A13 ) if M-K-L < 0;
39 * M-K ( 0 0 A23 )
40 *
41 * N-K-L K L
42 * B = L ( 0 0 B13 )
43 * P-L ( 0 0 0 )
44 *
45 * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
46 * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
47 * otherwise A23 is (M-K)-by-L upper trapezoidal.
48 *
49 * On exit,
50 *
51 * U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
52 *
53 * where U, V and Q are orthogonal matrices.
54 * R is a nonsingular upper triangular matrix, and D1 and D2 are
55 * ``diagonal'' matrices, which are of the following structures:
56 *
57 * If M-K-L >= 0,
58 *
59 * K L
60 * D1 = K ( I 0 )
61 * L ( 0 C )
62 * M-K-L ( 0 0 )
63 *
64 * K L
65 * D2 = L ( 0 S )
66 * P-L ( 0 0 )
67 *
68 * N-K-L K L
69 * ( 0 R ) = K ( 0 R11 R12 ) K
70 * L ( 0 0 R22 ) L
71 *
72 * where
73 *
74 * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
75 * S = diag( BETA(K+1), ... , BETA(K+L) ),
76 * C**2 + S**2 = I.
77 *
78 * R is stored in A(1:K+L,N-K-L+1:N) on exit.
79 *
80 * If M-K-L < 0,
81 *
82 * K M-K K+L-M
83 * D1 = K ( I 0 0 )
84 * M-K ( 0 C 0 )
85 *
86 * K M-K K+L-M
87 * D2 = M-K ( 0 S 0 )
88 * K+L-M ( 0 0 I )
89 * P-L ( 0 0 0 )
90 *
91 * N-K-L K M-K K+L-M
92 * ( 0 R ) = K ( 0 R11 R12 R13 )
93 * M-K ( 0 0 R22 R23 )
94 * K+L-M ( 0 0 0 R33 )
95 *
96 * where
97 * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
98 * S = diag( BETA(K+1), ... , BETA(M) ),
99 * C**2 + S**2 = I.
100 *
101 * R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
102 * ( 0 R22 R23 )
103 * in B(M-K+1:L,N+M-K-L+1:N) on exit.
104 *
105 * The computation of the orthogonal transformation matrices U, V or Q
106 * is optional. These matrices may either be formed explicitly, or they
107 * may be postmultiplied into input matrices U1, V1, or Q1.
108 *
109 * Arguments
110 * =========
111 *
112 * JOBU (input) CHARACTER*1
113 * = 'U': U must contain an orthogonal matrix U1 on entry, and
114 * the product U1*U is returned;
115 * = 'I': U is initialized to the unit matrix, and the
116 * orthogonal matrix U is returned;
117 * = 'N': U is not computed.
118 *
119 * JOBV (input) CHARACTER*1
120 * = 'V': V must contain an orthogonal matrix V1 on entry, and
121 * the product V1*V is returned;
122 * = 'I': V is initialized to the unit matrix, and the
123 * orthogonal matrix V is returned;
124 * = 'N': V is not computed.
125 *
126 * JOBQ (input) CHARACTER*1
127 * = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
128 * the product Q1*Q is returned;
129 * = 'I': Q is initialized to the unit matrix, and the
130 * orthogonal matrix Q is returned;
131 * = 'N': Q is not computed.
132 *
133 * M (input) INTEGER
134 * The number of rows of the matrix A. M >= 0.
135 *
136 * P (input) INTEGER
137 * The number of rows of the matrix B. P >= 0.
138 *
139 * N (input) INTEGER
140 * The number of columns of the matrices A and B. N >= 0.
141 *
142 * K (input) INTEGER
143 * L (input) INTEGER
144 * K and L specify the subblocks in the input matrices A and B:
145 * A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
146 * of A and B, whose GSVD is going to be computed by DTGSJA.
147 * See Further Details.
148 *
149 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
150 * On entry, the M-by-N matrix A.
151 * On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
152 * matrix R or part of R. See Purpose for details.
