1 SUBROUTINE ZTGSJA( 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 ALPHA( * ), BETA( * )
18 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
19 $ U( LDU, * ), V( LDV, * ), WORK( * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZTGSJA computes the generalized singular value decomposition (GSVD)
26 * of two complex 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 ZGGSVP
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**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ),
52 *
53 * where U, V and Q are unitary matrices.
54 * R is a nonsingular upper triangular matrix, and D1
55 * and D2 are ``diagonal'' matrices, which are of the following
56 * structures:
57 *
58 * If M-K-L >= 0,
59 *
60 * K L
61 * D1 = K ( I 0 )
62 * L ( 0 C )
63 * M-K-L ( 0 0 )
64 *
65 * K L
66 * D2 = L ( 0 S )
67 * P-L ( 0 0 )
68 *
69 * N-K-L K L
70 * ( 0 R ) = K ( 0 R11 R12 ) K
71 * L ( 0 0 R22 ) L
72 *
73 * where
74 *
75 * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
76 * S = diag( BETA(K+1), ... , BETA(K+L) ),
77 * C**2 + S**2 = I.
78 *
79 * R is stored in A(1:K+L,N-K-L+1:N) on exit.
80 *
81 * If M-K-L < 0,
82 *
83 * K M-K K+L-M
84 * D1 = K ( I 0 0 )
85 * M-K ( 0 C 0 )
86 *
87 * K M-K K+L-M
88 * D2 = M-K ( 0 S 0 )
89 * K+L-M ( 0 0 I )
90 * P-L ( 0 0 0 )
91 *
92 * N-K-L K M-K K+L-M
93 * ( 0 R ) = K ( 0 R11 R12 R13 )
94 * M-K ( 0 0 R22 R23 )
95 * K+L-M ( 0 0 0 R33 )
96 *
97 * where
98 * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
99 * S = diag( BETA(K+1), ... , BETA(M) ),
100 * C**2 + S**2 = I.
101 *
102 * R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
103 * ( 0 R22 R23 )
104 * in B(M-K+1:L,N+M-K-L+1:N) on exit.
105 *
106 * The computation of the unitary transformation matrices U, V or Q
107 * is optional. These matrices may either be formed explicitly, or they
108 * may be postmultiplied into input matrices U1, V1, or Q1.
109 *
110 * Arguments
111 * =========
112 *
113 * JOBU (input) CHARACTER*1
114 * = 'U': U must contain a unitary matrix U1 on entry, and
115 * the product U1*U is returned;
116 * = 'I': U is initialized to the unit matrix, and the
117 * unitary matrix U is returned;
118 * = 'N': U is not computed.
119 *
120 * JOBV (input) CHARACTER*1
121 * = 'V': V must contain a unitary matrix V1 on entry, and
122 * the product V1*V is returned;
123 * = 'I': V is initialized to the unit matrix, and the
124 * unitary matrix V is returned;
125 * = 'N': V is not computed.
126 *
127 * JOBQ (input) CHARACTER*1
128 * = 'Q': Q must contain a unitary matrix Q1 on entry, and
129 * the product Q1*Q is returned;
130 * = 'I': Q is initialized to the unit matrix, and the
131 * unitary matrix Q is returned;
132 * = 'N': Q is not computed.
133 *
134 * M (input) INTEGER
135 * The number of rows of the matrix A. M >= 0.
136 *
137 * P (input) INTEGER
138 * The number of rows of the matrix B. P >= 0.
139 *
140 * N (input) INTEGER
141 * The number of columns of the matrices A and B. N >= 0.
142 *
143 * K (input) INTEGER
144 * L (input) INTEGER
145 * K and L specify the subblocks in the input matrices A and B:
146 * A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
147 * of A and B, whose GSVD is going to be computed by ZTGSJA.
148 * See Further Details.
149 *
150 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
151 * On entry, the M-by-N matrix A.
152 * On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
153 * matrix R or part of R. See Purpose for details.
154 *
155 * LDA (input) INTEGER
156 * The leading dimension of the array A. LDA >= max(1,M).
157 *
158 * B (input/output) COMPLEX*16 array, dimension (LDB,N)
159 * On entry, the P-by-N matrix B.
