 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

◆ dggsvp()

 subroutine dggsvp ( character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision TOLA, double precision TOLB, integer K, integer L, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldq, * ) Q, integer LDQ, integer, dimension( * ) IWORK, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO )

DGGSVP

Purpose:
``` This routine is deprecated and has been replaced by routine DGGSVP3.

DGGSVP computes orthogonal matrices U, V and Q such that

N-K-L  K    L
U**T*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**T*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**T,B**T)**T.

This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
DGGSVD.```
Parameters
 [in] JOBU ``` JOBU is CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed.``` [in] JOBV ``` JOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed.``` [in] JOBQ ``` JOBQ is CHARACTER*1 = 'Q': Orthogonal 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS. 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**T,B**T)**T.``` [out] U ``` U is DOUBLE PRECISION array, dimension (LDU,M) If JOBU = 'U', U contains the orthogonal 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 DOUBLE PRECISION array, dimension (LDV,P) If JOBV = 'V', V contains the orthogonal 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 DOUBLE PRECISION array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the orthogonal 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] TAU ` TAU is DOUBLE PRECISION array, dimension (N)` [out] WORK ` WORK is DOUBLE PRECISION 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.```
Date
December 2016
Further Details:
The subroutine uses LAPACK subroutine DGEQPF 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 258 of file dggsvp.f.

258 *
259 * -- LAPACK computational routine (version 3.7.0) --
260 * -- LAPACK is a software package provided by Univ. of Tennessee, --
261 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
262 * December 2016
263 *
264 * .. Scalar Arguments ..
265  CHARACTER jobq, jobu, jobv
266  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p
267  DOUBLE PRECISION tola, tolb
268 * ..
269 * .. Array Arguments ..
270  INTEGER iwork( * )
271  DOUBLE PRECISION a( lda, * ), b( ldb, * ), q( ldq, * ),
272  \$ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
273 * ..
274 *
275 * =====================================================================
276 *
277 * .. Parameters ..
278  DOUBLE PRECISION zero, one
279  parameter( zero = 0.0d+0, one = 1.0d+0 )
280 * ..
281 * .. Local Scalars ..
282  LOGICAL forwrd, wantq, wantu, wantv
283  INTEGER i, j
284 * ..
285 * .. External Functions ..
286  LOGICAL lsame
287  EXTERNAL lsame
288 * ..
289 * .. External Subroutines ..
290  EXTERNAL dgeqpf, dgeqr2, dgerq2, dlacpy, dlapmt, dlaset,
292 * ..
293 * .. Intrinsic Functions ..
294  INTRINSIC abs, max, min
295 * ..
296 * .. Executable Statements ..
297 *
298 * Test the input parameters
299 *
300  wantu = lsame( jobu, 'U' )
301  wantv = lsame( jobv, 'V' )
302  wantq = lsame( jobq, 'Q' )
303  forwrd = .true.
304 *
305  info = 0
306  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
307  info = -1
308  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
309  info = -2
310  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
311  info = -3
312  ELSE IF( m.LT.0 ) THEN
313  info = -4
314  ELSE IF( p.LT.0 ) THEN
315  info = -5
316  ELSE IF( n.LT.0 ) THEN
317  info = -6
318  ELSE IF( lda.LT.max( 1, m ) ) THEN
319  info = -8
320  ELSE IF( ldb.LT.max( 1, p ) ) THEN
321  info = -10
322  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
323  info = -16
324  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
325  info = -18
326  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
327  info = -20
328  END IF
329  IF( info.NE.0 ) THEN
330  CALL xerbla( 'DGGSVP', -info )
331  RETURN
332  END IF
333 *
334 * QR with column pivoting of B: B*P = V*( S11 S12 )