153 *
154 * LDA (input) INTEGER
155 * The leading dimension of the array A. LDA >= max(1,M).
156 *
157 * B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
158 * On entry, the P-by-N matrix B.
159 * On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
160 * a part of R. See Purpose for details.
161 *
162 * LDB (input) INTEGER
163 * The leading dimension of the array B. LDB >= max(1,P).
164 *
165 * TOLA (input) DOUBLE PRECISION
166 * TOLB (input) DOUBLE PRECISION
167 * TOLA and TOLB are the convergence criteria for the Jacobi-
168 * Kogbetliantz iteration procedure. Generally, they are the
169 * same as used in the preprocessing step, say
170 * TOLA = max(M,N)*norm(A)*MAZHEPS,
171 * TOLB = max(P,N)*norm(B)*MAZHEPS.
172 *
173 * ALPHA (output) DOUBLE PRECISION array, dimension (N)
174 * BETA (output) DOUBLE PRECISION array, dimension (N)
175 * On exit, ALPHA and BETA contain the generalized singular
176 * value pairs of A and B;
177 * ALPHA(1:K) = 1,
178 * BETA(1:K) = 0,
179 * and if M-K-L >= 0,
180 * ALPHA(K+1:K+L) = diag(C),
181 * BETA(K+1:K+L) = diag(S),
182 * or if M-K-L < 0,
183 * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
184 * BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
185 * Furthermore, if K+L < N,
186 * ALPHA(K+L+1:N) = 0 and
187 * BETA(K+L+1:N) = 0.
188 *
189 * U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
190 * On entry, if JOBU = 'U', U must contain a matrix U1 (usually
191 * the orthogonal matrix returned by DGGSVP).
192 * On exit,
193 * if JOBU = 'I', U contains the orthogonal matrix U;
194 * if JOBU = 'U', U contains the product U1*U.
195 * If JOBU = 'N', U is not referenced.
196 *
197 * LDU (input) INTEGER
198 * The leading dimension of the array U. LDU >= max(1,M) if
199 * JOBU = 'U'; LDU >= 1 otherwise.
200 *
201 * V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
202 * On entry, if JOBV = 'V', V must contain a matrix V1 (usually
203 * the orthogonal matrix returned by DGGSVP).
204 * On exit,
205 * if JOBV = 'I', V contains the orthogonal matrix V;
206 * if JOBV = 'V', V contains the product V1*V.
207 * If JOBV = 'N', V is not referenced.
208 *
209 * LDV (input) INTEGER
210 * The leading dimension of the array V. LDV >= max(1,P) if
211 * JOBV = 'V'; LDV >= 1 otherwise.
212 *
213 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
214 * On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
215 * the orthogonal matrix returned by DGGSVP).
216 * On exit,
217 * if JOBQ = 'I', Q contains the orthogonal matrix Q;
218 * if JOBQ = 'Q', Q contains the product Q1*Q.
219 * If JOBQ = 'N', Q is not referenced.
220 *
221 * LDQ (input) INTEGER
222 * The leading dimension of the array Q. LDQ >= max(1,N) if
223 * JOBQ = 'Q'; LDQ >= 1 otherwise.
224 *
225 * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
226 *
227 * NCYCLE (output) INTEGER
228 * The number of cycles required for convergence.
229 *
230 * INFO (output) INTEGER
231 * = 0: successful exit
232 * < 0: if INFO = -i, the i-th argument had an illegal value.
233 * = 1: the procedure does not converge after MAXIT cycles.
234 *
235 * Internal Parameters
236 * ===================
237 *
238 * MAXIT INTEGER
239 * MAXIT specifies the total loops that the iterative procedure
240 * may take. If after MAXIT cycles, the routine fails to
241 * converge, we return INFO = 1.
