LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ 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

Download ZGGSVP + dependencies [TGZ] [ZIP] [TXT]

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 This routine is deprecated and has been replaced by routine ZGGSVP3.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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016
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 267 of file zggsvp.f.

267 *
268 * -- LAPACK computational routine (version 3.7.0) --
269 * -- LAPACK is a software package provided by Univ. of Tennessee, --
270 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
271 * December 2016
272 *
273 * .. Scalar Arguments ..
274  CHARACTER jobq, jobu, jobv
275  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p
276  DOUBLE PRECISION tola, tolb
277 * ..
278 * .. Array Arguments ..
279  INTEGER iwork( * )
280  DOUBLE PRECISION rwork( * )
281  COMPLEX*16 a( lda, * ), b( ldb, * ), q( ldq, * ),
282  $ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
283 * ..
284 *
285 * =====================================================================
286 *
287 * .. Parameters ..
288  COMPLEX*16 czero, cone
289  parameter( czero = ( 0.0d+0, 0.0d+0 ),
290  $ cone = ( 1.0d+0, 0.0d+0 ) )
291 * ..
292 * .. Local Scalars ..
293  LOGICAL forwrd, wantq, wantu, wantv
294  INTEGER i, j
295  COMPLEX*16 t
296 * ..
297 * .. External Functions ..
298  LOGICAL lsame
299  EXTERNAL lsame
300 * ..
301 * .. External Subroutines ..
302  EXTERNAL xerbla, zgeqpf, zgeqr2, zgerq2, zlacpy, zlapmt,
304 * ..
305 * .. Intrinsic Functions ..
306  INTRINSIC abs, dble, dimag, max, min
307 * ..
308 * .. Statement Functions ..
309  DOUBLE PRECISION cabs1
310 * ..
311 * .. Statement Function definitions ..
312  cabs1( t ) = abs( dble( t ) ) + abs( dimag( t ) )
313 * ..
314 * .. Executable Statements ..
315 *
316 * Test the input parameters
317 *
318  wantu = lsame( jobu, 'U' )
319  wantv = lsame( jobv, 'V' )
320  wantq = lsame( jobq, 'Q' )
321  forwrd = .true.
322 *
323  info = 0
324  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
325  info = -1
326  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
327  info = -2
328  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
329  info = -3
330  ELSE IF( m.LT.0 ) THEN
331  info = -4
332  ELSE IF( p.LT.0 ) THEN
333  info = -5
334  ELSE IF( n.LT.0 ) THEN
335  info = -6
336  ELSE IF( lda.LT.max( 1, m ) ) THEN
337  info = -8
338  ELSE IF( ldb.LT.max( 1, p ) ) THEN
339  info = -10
340  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
341  info = -16
342  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
343  info = -18
344  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
345  info = -20
346  END IF
347  IF( info.NE.0 ) THEN
348  CALL xerbla( 'ZGGSVP', -info )
349  RETURN
350  END IF
351 *
352 * QR with column pivoting of B: B*P = V*( S11 S12 )
353 * ( 0 0 )
354 *
355  DO 10 i = 1, n
356  iwork( i ) = 0
357  10 CONTINUE
358  CALL zgeqpf( p, n, b, ldb, iwork, tau, work, rwork, info )
359 *
360 * Update A := A*P
361 *
362  CALL zlapmt( forwrd, m, n, a, lda, iwork )
363 *
364 * Determine the effective rank of matrix B.
365 *
366  l = 0
367  DO 20 i = 1, min( p, n )
368  IF( cabs1( b( i, i ) ).GT.tolb )
369  $ l = l + 1
370  20 CONTINUE
371 *
372  IF( wantv ) THEN
373 *
374 * Copy the details of V, and form V.
375 *
376  CALL zlaset( 'Full', p, p, czero, czero, v, ldv )
377  IF( p.GT.1 )
378  $ CALL zlacpy( 'Lower', p-1, n, b( 2, 1 ), ldb, v( 2, 1 ),
379  $ ldv )
380  CALL zung2r( p, p, min( p, n ), v, ldv, tau, work, info )
381  END IF
382 *
383 * Clean up B
384 *
385  DO 40 j = 1, l - 1
386  DO 30 i = j + 1, l
387  b( i, j ) = czero
388  30 CONTINUE
389  40 CONTINUE
390  IF( p.GT.l )
391  $ CALL zlaset( 'Full', p-l, n, czero, czero, b( l+1, 1 ), ldb )
392 *
393  IF( wantq ) THEN
394 *
395 * Set Q = I and Update Q := Q*P
396 *
397  CALL zlaset( 'Full', n, n, czero, cone, q, ldq )
398  CALL zlapmt( forwrd, n, n, q, ldq, iwork )
399  END IF
400 *
401  IF( p.GE.l .AND. n.NE.l ) THEN
402 *