160 * On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
161 * a part of R. See Purpose for details.
162 *
163 * LDB (input) INTEGER
164 * The leading dimension of the array B. LDB >= max(1,P).
165 *
166 * TOLA (input) DOUBLE PRECISION
167 * TOLB (input) DOUBLE PRECISION
168 * TOLA and TOLB are the convergence criteria for the Jacobi-
169 * Kogbetliantz iteration procedure. Generally, they are the
170 * same as used in the preprocessing step, say
171 * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
172 * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
173 *
174 * ALPHA (output) DOUBLE PRECISION array, dimension (N)
175 * BETA (output) DOUBLE PRECISION array, dimension (N)
176 * On exit, ALPHA and BETA contain the generalized singular
177 * value pairs of A and B;
178 * ALPHA(1:K) = 1,
179 * BETA(1:K) = 0,
180 * and if M-K-L >= 0,
181 * ALPHA(K+1:K+L) = diag(C),
182 * BETA(K+1:K+L) = diag(S),
183 * or if M-K-L < 0,
184 * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
185 * BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
186 * Furthermore, if K+L < N,
187 * ALPHA(K+L+1:N) = 0 and
188 * BETA(K+L+1:N) = 0.
189 *
190 * U (input/output) COMPLEX*16 array, dimension (LDU,M)
191 * On entry, if JOBU = 'U', U must contain a matrix U1 (usually
192 * the unitary matrix returned by ZGGSVP).
193 * On exit,
194 * if JOBU = 'I', U contains the unitary matrix U;
195 * if JOBU = 'U', U contains the product U1*U.
196 * If JOBU = 'N', U is not referenced.
197 *
198 * LDU (input) INTEGER
199 * The leading dimension of the array U. LDU >= max(1,M) if
200 * JOBU = 'U'; LDU >= 1 otherwise.
201 *
202 * V (input/output) COMPLEX*16 array, dimension (LDV,P)
203 * On entry, if JOBV = 'V', V must contain a matrix V1 (usually
204 * the unitary matrix returned by ZGGSVP).
205 * On exit,
206 * if JOBV = 'I', V contains the unitary matrix V;
207 * if JOBV = 'V', V contains the product V1*V.
208 * If JOBV = 'N', V is not referenced.
209 *
210 * LDV (input) INTEGER
211 * The leading dimension of the array V. LDV >= max(1,P) if
212 * JOBV = 'V'; LDV >= 1 otherwise.
213 *
214 * Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
215 * On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
216 * the unitary matrix returned by ZGGSVP).
217 * On exit,
218 * if JOBQ = 'I', Q contains the unitary matrix Q;
219 * if JOBQ = 'Q', Q contains the product Q1*Q.
220 * If JOBQ = 'N', Q is not referenced.
221 *
222 * LDQ (input) INTEGER
223 * The leading dimension of the array Q. LDQ >= max(1,N) if
224 * JOBQ = 'Q'; LDQ >= 1 otherwise.
225 *
226 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
227 *
228 * NCYCLE (output) INTEGER
229 * The number of cycles required for convergence.
230 *
231 * INFO (output) INTEGER
232 * = 0: successful exit
233 * < 0: if INFO = -i, the i-th argument had an illegal value.
234 * = 1: the procedure does not converge after MAXIT cycles.
235 *
236 * Internal Parameters
237 * ===================
238 *
239 * MAXIT INTEGER
240 * MAXIT specifies the total loops that the iterative procedure
241 * may take. If after MAXIT cycles, the routine fails to
242 * converge, we return INFO = 1.
243 *
244 * Further Details
245 * ===============
246 *
247 * ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
248 * min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
249 * matrix B13 to the form:
250 *
251 * U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,
252 *
253 * where U1, V1 and Q1 are unitary matrix.
254 * C1 and S1 are diagonal matrices satisfying
255 *
256 * C1**2 + S1**2 = I,
257 *
258 * and R1 is an L-by-L nonsingular upper triangular matrix.
259 *
260 * =====================================================================
261 *
262 * .. Parameters ..
263 INTEGER MAXIT
264 PARAMETER ( MAXIT = 40 )
265 DOUBLE PRECISION ZERO, ONE
266 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
267 COMPLEX*16 CZERO, CONE
268 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
269 $ CONE = ( 1.0D+0, 0.0D+0 ) )
270 * ..