242 *
243 * Further Details
244 * ===============
245 *
246 * DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
247 * min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
248 * matrix B13 to the form:
249 *
250 * U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
251 *
252 * where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
253 * of Z. C1 and S1 are diagonal matrices satisfying
254 *
255 * C1**2 + S1**2 = I,
256 *
257 * and R1 is an L-by-L nonsingular upper triangular matrix.
258 *
259 * =====================================================================
260 *
261 * .. Parameters ..
262 INTEGER MAXIT
263 PARAMETER ( MAXIT = 40 )
264 DOUBLE PRECISION ZERO, ONE
265 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
266 * ..
267 * .. Local Scalars ..
268 *
269 LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
270 INTEGER I, J, KCYCLE
271 DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
272 $ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
273 * ..
274 * .. External Functions ..
275 LOGICAL LSAME
276 EXTERNAL LSAME
277 * ..
278 * .. External Subroutines ..
279 EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET, DROT,
280 $ DSCAL, XERBLA
281 * ..
282 * .. Intrinsic Functions ..
283 INTRINSIC ABS, MAX, MIN
284 * ..
285 * .. Executable Statements ..
286 *
287 * Decode and test the input parameters
288 *
289 INITU = LSAME( JOBU, 'I' )
290 WANTU = INITU .OR. LSAME( JOBU, 'U' )
291 *
292 INITV = LSAME( JOBV, 'I' )
293 WANTV = INITV .OR. LSAME( JOBV, 'V' )
294 *
295 INITQ = LSAME( JOBQ, 'I' )
296 WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
297 *
298 INFO = 0
299 IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
300 INFO = -1
301 ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
302 INFO = -2
303 ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
304 INFO = -3
305 ELSE IF( M.LT.0 ) THEN
306 INFO = -4
307 ELSE IF( P.LT.0 ) THEN
308 INFO = -5
309 ELSE IF( N.LT.0 ) THEN
310 INFO = -6
311 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
312 INFO = -10
313 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
314 INFO = -12
315 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
316 INFO = -18
317 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
318 INFO = -20
319 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
320 INFO = -22
321 END IF
322 IF( INFO.NE.0 ) THEN
323 CALL XERBLA( 'DTGSJA', -INFO )
324 RETURN
325 END IF
326 *
327 * Initialize U, V and Q, if necessary
328 *
329 IF( INITU )
330 $ CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
331 IF( INITV )
332 $ CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
333 IF( INITQ )
334 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
335 *
336 * Loop until convergence
337 *
338 UPPER = .FALSE.
339 DO 40 KCYCLE = 1, MAXIT
340 *
341 UPPER = .NOT.UPPER
342 *
343 DO 20 I = 1, L - 1
344 DO 10 J = I + 1, L
345 *
346 A1 = ZERO
347 A2 = ZERO
348 A3 = ZERO
349 IF( K+I.LE.M )
350 $ A1 = A( K+I, N-L+I )
351 IF( K+J.LE.M )
352 $ A3 = A( K+J, N-L+J )
353 *
354 B1 = B( I, N-L+I )
355 B3 = B( J, N-L+J )
356 *
357 IF( UPPER ) THEN
358 IF( K+I.LE.M )
359 $ A2 = A( K+I, N-L+J )
360 B2 = B( I, N-L+J )
361 ELSE
362 IF( K+J.LE.M )
363 $ A2 = A( K+J, N-L+I )
364 B2 = B( J, N-L+I )
365 END IF
366 *
367 CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
368 $ CSV, SNV, CSQ, SNQ )
369 *
370 * Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A
371 *
372 IF( K+J.LE.M )
373 $ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
374 $ LDA, CSU, SNU )
375 *
376 * Update I-th and J-th rows of matrix B: V**T *B
377 *
378 CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
379 $ CSV, SNV )
380 *
381 * Update (N-L+I)-th and (N-L+J)-th columns of matrices
382 * A and B: A*Q and B*Q
383 *
384 CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
385 $ A( 1, N-L+I ), 1, CSQ, SNQ )
386 *
387 CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
388 $ SNQ )
389 *
390 IF( UPPER ) THEN
391 IF( K+I.LE.M )
392 $ A( K+I, N-L+J ) = ZERO
393 B( I, N-L+J ) = ZERO
394 ELSE
395 IF( K+J.LE.M )
396 $ A( K+J, N-L+I ) = ZERO
397 B( J, N-L+I ) = ZERO
398 END IF
399 *
400 * Update orthogonal matrices U, V, Q, if desired.
