LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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◆ zggsvp()

 subroutine zggsvp ( character jobu, character jobv, character jobq, integer m, integer p, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision tola, double precision tolb, integer k, integer l, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) iwork, double precision, dimension( * ) rwork, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info )

ZGGSVP

Purpose:
``` This routine is deprecated and has been replaced by routine ZGGSVP3.

ZGGSVP computes unitary matrices U, V and Q such that

N-K-L  K    L
U**H*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
L ( 0     0   A23 )
M-K-L ( 0     0    0  )

N-K-L  K    L
=     K ( 0    A12  A13 )  if M-K-L < 0;
M-K ( 0     0   A23 )

N-K-L  K    L
V**H*B*Q =   L ( 0     0   B13 )
P-L ( 0     0    0  )

where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
numerical rank of the (M+P)-by-N matrix (A**H,B**H)**H.

This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
ZGGSVD.```
Parameters
 [in] JOBU ``` JOBU is CHARACTER*1 = 'U': Unitary matrix U is computed; = 'N': U is not computed.``` [in] JOBV ``` JOBV is CHARACTER*1 = 'V': Unitary matrix V is computed; = 'N': V is not computed.``` [in] JOBQ ``` JOBQ is CHARACTER*1 = 'Q': Unitary matrix Q is computed; = 'N': Q is not computed.``` [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] P ``` P is INTEGER The number of rows of the matrix B. P >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrices A and B. N >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in,out] B ``` B is COMPLEX*16 array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, B contains the triangular matrix described in the Purpose section.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).``` [in] TOLA ` TOLA is DOUBLE PRECISION` [in] TOLB ``` TOLB is DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective numerical rank of matrix B and a subblock of A. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition.``` [out] K ` K is INTEGER` [out] L ``` L is INTEGER On exit, K and L specify the dimension of the subblocks described in Purpose section. K + L = effective numerical rank of (A**H,B**H)**H.``` [out] U ``` U is COMPLEX*16 array, dimension (LDU,M) If JOBU = 'U', U contains the unitary matrix U. If JOBU = 'N', U is not referenced.``` [in] LDU ``` LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.``` [out] V ``` V is COMPLEX*16 array, dimension (LDV,P) If JOBV = 'V', V contains the unitary matrix V. If JOBV = 'N', V is not referenced.``` [in] LDV ``` LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.``` [out] Q ``` Q is COMPLEX*16 array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the unitary matrix Q. If JOBQ = 'N', Q is not referenced.``` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.``` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] RWORK ` RWORK is DOUBLE PRECISION array, dimension (2*N)` [out] TAU ` TAU is COMPLEX*16 array, dimension (N)` [out] WORK ` WORK is COMPLEX*16 array, dimension (max(3*N,M,P))` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.```
Further Details:
```  The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.```

Definition at line 262 of file zggsvp.f.