403 * RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
404 *
405  CALL zgerq2( l, n, b, ldb, tau, work, info )
406 *
407 * Update A := A*Z**H
408 *
409  CALL zunmr2( 'Right', 'Conjugate transpose', m, n, l, b, ldb,
410  $ tau, a, lda, work, info )
411  IF( wantq ) THEN
412 *
413 * Update Q := Q*Z**H
414 *
415  CALL zunmr2( 'Right', 'Conjugate transpose', n, n, l, b,
416  $ ldb, tau, q, ldq, work, info )
417  END IF
418 *
419 * Clean up B
420 *
421  CALL zlaset( 'Full', l, n-l, czero, czero, b, ldb )
422  DO 60 j = n - l + 1, n
423  DO 50 i = j - n + l + 1, l
424  b( i, j ) = czero
425  50 CONTINUE
426  60 CONTINUE
427 *
428  END IF
429 *
430 * Let N-L L
431 * A = ( A11 A12 ) M,
432 *
433 * then the following does the complete QR decomposition of A11:
434 *
435 * A11 = U*( 0 T12 )*P1**H
436 * ( 0 0 )
437 *
438  DO 70 i = 1, n - l
439  iwork( i ) = 0
440  70 CONTINUE
441  CALL zgeqpf( m, n-l, a, lda, iwork, tau, work, rwork, info )
442 *
443 * Determine the effective rank of A11
444 *
445  k = 0
446  DO 80 i = 1, min( m, n-l )
447  IF( cabs1( a( i, i ) ).GT.tola )
448  $ k = k + 1
449  80 CONTINUE
450 *
451 * Update A12 := U**H*A12, where A12 = A( 1:M, N-L+1:N )
452 *
453  CALL zunm2r( 'Left', 'Conjugate transpose', m, l, min( m, n-l ),
454  $ a, lda, tau, a( 1, n-l+1 ), lda, work, info )
455 *
456  IF( wantu ) THEN
457 *
458 * Copy the details of U, and form U
459 *
460  CALL zlaset( 'Full', m, m, czero, czero, u, ldu )
461  IF( m.GT.1 )
462  $ CALL zlacpy( 'Lower', m-1, n-l, a( 2, 1 ), lda, u( 2, 1 ),
463  $ ldu )
464  CALL zung2r( m, m, min( m, n-l ), u, ldu, tau, work, info )
465  END IF
466 *
467  IF( wantq ) THEN
468 *
469 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
470 *
471  CALL zlapmt( forwrd, n, n-l, q, ldq, iwork )
472  END IF
473 *
474 * Clean up A: set the strictly lower triangular part of
475 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
476 *
477  DO 100 j = 1, k - 1
478  DO 90 i = j + 1, k
479  a( i, j ) = czero
480  90 CONTINUE
481  100 CONTINUE
482  IF( m.GT.k )
483  $ CALL zlaset( 'Full', m-k, n-l, czero, czero, a( k+1, 1 ), lda )
484 *
485  IF( n-l.GT.k ) THEN
486 *
487 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
488 *
489  CALL zgerq2( k, n-l, a, lda, tau, work, info )
490 *
491  IF( wantq ) THEN
492 *
493 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**H
494 *
495  CALL zunmr2( 'Right', 'Conjugate transpose', n, n-l, k, a,
496  $ lda, tau, q, ldq, work, info )
497  END IF
498 *
499 * Clean up A
500 *
501  CALL zlaset( 'Full', k, n-l-k, czero, czero, a, lda )
502  DO 120 j = n - l - k + 1, n - l
503  DO 110 i = j - n + l + k + 1, k
504  a( i, j ) = czero
505  110 CONTINUE
506  120 CONTINUE
507 *
508  END IF
509 *
510  IF( m.GT.k ) THEN
511 *
512 * QR factorization of A( K+1:M,N-L+1:N )
513 *
514  CALL zgeqr2( m-k, l, a( k+1, n-l+1 ), lda, tau, work, info )
515 *
516  IF( wantu ) THEN
517 *
518 * Update U(:,K+1:M) := U(:,K+1:M)*U1
519 *
520  CALL zunm2r( 'Right', 'No transpose', m, m-k, min( m-k, l ),
521  $ a( k+1, n-l+1 ), lda, tau, u( 1, k+1 ), ldu,
522  $ work, info )
523  END IF
524 *
525 * Clean up
526 *
527  DO 140 j = n - l + 1, n
528  DO 130 i = j - n + k + l + 1, m
529  a( i, j ) = czero
530  130 CONTINUE
531  140 CONTINUE
532 *
533  END IF
534 *
535  RETURN
536 *
537 * End of ZGGSVP
538 *
subroutine zgeqpf(M, N, A, LDA, JPVT, TAU, WORK, RWORK, INFO)
ZGEQPF
Definition: zgeqpf.f:150
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:161
subroutine zung2r(M, N, K, A, LDA, TAU, WORK, INFO)
ZUNG2R
Definition: zung2r.f:116
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:161
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
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:123
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:108
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zlapmt(FORWRD, M, N, X, LDX, K)
ZLAPMT performs a forward or backward permutation of the columns of a matrix.
Definition: zlapmt.f:106
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:125
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