271 * .. Local Scalars ..
272 *
273 LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
274 INTEGER I, J, KCYCLE
275 DOUBLE PRECISION A1, A3, B1, B3, CSQ, CSU, CSV, ERROR, GAMMA,
276 $ RWK, SSMIN
277 COMPLEX*16 A2, B2, SNQ, SNU, SNV
278 * ..
279 * .. External Functions ..
280 LOGICAL LSAME
281 EXTERNAL LSAME
282 * ..
283 * .. External Subroutines ..
284 EXTERNAL DLARTG, XERBLA, ZCOPY, ZDSCAL, ZLAGS2, ZLAPLL,
285 $ ZLASET, ZROT
286 * ..
287 * .. Intrinsic Functions ..
288 INTRINSIC ABS, DBLE, DCONJG, MAX, MIN
289 * ..
290 * .. Executable Statements ..
291 *
292 * Decode and test the input parameters
293 *
294 INITU = LSAME( JOBU, 'I' )
295 WANTU = INITU .OR. LSAME( JOBU, 'U' )
296 *
297 INITV = LSAME( JOBV, 'I' )
298 WANTV = INITV .OR. LSAME( JOBV, 'V' )
299 *
300 INITQ = LSAME( JOBQ, 'I' )
301 WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
302 *
303 INFO = 0
304 IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
305 INFO = -1
306 ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
307 INFO = -2
308 ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
309 INFO = -3
310 ELSE IF( M.LT.0 ) THEN
311 INFO = -4
312 ELSE IF( P.LT.0 ) THEN
313 INFO = -5
314 ELSE IF( N.LT.0 ) THEN
315 INFO = -6
316 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
317 INFO = -10
318 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
319 INFO = -12
320 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
321 INFO = -18
322 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
323 INFO = -20
324 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
325 INFO = -22
326 END IF
327 IF( INFO.NE.0 ) THEN
328 CALL XERBLA( 'ZTGSJA', -INFO )
329 RETURN
330 END IF
331 *
332 * Initialize U, V and Q, if necessary
333 *
334 IF( INITU )
335 $ CALL ZLASET( 'Full', M, M, CZERO, CONE, U, LDU )
336 IF( INITV )
337 $ CALL ZLASET( 'Full', P, P, CZERO, CONE, V, LDV )
338 IF( INITQ )
339 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
340 *
341 * Loop until convergence
342 *
343 UPPER = .FALSE.
344 DO 40 KCYCLE = 1, MAXIT
345 *
346 UPPER = .NOT.UPPER
347 *
348 DO 20 I = 1, L - 1
349 DO 10 J = I + 1, L
350 *
351 A1 = ZERO
352 A2 = CZERO
353 A3 = ZERO
354 IF( K+I.LE.M )
355 $ A1 = DBLE( A( K+I, N-L+I ) )
356 IF( K+J.LE.M )
357 $ A3 = DBLE( A( K+J, N-L+J ) )
358 *
359 B1 = DBLE( B( I, N-L+I ) )
360 B3 = DBLE( B( J, N-L+J ) )
361 *
362 IF( UPPER ) THEN
363 IF( K+I.LE.M )
364 $ A2 = A( K+I, N-L+J )
365 B2 = B( I, N-L+J )
366 ELSE
367 IF( K+J.LE.M )
368 $ A2 = A( K+J, N-L+I )
369 B2 = B( J, N-L+I )
370 END IF
371 *
372 CALL ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
373 $ CSV, SNV, CSQ, SNQ )
374 *
375 * Update (K+I)-th and (K+J)-th rows of matrix A: U**H *A
376 *
377 IF( K+J.LE.M )
378 $ CALL ZROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
379 $ LDA, CSU, DCONJG( SNU ) )
380 *
381 * Update I-th and J-th rows of matrix B: V**H *B
382 *
383 CALL ZROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
384 $ CSV, DCONJG( SNV ) )
385 *
386 * Update (N-L+I)-th and (N-L+J)-th columns of matrices
387 * A and B: A*Q and B*Q
388 *
389 CALL ZROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
390 $ A( 1, N-L+I ), 1, CSQ, SNQ )
391 *
392 CALL ZROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
393 $ SNQ )
394 *
395 IF( UPPER ) THEN
396 IF( K+I.LE.M )
397 $ A( K+I, N-L+J ) = CZERO
398 B( I, N-L+J ) = CZERO
399 ELSE
400 IF( K+J.LE.M )
401 $ A( K+J, N-L+I ) = CZERO
402 B( J, N-L+I ) = CZERO
403 END IF
404 *
405 * Ensure that the diagonal elements of A and B are real.