401 *
402 IF( WANTU .AND. K+J.LE.M )
403 $ CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
404 $ SNU )
405 *
406 IF( WANTV )
407 $ CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
408 *
409 IF( WANTQ )
410 $ CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
411 $ SNQ )
412 *
413 10 CONTINUE
414 20 CONTINUE
415 *
416 IF( .NOT.UPPER ) THEN
417 *
418 * The matrices A13 and B13 were lower triangular at the start
419 * of the cycle, and are now upper triangular.
420 *
421 * Convergence test: test the parallelism of the corresponding
422 * rows of A and B.
423 *
424 ERROR = ZERO
425 DO 30 I = 1, MIN( L, M-K )
426 CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
427 CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
428 CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
429 ERROR = MAX( ERROR, SSMIN )
430 30 CONTINUE
431 *
432 IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
433 $ GO TO 50
434 END IF
435 *
436 * End of cycle loop
437 *
438 40 CONTINUE
439 *
440 * The algorithm has not converged after MAXIT cycles.
441 *
442 INFO = 1
443 GO TO 100
444 *
445 50 CONTINUE
446 *
447 * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
448 * Compute the generalized singular value pairs (ALPHA, BETA), and
449 * set the triangular matrix R to array A.
450 *
451 DO 60 I = 1, K
452 ALPHA( I ) = ONE
453 BETA( I ) = ZERO
454 60 CONTINUE
455 *
456 DO 70 I = 1, MIN( L, M-K )
457 *
458 A1 = A( K+I, N-L+I )
459 B1 = B( I, N-L+I )
460 *
461 IF( A1.NE.ZERO ) THEN
462 GAMMA = B1 / A1
463 *
464 * change sign if necessary
465 *
466 IF( GAMMA.LT.ZERO ) THEN
467 CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
468 IF( WANTV )
469 $ CALL DSCAL( P, -ONE, V( 1, I ), 1 )
470 END IF
471 *
472 CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
473 $ RWK )
474 *
475 IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
476 CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
477 $ LDA )
478 ELSE
479 CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
480 $ LDB )
481 CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
482 $ LDA )
483 END IF
484 *
485 ELSE
486 *
487 ALPHA( K+I ) = ZERO
488 BETA( K+I ) = ONE
489 CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
490 $ LDA )
491 *
492 END IF
493 *
494 70 CONTINUE
495 *
496 * Post-assignment
497 *
498 DO 80 I = M + 1, K + L
499 ALPHA( I ) = ZERO
500 BETA( I ) = ONE
501 80 CONTINUE
502 *
503 IF( K+L.LT.N ) THEN
504 DO 90 I = K + L + 1, N
505 ALPHA( I ) = ZERO
506 BETA( I ) = ZERO
507 90 CONTINUE
508 END IF
509 *
510 100 CONTINUE
511 NCYCLE = KCYCLE
512 RETURN
513 *
514 * End of DTGSJA
515 *
516 END
2 $ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
3 $ Q, LDQ, WORK, NCYCLE, INFO )
4 *
5 * -- LAPACK routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2009 --
9 *
10 * .. Scalar Arguments ..
11 CHARACTER JOBQ, JOBU, JOBV
12 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
13 $ NCYCLE, P
14 DOUBLE PRECISION TOLA, TOLB
15 * ..
16 * .. Array Arguments ..
17 DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
18 $ BETA( * ), Q( LDQ, * ), U( LDU, * ),
19 $ V( LDV, * ), WORK( * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * DTGSJA computes the generalized singular value decomposition (GSVD)
26 * of two real upper triangular (or trapezoidal) matrices A and B.