265*
266* -- LAPACK computational routine --
267* -- LAPACK is a software package provided by Univ. of Tennessee, --
268* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
269*
270* .. Scalar Arguments ..
271 CHARACTER JOBQ, JOBU, JOBV
272 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
273 DOUBLE PRECISION TOLA, TOLB
274* ..
275* .. Array Arguments ..
276 INTEGER IWORK( * )
277 DOUBLE PRECISION RWORK( * )
278 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
279 \$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
280* ..
281*
282* =====================================================================
283*
284* .. Parameters ..
285 COMPLEX*16 CZERO, CONE
286 parameter( czero = ( 0.0d+0, 0.0d+0 ),
287 \$ cone = ( 1.0d+0, 0.0d+0 ) )
288* ..
289* .. Local Scalars ..
290 LOGICAL FORWRD, WANTQ, WANTU, WANTV
291 INTEGER I, J
292 COMPLEX*16 T
293* ..
294* .. External Functions ..
295 LOGICAL LSAME
296 EXTERNAL lsame
297* ..
298* .. External Subroutines ..
299 EXTERNAL xerbla, zgeqpf, zgeqr2, zgerq2, zlacpy, zlapmt,
301* ..
302* .. Intrinsic Functions ..
303 INTRINSIC abs, dble, dimag, max, min
304* ..
305* .. Statement Functions ..
306 DOUBLE PRECISION CABS1
307* ..
308* .. Statement Function definitions ..
309 cabs1( t ) = abs( dble( t ) ) + abs( dimag( t ) )
310* ..
311* .. Executable Statements ..
312*
313* Test the input parameters
314*
315 wantu = lsame( jobu, 'U' )
316 wantv = lsame( jobv, 'V' )
317 wantq = lsame( jobq, 'Q' )
318 forwrd = .true.
319*
320 info = 0
321 IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
322 info = -1
323 ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
324 info = -2
325 ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
326 info = -3
327 ELSE IF( m.LT.0 ) THEN
328 info = -4
329 ELSE IF( p.LT.0 ) THEN
330 info = -5
331 ELSE IF( n.LT.0 ) THEN
332 info = -6
333 ELSE IF( lda.LT.max( 1, m ) ) THEN
334 info = -8
335 ELSE IF( ldb.LT.max( 1, p ) ) THEN
336 info = -10
337 ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
338 info = -16
339 ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
340 info = -18
341 ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
342 info = -20
343 END IF
344 IF( info.NE.0 ) THEN
345 CALL xerbla( 'ZGGSVP', -info )
346 RETURN
347 END IF
348*
349* QR with column pivoting of B: B*P = V*( S11 S12 )
350* ( 0 0 )
351*
352 DO 10 i = 1, n
353 iwork( i ) = 0
354 10 CONTINUE
355 CALL zgeqpf( p, n, b, ldb, iwork, tau, work, rwork, info )
356*
357* Update A := A*P
358*
359 CALL zlapmt( forwrd, m, n, a, lda, iwork )
360*
361* Determine the effective rank of matrix B.
362*
363 l = 0
364 DO 20 i = 1, min( p, n )
365 IF( cabs1( b( i, i ) ).GT.tolb )
366 \$ l = l + 1
367 20 CONTINUE
368*
369 IF( wantv ) THEN
370*
371* Copy the details of V, and form V.
372*
373 CALL zlaset( 'Full', p, p, czero, czero, v, ldv )
374 IF( p.GT.1 )
375 \$ CALL zlacpy( 'Lower', p-1, n, b( 2, 1 ), ldb, v( 2, 1 ),
376 \$ ldv )
377 CALL zung2r( p, p, min( p, n ), v, ldv, tau, work, info )
378 END IF
379*
380* Clean up B
381*
382 DO 40 j = 1, l - 1
383 DO 30 i = j + 1, l
384 b( i, j ) = czero
385 30 CONTINUE
386 40 CONTINUE
387 IF( p.GT.l )
388 \$ CALL zlaset( 'Full', p-l, n, czero, czero, b( l+1, 1 ), ldb )
389*
390 IF( wantq ) THEN
391*
392* Set Q = I and Update Q := Q*P
393*
394 CALL zlaset( 'Full', n, n, czero, cone, q, ldq )
395 CALL zlapmt( forwrd, n, n, q, ldq, iwork )
396 END IF
397*
398 IF( p.GE.l .AND. n.NE.l ) THEN
399*
400* RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
401*
402 CALL zgerq2( l, n, b, ldb, tau, work, info )
403*
404* Update A := A*Z**H
405*
406 CALL zunmr2( 'Right', 'Conjugate transpose', m, n, l, b, ldb,
407 \$ tau, a, lda, work, info )
408 IF( wantq ) THEN
409*
410* Update Q := Q*Z**H
411*
412 CALL zunmr2( 'Right', 'Conjugate transpose', n, n, l, b,
413 \$ ldb, tau, q, ldq, work, info )
414 END IF
415*
416* Clean up B
417*
418 CALL zlaset( 'Full', l, n-l, czero, czero, b, ldb )
419 DO 60 j = n - l + 1, n