406 *
407 IF( K+I.LE.M )
408 $ A( K+I, N-L+I ) = DBLE( A( K+I, N-L+I ) )
409 IF( K+J.LE.M )
410 $ A( K+J, N-L+J ) = DBLE( A( K+J, N-L+J ) )
411 B( I, N-L+I ) = DBLE( B( I, N-L+I ) )
412 B( J, N-L+J ) = DBLE( B( J, N-L+J ) )
413 *
414 * Update unitary matrices U, V, Q, if desired.
415 *
416 IF( WANTU .AND. K+J.LE.M )
417 $ CALL ZROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
418 $ SNU )
419 *
420 IF( WANTV )
421 $ CALL ZROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
422 *
423 IF( WANTQ )
424 $ CALL ZROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
425 $ SNQ )
426 *
427 10 CONTINUE
428 20 CONTINUE
429 *
430 IF( .NOT.UPPER ) THEN
431 *
432 * The matrices A13 and B13 were lower triangular at the start
433 * of the cycle, and are now upper triangular.
434 *
435 * Convergence test: test the parallelism of the corresponding
436 * rows of A and B.
437 *
438 ERROR = ZERO
439 DO 30 I = 1, MIN( L, M-K )
440 CALL ZCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
441 CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
442 CALL ZLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
443 ERROR = MAX( ERROR, SSMIN )
444 30 CONTINUE
445 *
446 IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
447 $ GO TO 50
448 END IF
449 *
450 * End of cycle loop
451 *
452 40 CONTINUE
453 *
454 * The algorithm has not converged after MAXIT cycles.
455 *
456 INFO = 1
457 GO TO 100
458 *
459 50 CONTINUE
460 *
461 * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
462 * Compute the generalized singular value pairs (ALPHA, BETA), and
463 * set the triangular matrix R to array A.
464 *
465 DO 60 I = 1, K
466 ALPHA( I ) = ONE
467 BETA( I ) = ZERO
468 60 CONTINUE
469 *
470 DO 70 I = 1, MIN( L, M-K )
471 *
472 A1 = DBLE( A( K+I, N-L+I ) )
473 B1 = DBLE( B( I, N-L+I ) )
474 *
475 IF( A1.NE.ZERO ) THEN
476 GAMMA = B1 / A1
477 *
478 IF( GAMMA.LT.ZERO ) THEN
479 CALL ZDSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
480 IF( WANTV )
481 $ CALL ZDSCAL( P, -ONE, V( 1, I ), 1 )
482 END IF
483 *
484 CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
485 $ RWK )
486 *
487 IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
488 CALL ZDSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
489 $ LDA )
490 ELSE
491 CALL ZDSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
492 $ LDB )
493 CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
494 $ LDA )
495 END IF
496 *
497 ELSE
498 *
499 ALPHA( K+I ) = ZERO
500 BETA( K+I ) = ONE
501 CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
502 $ LDA )
503 END IF
504 70 CONTINUE
505 *
506 * Post-assignment
507 *
508 DO 80 I = M + 1, K + L
509 ALPHA( I ) = ZERO
510 BETA( I ) = ONE
511 80 CONTINUE
512 *
513 IF( K+L.LT.N ) THEN
514 DO 90 I = K + L + 1, N
515 ALPHA( I ) = ZERO
516 BETA( I ) = ZERO
517 90 CONTINUE
518 END IF
519 *
520 100 CONTINUE
521 NCYCLE = KCYCLE
522 *
523 RETURN
524 *
525 * End of ZTGSJA
526 *
527 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 ALPHA( * ), BETA( * )
18 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
19 $ U( LDU, * ), V( LDV, * ), WORK( * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZTGSJA computes the generalized singular value decomposition (GSVD)
26 * of two complex 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 ZGGSVP
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**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ),
52 *
53 * where U, V and Q are unitary matrices.