27 *
28 * On entry, it is assumed that matrices A and B have the following
29 * forms, which may be obtained by the preprocessing subroutine DGGSVP
30 * from a general M-by-N matrix A and P-by-N matrix B:
31 *
32 * N-K-L K L
33 * A = K ( 0 A12 A13 ) if M-K-L >= 0;
34 * L ( 0 0 A23 )
35 * M-K-L ( 0 0 0 )
36 *
37 * N-K-L K L
38 * A = K ( 0 A12 A13 ) if M-K-L < 0;
39 * M-K ( 0 0 A23 )
40 *
41 * N-K-L K L
42 * B = L ( 0 0 B13 )
43 * P-L ( 0 0 0 )
44 *
45 * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
46 * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
47 * otherwise A23 is (M-K)-by-L upper trapezoidal.
48 *
49 * On exit,
50 *
51 * U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ),
52 *
53 * where U, V and Q are orthogonal matrices.
54 * R is a nonsingular upper triangular matrix, and D1 and D2 are
55 * ``diagonal'' matrices, which are of the following structures:
56 *
57 * If M-K-L >= 0,
58 *
59 * K L
60 * D1 = K ( I 0 )
61 * L ( 0 C )
62 * M-K-L ( 0 0 )
63 *
64 * K L
65 * D2 = L ( 0 S )
66 * P-L ( 0 0 )
67 *
68 * N-K-L K L
69 * ( 0 R ) = K ( 0 R11 R12 ) K
70 * L ( 0 0 R22 ) L
71 *
72 * where
73 *
74 * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
75 * S = diag( BETA(K+1), ... , BETA(K+L) ),
76 * C**2 + S**2 = I.
77 *
78 * R is stored in A(1:K+L,N-K-L+1:N) on exit.
79 *
80 * If M-K-L < 0,
81 *
82 * K M-K K+L-M
83 * D1 = K ( I 0 0 )
84 * M-K ( 0 C 0 )
85 *
86 * K M-K K+L-M
87 * D2 = M-K ( 0 S 0 )
88 * K+L-M ( 0 0 I )
89 * P-L ( 0 0 0 )
90 *
91 * N-K-L K M-K K+L-M
92 * ( 0 R ) = K ( 0 R11 R12 R13 )
93 * M-K ( 0 0 R22 R23 )
94 * K+L-M ( 0 0 0 R33 )
95 *
96 * where
97 * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
98 * S = diag( BETA(K+1), ... , BETA(M) ),
99 * C**2 + S**2 = I.
100 *
101 * R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
102 * ( 0 R22 R23 )
103 * in B(M-K+1:L,N+M-K-L+1:N) on exit.
104 *
105 * The computation of the orthogonal transformation matrices U, V or Q
106 * is optional. These matrices may either be formed explicitly, or they
107 * may be postmultiplied into input matrices U1, V1, or Q1.
108 *
109 * Arguments
110 * =========
111 *
112 * JOBU (input) CHARACTER*1
113 * = 'U': U must contain an orthogonal matrix U1 on entry, and
114 * the product U1*U is returned;
115 * = 'I': U is initialized to the unit matrix, and the
116 * orthogonal matrix U is returned;
117 * = 'N': U is not computed.
118 *
119 * JOBV (input) CHARACTER*1
120 * = 'V': V must contain an orthogonal matrix V1 on entry, and
121 * the product V1*V is returned;
122 * = 'I': V is initialized to the unit matrix, and the
123 * orthogonal matrix V is returned;
124 * = 'N': V is not computed.
125 *
126 * JOBQ (input) CHARACTER*1
127 * = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
128 * the product Q1*Q is returned;
129 * = 'I': Q is initialized to the unit matrix, and the
130 * orthogonal matrix Q is returned;
131 * = 'N': Q is not computed.
132 *
133 * M (input) INTEGER
134 * The number of rows of the matrix A. M >= 0.