420 DO 50 i = j - n + l + 1, l
421 b( i, j ) = czero
422 50 CONTINUE
423 60 CONTINUE
424*
425 END IF
426*
427* Let N-L L
428* A = ( A11 A12 ) M,
429*
430* then the following does the complete QR decomposition of A11:
431*
432* A11 = U*( 0 T12 )*P1**H
433* ( 0 0 )
434*
435 DO 70 i = 1, n - l
436 iwork( i ) = 0
437 70 CONTINUE
438 CALL zgeqpf( m, n-l, a, lda, iwork, tau, work, rwork, info )
439*
440* Determine the effective rank of A11
441*
442 k = 0
443 DO 80 i = 1, min( m, n-l )
444 IF( cabs1( a( i, i ) ).GT.tola )
445 \$ k = k + 1
446 80 CONTINUE
447*
448* Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N )
449*
450 CALL zunm2r( 'Left', 'Conjugate transpose', m, l, min( m, n-l ),
451 \$ a, lda, tau, a( 1, n-l+1 ), lda, work, info )
452*
453 IF( wantu ) THEN
454*
455* Copy the details of U, and form U
456*
457 CALL zlaset( 'Full', m, m, czero, czero, u, ldu )
458 IF( m.GT.1 )
459 \$ CALL zlacpy( 'Lower', m-1, n-l, a( 2, 1 ), lda, u( 2, 1 ),
460 \$ ldu )
461 CALL zung2r( m, m, min( m, n-l ), u, ldu, tau, work, info )
462 END IF
463*
464 IF( wantq ) THEN
465*
466* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
467*
468 CALL zlapmt( forwrd, n, n-l, q, ldq, iwork )
469 END IF
470*
471* Clean up A: set the strictly lower triangular part of
472* A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
473*
474 DO 100 j = 1, k - 1
475 DO 90 i = j + 1, k
476 a( i, j ) = czero
477 90 CONTINUE
478 100 CONTINUE
479 IF( m.GT.k )
480 \$ CALL zlaset( 'Full', m-k, n-l, czero, czero, a( k+1, 1 ), lda )
481*
482 IF( n-l.GT.k ) THEN
483*
484* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
485*
486 CALL zgerq2( k, n-l, a, lda, tau, work, info )
487*
488 IF( wantq ) THEN
489*
490* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H
491*
492 CALL zunmr2( 'Right', 'Conjugate transpose', n, n-l, k, a,
493 \$ lda, tau, q, ldq, work, info )
494 END IF
495*
496* Clean up A
497*
498 CALL zlaset( 'Full', k, n-l-k, czero, czero, a, lda )
499 DO 120 j = n - l - k + 1, n - l
500 DO 110 i = j - n + l + k + 1, k
501 a( i, j ) = czero
502 110 CONTINUE
503 120 CONTINUE
504*
505 END IF
506*
507 IF( m.GT.k ) THEN
508*
509* QR factorization of A( K+1:M,N-L+1:N )
510*
511 CALL zgeqr2( m-k, l, a( k+1, n-l+1 ), lda, tau, work, info )
512*
513 IF( wantu ) THEN
514*
515* Update U(:,K+1:M) := U(:,K+1:M)*U1
516*
517 CALL zunm2r( 'Right', 'No transpose', m, m-k, min( m-k, l ),
518 \$ a( k+1, n-l+1 ), lda, tau, u( 1, k+1 ), ldu,
519 \$ work, info )
520 END IF
521*
522* Clean up
523*
524 DO 140 j = n - l + 1, n
525 DO 130 i = j - n + k + l + 1, m
526 a( i, j ) = czero
527 130 CONTINUE
528 140 CONTINUE
529*
530 END IF
531*
532 RETURN
533*
534* End of ZGGSVP
535*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zgeqr2(m, n, a, lda, tau, work, info)
ZGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Definition zgeqr2.f:130
subroutine zgerq2(m, n, a, lda, tau, work, info)
ZGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
Definition zgerq2.f:123
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:103
subroutine zlapmt(forwrd, m, n, x, ldx, k)
ZLAPMT performs a forward or backward permutation of the columns of a matrix.
Definition zlapmt.f:104
subroutine zlaset(uplo, m, n, alpha, beta, a, lda)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition zlaset.f:106
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine zung2r(m, n, k, a, lda, tau, work, info)
ZUNG2R
Definition zung2r.f:114
subroutine zunm2r(side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
ZUNM2R multiplies a general matrix by the unitary matrix from a QR factorization determined by cgeqrf...
Definition zunm2r.f:159
subroutine zunmr2(side, trans, m, n, k, a, lda, tau, c, ldc, work, info)
ZUNMR2 multiplies a general matrix by the unitary matrix from a RQ factorization determined by cgerqf...
Definition zunmr2.f:159
subroutine zgeqpf(m, n, a, lda, jpvt, tau, work, rwork, info)
ZGEQPF
Definition zgeqpf.f:148
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