54 * R is a nonsingular upper triangular matrix, and D1
55 * and D2 are ``diagonal'' matrices, which are of the following
56 * structures:
57 *
58 * If M-K-L >= 0,
59 *
60 * K L
61 * D1 = K ( I 0 )
62 * L ( 0 C )
63 * M-K-L ( 0 0 )
64 *
65 * K L
66 * D2 = L ( 0 S )
67 * P-L ( 0 0 )
68 *
69 * N-K-L K L
70 * ( 0 R ) = K ( 0 R11 R12 ) K
71 * L ( 0 0 R22 ) L
72 *
73 * where
74 *
75 * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
76 * S = diag( BETA(K+1), ... , BETA(K+L) ),
77 * C**2 + S**2 = I.
78 *
79 * R is stored in A(1:K+L,N-K-L+1:N) on exit.
80 *
81 * If M-K-L < 0,
82 *
83 * K M-K K+L-M
84 * D1 = K ( I 0 0 )
85 * M-K ( 0 C 0 )
86 *
87 * K M-K K+L-M
88 * D2 = M-K ( 0 S 0 )
89 * K+L-M ( 0 0 I )
90 * P-L ( 0 0 0 )
91 *
92 * N-K-L K M-K K+L-M
93 * ( 0 R ) = K ( 0 R11 R12 R13 )
94 * M-K ( 0 0 R22 R23 )
95 * K+L-M ( 0 0 0 R33 )
96 *
97 * where
98 * C = diag( ALPHA(K+1), ... , ALPHA(M) ),
99 * S = diag( BETA(K+1), ... , BETA(M) ),
100 * C**2 + S**2 = I.
101 *
102 * R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
103 * ( 0 R22 R23 )
104 * in B(M-K+1:L,N+M-K-L+1:N) on exit.
105 *
106 * The computation of the unitary transformation matrices U, V or Q
107 * is optional. These matrices may either be formed explicitly, or they
108 * may be postmultiplied into input matrices U1, V1, or Q1.
109 *
110 * Arguments
111 * =========
112 *
113 * JOBU (input) CHARACTER*1
114 * = 'U': U must contain a unitary matrix U1 on entry, and
115 * the product U1*U is returned;
116 * = 'I': U is initialized to the unit matrix, and the
117 * unitary matrix U is returned;
118 * = 'N': U is not computed.
119 *
120 * JOBV (input) CHARACTER*1
121 * = 'V': V must contain a unitary matrix V1 on entry, and
122 * the product V1*V is returned;
123 * = 'I': V is initialized to the unit matrix, and the
124 * unitary matrix V is returned;
125 * = 'N': V is not computed.
126 *
127 * JOBQ (input) CHARACTER*1
128 * = 'Q': Q must contain a unitary matrix Q1 on entry, and
129 * the product Q1*Q is returned;
130 * = 'I': Q is initialized to the unit matrix, and the
131 * unitary matrix Q is returned;
132 * = 'N': Q is not computed.
133 *
134 * M (input) INTEGER
135 * The number of rows of the matrix A. M >= 0.
136 *
137 * P (input) INTEGER
138 * The number of rows of the matrix B. P >= 0.
139 *
140 * N (input) INTEGER
141 * The number of columns of the matrices A and B. N >= 0.
142 *
143 * K (input) INTEGER
144 * L (input) INTEGER
145 * K and L specify the subblocks in the input matrices A and B:
146 * A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
147 * of A and B, whose GSVD is going to be computed by ZTGSJA.
148 * See Further Details.
149 *
150 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
151 * On entry, the M-by-N matrix A.
152 * On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
153 * matrix R or part of R. See Purpose for details.
154 *
155 * LDA (input) INTEGER
156 * The leading dimension of the array A. LDA >= max(1,M).
157 *
158 * B (input/output) COMPLEX*16 array, dimension (LDB,N)
159 * On entry, the P-by-N matrix B.
160 * On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
161 * a part of R. See Purpose for details.