135 *
136 * P (input) INTEGER
137 * The number of rows of the matrix B. P >= 0.
138 *
139 * N (input) INTEGER
140 * The number of columns of the matrices A and B. N >= 0.
141 *
142 * K (input) INTEGER
143 * L (input) INTEGER
144 * K and L specify the subblocks in the input matrices A and B:
145 * A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
146 * of A and B, whose GSVD is going to be computed by DTGSJA.
147 * See Further Details.
148 *
149 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
150 * On entry, the M-by-N matrix A.
151 * On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
152 * matrix R or part of R. See Purpose for details.
153 *
154 * LDA (input) INTEGER
155 * The leading dimension of the array A. LDA >= max(1,M).
156 *
157 * B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
158 * On entry, the P-by-N matrix B.
159 * On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
160 * a part of R. See Purpose for details.
161 *
162 * LDB (input) INTEGER
163 * The leading dimension of the array B. LDB >= max(1,P).
164 *
165 * TOLA (input) DOUBLE PRECISION
166 * TOLB (input) DOUBLE PRECISION
167 * TOLA and TOLB are the convergence criteria for the Jacobi-
168 * Kogbetliantz iteration procedure. Generally, they are the
169 * same as used in the preprocessing step, say
170 * TOLA = max(M,N)*norm(A)*MAZHEPS,
171 * TOLB = max(P,N)*norm(B)*MAZHEPS.
172 *
173 * ALPHA (output) DOUBLE PRECISION array, dimension (N)
174 * BETA (output) DOUBLE PRECISION array, dimension (N)
175 * On exit, ALPHA and BETA contain the generalized singular
176 * value pairs of A and B;
177 * ALPHA(1:K) = 1,
178 * BETA(1:K) = 0,
179 * and if M-K-L >= 0,
180 * ALPHA(K+1:K+L) = diag(C),
181 * BETA(K+1:K+L) = diag(S),
182 * or if M-K-L < 0,
183 * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
184 * BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
185 * Furthermore, if K+L < N,
186 * ALPHA(K+L+1:N) = 0 and
187 * BETA(K+L+1:N) = 0.
188 *
189 * U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
190 * On entry, if JOBU = 'U', U must contain a matrix U1 (usually
191 * the orthogonal matrix returned by DGGSVP).
192 * On exit,
193 * if JOBU = 'I', U contains the orthogonal matrix U;
194 * if JOBU = 'U', U contains the product U1*U.
195 * If JOBU = 'N', U is not referenced.
196 *
197 * LDU (input) INTEGER
198 * The leading dimension of the array U. LDU >= max(1,M) if
199 * JOBU = 'U'; LDU >= 1 otherwise.
200 *
201 * V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
202 * On entry, if JOBV = 'V', V must contain a matrix V1 (usually
203 * the orthogonal matrix returned by DGGSVP).
204 * On exit,
205 * if JOBV = 'I', V contains the orthogonal matrix V;
206 * if JOBV = 'V', V contains the product V1*V.
207 * If JOBV = 'N', V is not referenced.
208 *
209 * LDV (input) INTEGER
210 * The leading dimension of the array V. LDV >= max(1,P) if
211 * JOBV = 'V'; LDV >= 1 otherwise.
212 *
213 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
214 * On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
215 * the orthogonal matrix returned by DGGSVP).
216 * On exit,
217 * if JOBQ = 'I', Q contains the orthogonal matrix Q;
218 * if JOBQ = 'Q', Q contains the product Q1*Q.
219 * If JOBQ = 'N', Q is not referenced.
220 *
221 * LDQ (input) INTEGER
222 * The leading dimension of the array Q. LDQ >= max(1,N) if
223 * JOBQ = 'Q'; LDQ >= 1 otherwise.
224 *
225 * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
226 *
227 * NCYCLE (output) INTEGER
228 * The number of cycles required for convergence.
229 *
230 * INFO (output) INTEGER
231 * = 0: successful exit
232 * < 0: if INFO = -i, the i-th argument had an illegal value.