162 *
163 * LDB (input) INTEGER
164 * The leading dimension of the array B. LDB >= max(1,P).
165 *
166 * TOLA (input) DOUBLE PRECISION
167 * TOLB (input) DOUBLE PRECISION
168 * TOLA and TOLB are the convergence criteria for the Jacobi-
169 * Kogbetliantz iteration procedure. Generally, they are the
170 * same as used in the preprocessing step, say
171 * TOLA = MAX(M,N)*norm(A)*MAZHEPS,
172 * TOLB = MAX(P,N)*norm(B)*MAZHEPS.
173 *
174 * ALPHA (output) DOUBLE PRECISION array, dimension (N)
175 * BETA (output) DOUBLE PRECISION array, dimension (N)
176 * On exit, ALPHA and BETA contain the generalized singular
177 * value pairs of A and B;
178 * ALPHA(1:K) = 1,
179 * BETA(1:K) = 0,
180 * and if M-K-L >= 0,
181 * ALPHA(K+1:K+L) = diag(C),
182 * BETA(K+1:K+L) = diag(S),
183 * or if M-K-L < 0,
184 * ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
185 * BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
186 * Furthermore, if K+L < N,
187 * ALPHA(K+L+1:N) = 0 and
188 * BETA(K+L+1:N) = 0.
189 *
190 * U (input/output) COMPLEX*16 array, dimension (LDU,M)
191 * On entry, if JOBU = 'U', U must contain a matrix U1 (usually
192 * the unitary matrix returned by ZGGSVP).
193 * On exit,
194 * if JOBU = 'I', U contains the unitary matrix U;
195 * if JOBU = 'U', U contains the product U1*U.
196 * If JOBU = 'N', U is not referenced.
197 *
198 * LDU (input) INTEGER
199 * The leading dimension of the array U. LDU >= max(1,M) if
200 * JOBU = 'U'; LDU >= 1 otherwise.
201 *
202 * V (input/output) COMPLEX*16 array, dimension (LDV,P)
203 * On entry, if JOBV = 'V', V must contain a matrix V1 (usually
204 * the unitary matrix returned by ZGGSVP).
205 * On exit,
206 * if JOBV = 'I', V contains the unitary matrix V;
207 * if JOBV = 'V', V contains the product V1*V.
208 * If JOBV = 'N', V is not referenced.
209 *
210 * LDV (input) INTEGER
211 * The leading dimension of the array V. LDV >= max(1,P) if
212 * JOBV = 'V'; LDV >= 1 otherwise.
213 *
214 * Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
215 * On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
216 * the unitary matrix returned by ZGGSVP).
217 * On exit,
218 * if JOBQ = 'I', Q contains the unitary matrix Q;
219 * if JOBQ = 'Q', Q contains the product Q1*Q.
220 * If JOBQ = 'N', Q is not referenced.
221 *
222 * LDQ (input) INTEGER
223 * The leading dimension of the array Q. LDQ >= max(1,N) if
224 * JOBQ = 'Q'; LDQ >= 1 otherwise.
225 *
226 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
227 *
228 * NCYCLE (output) INTEGER
229 * The number of cycles required for convergence.
230 *
231 * INFO (output) INTEGER
232 * = 0: successful exit
233 * < 0: if INFO = -i, the i-th argument had an illegal value.
234 * = 1: the procedure does not converge after MAXIT cycles.
235 *
236 * Internal Parameters
237 * ===================
238 *
239 * MAXIT INTEGER
240 * MAXIT specifies the total loops that the iterative procedure
241 * may take. If after MAXIT cycles, the routine fails to
242 * converge, we return INFO = 1.
243 *
244 * Further Details
245 * ===============
246 *
247 * ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
248 * min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
249 * matrix B13 to the form:
250 *
251 * U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,
252 *
253 * where U1, V1 and Q1 are unitary matrix.
254 * C1 and S1 are diagonal matrices satisfying
255 *
256 * C1**2 + S1**2 = I,
257 *
258 * and R1 is an L-by-L nonsingular upper triangular matrix.
259 *
260 * =====================================================================
261 *
262 * .. Parameters ..
263 INTEGER MAXIT
264 PARAMETER ( MAXIT = 40 )
265 DOUBLE PRECISION ZERO, ONE
266 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
267 COMPLEX*16 CZERO, CONE
268 PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
269 $ CONE = ( 1.0D+0, 0.0D+0 ) )
270 * ..
271 * .. Local Scalars ..
272 *
273 LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
274 INTEGER I, J, KCYCLE
275 DOUBLE PRECISION A1, A3, B1, B3, CSQ, CSU, CSV, ERROR, GAMMA,
276 $ RWK, SSMIN
277 COMPLEX*16 A2, B2, SNQ, SNU, SNV
278 * ..
279 * .. External Functions ..
280 LOGICAL LSAME
281 EXTERNAL LSAME
282 * ..
283 * .. External Subroutines ..
284 EXTERNAL DLARTG, XERBLA, ZCOPY, ZDSCAL, ZLAGS2, ZLAPLL,
285 $ ZLASET, ZROT
286 * ..
287 * .. Intrinsic Functions ..
288 INTRINSIC ABS, DBLE, DCONJG, MAX, MIN
289 * ..
290 * .. Executable Statements ..
291 *
292 * Decode and test the input parameters
293 *
294 INITU = LSAME( JOBU, 'I' )
295 WANTU = INITU .OR. LSAME( JOBU, 'U' )
296 *
297 INITV = LSAME( JOBV, 'I' )
298 WANTV = INITV .OR. LSAME( JOBV, 'V' )
299 *
300 INITQ = LSAME( JOBQ, 'I' )
301 WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
302 *
303 INFO = 0
304 IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
305 INFO = -1
306 ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
307 INFO = -2
308 ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
309 INFO = -3
310 ELSE IF( M.LT.0 ) THEN
311 INFO = -4
312 ELSE IF( P.LT.0 ) THEN
313 INFO = -5
314 ELSE IF( N.LT.0 ) THEN
315 INFO = -6
316 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
317 INFO = -10
318 ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
319 INFO = -12
320 ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
321 INFO = -18
322 ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
323 INFO = -20
324 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
325 INFO = -22
326 END IF
327 IF( INFO.NE.0 ) THEN
328 CALL XERBLA( 'ZTGSJA', -INFO )
329 RETURN
330 END IF
331 *
332 * Initialize U, V and Q, if necessary
333 *
334 IF( INITU )
335 $ CALL ZLASET( 'Full', M, M, CZERO, CONE, U, LDU )
336 IF( INITV )
337 $ CALL ZLASET( 'Full', P, P, CZERO, CONE, V, LDV )
338 IF( INITQ )
339 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
340 *
341 * Loop until convergence
342 *
343 UPPER = .FALSE.
344 DO 40 KCYCLE = 1, MAXIT
345 *
346 UPPER = .NOT.UPPER
347 *
348 DO 20 I = 1, L - 1
349 DO 10 J = I + 1, L
350 *
351 A1 = ZERO
352 A2 = CZERO
353 A3 = ZERO
354 IF( K+I.LE.M )
355 $ A1 = DBLE( A( K+I, N-L+I ) )
356 IF( K+J.LE.M )
357 $ A3 = DBLE( A( K+J, N-L+J ) )
358 *
359 B1 = DBLE( B( I, N-L+I ) )
360 B3 = DBLE( B( J, N-L+J ) )
361 *
362 IF( UPPER ) THEN
363 IF( K+I.LE.M )
364 $ A2 = A( K+I, N-L+J )
365 B2 = B( I, N-L+J )
366 ELSE
367 IF( K+J.LE.M )
368 $ A2 = A( K+J, N-L+I )
369 B2 = B( J, N-L+I )
370 END IF
371 *
372 CALL ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
373 $ CSV, SNV, CSQ, SNQ )
374 *
375 * Update (K+I)-th and (K+J)-th rows of matrix A: U**H *A
376 *
377 IF( K+J.LE.M )
378 $ CALL ZROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
379 $ LDA, CSU, DCONJG( SNU ) )
380 *
381 * Update I-th and J-th rows of matrix B: V**H *B
382 *
383 CALL ZROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
384 $ CSV, DCONJG( SNV ) )
385 *
386 * Update (N-L+I)-th and (N-L+J)-th columns of matrices
387 * A and B: A*Q and B*Q
388 *
389 CALL ZROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
390 $ A( 1, N-L+I ), 1, CSQ, SNQ )
391 *
392 CALL ZROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
393 $ SNQ )
394 *
395 IF( UPPER ) THEN
396 IF( K+I.LE.M )
397 $ A( K+I, N-L+J ) = CZERO
398 B( I, N-L+J ) = CZERO
399 ELSE
400 IF( K+J.LE.M )
401 $ A( K+J, N-L+I ) = CZERO
402 B( J, N-L+I ) = CZERO
403 END IF
404 *
405 * Ensure that the diagonal elements of A and B are real.