233 * = 1: the procedure does not converge after MAXIT cycles.
234 *
235 * Internal Parameters
236 * ===================
237 *
238 * MAXIT INTEGER
239 * MAXIT specifies the total loops that the iterative procedure
240 * may take. If after MAXIT cycles, the routine fails to
241 * converge, we return INFO = 1.
242 *
243 * Further Details
244 * ===============
245 *
246 * DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
247 * min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
248 * matrix B13 to the form:
249 *
250 * U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
251 *
252 * where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
253 * of Z. C1 and S1 are diagonal matrices satisfying
254 *
255 * C1**2 + S1**2 = I,
256 *
257 * and R1 is an L-by-L nonsingular upper triangular matrix.
258 *
259 * =====================================================================
260 *
261 * .. Parameters ..
262 INTEGER MAXIT
263 PARAMETER ( MAXIT = 40 )
264 DOUBLE PRECISION ZERO, ONE
265 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
266 * ..
267 * .. Local Scalars ..
268 *
269 LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
270 INTEGER I, J, KCYCLE
271 DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
272 $ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
273 * ..
274 * .. External Functions ..
275 LOGICAL LSAME
276 EXTERNAL LSAME
277 * ..
278 * .. External Subroutines ..
279 EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET, DROT,
280 $ DSCAL, XERBLA
281 * ..
282 * .. Intrinsic Functions ..
283 INTRINSIC ABS, MAX, MIN
284 * ..
285 * .. Executable Statements ..
286 *
287 * Decode and test the input parameters
288 *
289 INITU = LSAME( JOBU, 'I' )
290 WANTU = INITU .OR. LSAME( JOBU, 'U' )
291 *
292 INITV = LSAME( JOBV, 'I' )
293 WANTV = INITV .OR. LSAME( JOBV, 'V' )
294 *
295 INITQ = LSAME( JOBQ, 'I' )
296 WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
297 *
298 INFO = 0
299 IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
300 INFO = -1
301 ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
302 INFO = -2
303 ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
304 INFO = -3
305 ELSE IF( M.LT.0 ) THEN
306 INFO = -4
307 ELSE IF( P.LT.0 ) THEN
308 INFO = -5
309 ELSE IF( N.LT.0 ) THEN
310 INFO = -6
311 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
312 INFO = -10
313 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
314 INFO = -12
315 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
316 INFO = -18
317 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
318 INFO = -20
319 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
320 INFO = -22
321 END IF
322 IF( INFO.NE.0 ) THEN
323 CALL XERBLA( 'DTGSJA', -INFO )
324 RETURN
325 END IF
326 *
327 * Initialize U, V and Q, if necessary
328 *
329 IF( INITU )
330 $ CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
331 IF( INITV )
332 $ CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
333 IF( INITQ )
334 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
335 *
336 * Loop until convergence
337 *
338 UPPER = .FALSE.
339 DO 40 KCYCLE = 1, MAXIT
340 *
341 UPPER = .NOT.UPPER
342 *
343 DO 20 I = 1, L - 1
344 DO 10 J = I + 1, L
345 *
346 A1 = ZERO
347 A2 = ZERO
348 A3 = ZERO
349 IF( K+I.LE.M )
350 $ A1 = A( K+I, N-L+I )
351 IF( K+J.LE.M )
352 $ A3 = A( K+J, N-L+J )
353 *
354 B1 = B( I, N-L+I )
355 B3 = B( J, N-L+J )
356 *
357 IF( UPPER ) THEN
358 IF( K+I.LE.M )
359 $ A2 = A( K+I, N-L+J )
360 B2 = B( I, N-L+J )
361 ELSE
362 IF( K+J.LE.M )
363 $ A2 = A( K+J, N-L+I )
364 B2 = B( J, N-L+I )
365 END IF
366 *
367 CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
368 $ CSV, SNV, CSQ, SNQ )
369 *
370 * Update (K+I)-th and (K+J)-th rows of matrix A: U**T *A
371 *
372 IF( K+J.LE.M )
373 $ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
374 $ LDA, CSU, SNU )
375 *
376 * Update I-th and J-th rows of matrix B: V**T *B
377 *
378 CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
379 $ CSV, SNV )
380 *
381 * Update (N-L+I)-th and (N-L+J)-th columns of matrices
382 * A and B: A*Q and B*Q
383 *
384 CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
385 $ A( 1, N-L+I ), 1, CSQ, SNQ )
386 *
387 CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
388 $ SNQ )
389 *
390 IF( UPPER ) THEN
391 IF( K+I.LE.M )
392 $ A( K+I, N-L+J ) = ZERO
393 B( I, N-L+J ) = ZERO
394 ELSE
395 IF( K+J.LE.M )
396 $ A( K+J, N-L+I ) = ZERO
397 B( J, N-L+I ) = ZERO
398 END IF
399 *
400 * Update orthogonal matrices U, V, Q, if desired.
401 *
402 IF( WANTU .AND. K+J.LE.M )
403 $ CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
404 $ SNU )
405 *
406 IF( WANTV )
407 $ CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
408 *
409 IF( WANTQ )
410 $ CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
411 $ SNQ )
412 *
413 10 CONTINUE
414 20 CONTINUE
415 *
416 IF( .NOT.UPPER ) THEN
417 *
418 * The matrices A13 and B13 were lower triangular at the start
419 * of the cycle, and are now upper triangular.
420 *
421 * Convergence test: test the parallelism of the corresponding
422 * rows of A and B.
423 *
424 ERROR = ZERO
425 DO 30 I = 1, MIN( L, M-K )
426 CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
427 CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
428 CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
429 ERROR = MAX( ERROR, SSMIN )
430 30 CONTINUE
431 *
432 IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
433 $ GO TO 50
434 END IF
435 *
436 * End of cycle loop
437 *
438 40 CONTINUE
439 *
440 * The algorithm has not converged after MAXIT cycles.
441 *
442 INFO = 1
443 GO TO 100
444 *
445 50 CONTINUE
446 *
447 * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
448 * Compute the generalized singular value pairs (ALPHA, BETA), and
449 * set the triangular matrix R to array A.
450 *
451 DO 60 I = 1, K
452 ALPHA( I ) = ONE
453 BETA( I ) = ZERO
454 60 CONTINUE
455 *
456 DO 70 I = 1, MIN( L, M-K )
457 *
458 A1 = A( K+I, N-L+I )
459 B1 = B( I, N-L+I )
460 *
461 IF( A1.NE.ZERO ) THEN
462 GAMMA = B1 / A1
463 *
464 * change sign if necessary
465 *
466 IF( GAMMA.LT.ZERO ) THEN
467 CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
468 IF( WANTV )
469 $ CALL DSCAL( P, -ONE, V( 1, I ), 1 )
470 END IF
471 *
472 CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
473 $ RWK )
474 *
475 IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
476 CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
477 $ LDA )
478 ELSE
479 CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
480 $ LDB )
481 CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
482 $ LDA )
483 END IF
484 *
485 ELSE
486 *
487 ALPHA( K+I ) = ZERO
488 BETA( K+I ) = ONE
489 CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
490 $ LDA )
491 *
492 END IF
493 *
494 70 CONTINUE
495 *
496 * Post-assignment
497 *
498 DO 80 I = M + 1, K + L
499 ALPHA( I ) = ZERO
500 BETA( I ) = ONE
501 80 CONTINUE
502 *
503 IF( K+L.LT.N ) THEN
504 DO 90 I = K + L + 1, N
505 ALPHA( I ) = ZERO
506 BETA( I ) = ZERO
507 90 CONTINUE
508 END IF
509 *
510 100 CONTINUE
511 NCYCLE = KCYCLE
512 RETURN
513 *
514 * End of DTGSJA
515 *
516 END