406 *
407 IF( K+I.LE.M )
408 $ A( K+I, N-L+I ) = DBLE( A( K+I, N-L+I ) )
409 IF( K+J.LE.M )
410 $ A( K+J, N-L+J ) = DBLE( A( K+J, N-L+J ) )
411 B( I, N-L+I ) = DBLE( B( I, N-L+I ) )
412 B( J, N-L+J ) = DBLE( B( J, N-L+J ) )
413 *
414 * Update unitary matrices U, V, Q, if desired.
415 *
416 IF( WANTU .AND. K+J.LE.M )
417 $ CALL ZROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
418 $ SNU )
419 *
420 IF( WANTV )
421 $ CALL ZROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
422 *
423 IF( WANTQ )
424 $ CALL ZROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
425 $ SNQ )
426 *
427 10 CONTINUE
428 20 CONTINUE
429 *
430 IF( .NOT.UPPER ) THEN
431 *
432 * The matrices A13 and B13 were lower triangular at the start
433 * of the cycle, and are now upper triangular.
434 *
435 * Convergence test: test the parallelism of the corresponding
436 * rows of A and B.
437 *
438 ERROR = ZERO
439 DO 30 I = 1, MIN( L, M-K )
440 CALL ZCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
441 CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
442 CALL ZLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
443 ERROR = MAX( ERROR, SSMIN )
444 30 CONTINUE
445 *
446 IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
447 $ GO TO 50
448 END IF
449 *
450 * End of cycle loop
451 *
452 40 CONTINUE
453 *
454 * The algorithm has not converged after MAXIT cycles.
455 *
456 INFO = 1
457 GO TO 100
458 *
459 50 CONTINUE
460 *
461 * If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
462 * Compute the generalized singular value pairs (ALPHA, BETA), and
463 * set the triangular matrix R to array A.
464 *
465 DO 60 I = 1, K
466 ALPHA( I ) = ONE
467 BETA( I ) = ZERO
468 60 CONTINUE
469 *
470 DO 70 I = 1, MIN( L, M-K )
471 *
472 A1 = DBLE( A( K+I, N-L+I ) )
473 B1 = DBLE( B( I, N-L+I ) )
474 *
475 IF( A1.NE.ZERO ) THEN
476 GAMMA = B1 / A1
477 *
478 IF( GAMMA.LT.ZERO ) THEN
479 CALL ZDSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
480 IF( WANTV )
481 $ CALL ZDSCAL( P, -ONE, V( 1, I ), 1 )
482 END IF
483 *
484 CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
485 $ RWK )
486 *
487 IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
488 CALL ZDSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
489 $ LDA )
490 ELSE
491 CALL ZDSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
492 $ LDB )
493 CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
494 $ LDA )
495 END IF
496 *
497 ELSE
498 *
499 ALPHA( K+I ) = ZERO
500 BETA( K+I ) = ONE
501 CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
502 $ LDA )
503 END IF
504 70 CONTINUE
505 *
506 * Post-assignment
507 *
508 DO 80 I = M + 1, K + L
509 ALPHA( I ) = ZERO
510 BETA( I ) = ONE
511 80 CONTINUE
512 *
513 IF( K+L.LT.N ) THEN
514 DO 90 I = K + L + 1, N
515 ALPHA( I ) = ZERO
516 BETA( I ) = ZERO
517 90 CONTINUE
518 END IF
519 *
520 100 CONTINUE
521 NCYCLE = KCYCLE
522 *
523 RETURN
524 *
525 * End of ZTGSJA
526 *
527 END