335 * ( 0 0 )
336 *
337  DO 10 i = 1, n
338  iwork( i ) = 0
339  10 CONTINUE
340  CALL dgeqpf( p, n, b, ldb, iwork, tau, work, info )
341 *
342 * Update A := A*P
343 *
344  CALL dlapmt( forwrd, m, n, a, lda, iwork )
345 *
346 * Determine the effective rank of matrix B.
347 *
348  l = 0
349  DO 20 i = 1, min( p, n )
350  IF( abs( b( i, i ) ).GT.tolb )
351  \$ l = l + 1
352  20 CONTINUE
353 *
354  IF( wantv ) THEN
355 *
356 * Copy the details of V, and form V.
357 *
358  CALL dlaset( 'Full', p, p, zero, zero, v, ldv )
359  IF( p.GT.1 )
360  \$ CALL dlacpy( 'Lower', p-1, n, b( 2, 1 ), ldb, v( 2, 1 ),
361  \$ ldv )
362  CALL dorg2r( p, p, min( p, n ), v, ldv, tau, work, info )
363  END IF
364 *
365 * Clean up B
366 *
367  DO 40 j = 1, l - 1
368  DO 30 i = j + 1, l
369  b( i, j ) = zero
370  30 CONTINUE
371  40 CONTINUE
372  IF( p.GT.l )
373  \$ CALL dlaset( 'Full', p-l, n, zero, zero, b( l+1, 1 ), ldb )
374 *
375  IF( wantq ) THEN
376 *
377 * Set Q = I and Update Q := Q*P
378 *
379  CALL dlaset( 'Full', n, n, zero, one, q, ldq )
380  CALL dlapmt( forwrd, n, n, q, ldq, iwork )
381  END IF
382 *
383  IF( p.GE.l .AND. n.NE.l ) THEN
384 *
385 * RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
386 *
387  CALL dgerq2( l, n, b, ldb, tau, work, info )
388 *
389 * Update A := A*Z**T
390 *
391  CALL dormr2( 'Right', 'Transpose', m, n, l, b, ldb, tau, a,
392  \$ lda, work, info )
393 *
394  IF( wantq ) THEN
395 *
396 * Update Q := Q*Z**T
397 *
398  CALL dormr2( 'Right', 'Transpose', n, n, l, b, ldb, tau, q,
399  \$ ldq, work, info )
400  END IF
401 *
402 * Clean up B
403 *
404  CALL dlaset( 'Full', l, n-l, zero, zero, b, ldb )
405  DO 60 j = n - l + 1, n
406  DO 50 i = j - n + l + 1, l
407  b( i, j ) = zero
408  50 CONTINUE
409  60 CONTINUE
410 *
411  END IF
412 *
413 * Let N-L L
414 * A = ( A11 A12 ) M,
415 *
416 * then the following does the complete QR decomposition of A11:
417 *
418 * A11 = U*( 0 T12 )*P1**T
419 * ( 0 0 )
420 *
421  DO 70 i = 1, n - l
422  iwork( i ) = 0
423  70 CONTINUE
424  CALL dgeqpf( m, n-l, a, lda, iwork, tau, work, info )
425 *
426 * Determine the effective rank of A11
427 *
428  k = 0
429  DO 80 i = 1, min( m, n-l )
430  IF( abs( a( i, i ) ).GT.tola )
431  \$ k = k + 1
432  80 CONTINUE
433 *
434 * Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
435 *
436  CALL dorm2r( 'Left', 'Transpose', m, l, min( m, n-l ), a, lda,
437  \$ tau, a( 1, n-l+1 ), lda, work, info )
438 *
439  IF( wantu ) THEN
440 *
441 * Copy the details of U, and form U
442 *
443  CALL dlaset( 'Full', m, m, zero, zero, u, ldu )
444  IF( m.GT.1 )
445  \$ CALL dlacpy( 'Lower', m-1, n-l, a( 2, 1 ), lda, u( 2, 1 ),
446  \$ ldu )
447  CALL dorg2r( m, m, min( m, n-l ), u, ldu, tau, work, info )
448  END IF
449 *
450  IF( wantq ) THEN
451 *
452 * Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
453 *
454  CALL dlapmt( forwrd, n, n-l, q, ldq, iwork )
455  END IF
456 *
457 * Clean up A: set the strictly lower triangular part of
458 * A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
459 *
460  DO 100 j = 1, k - 1
461  DO 90 i = j + 1, k
462  a( i, j ) = zero
463  90 CONTINUE
464  100 CONTINUE
465  IF( m.GT.k )
466  \$ CALL dlaset( 'Full', m-k, n-l, zero, zero, a( k+1, 1 ), lda )
467 *
468  IF( n-l.GT.k ) THEN
469 *
470 * RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
471 *
472  CALL dgerq2( k, n-l, a, lda, tau, work, info )
473 *
474  IF( wantq ) THEN
475 *
476 * Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
477 *
478  CALL dormr2( 'Right', 'Transpose', n, n-l, k, a, lda, tau,
479  \$ q, ldq, work, info )
480  END IF
481 *
482 * Clean up A
483 *
484  CALL dlaset( 'Full', k, n-l-k, zero, zero, a, lda )
485  DO 120 j = n - l - k + 1, n - l
486  DO 110 i = j - n + l + k + 1, k
487  a( i, j ) = zero
488  110 CONTINUE
489  120 CONTINUE
490 *
491  END IF
492 *
493  IF( m.GT.k ) THEN
494 *
495 * QR factorization of A( K+1:M,N-L+1:N )
496 *
497  CALL dgeqr2( m-k, l, a( k+1, n-l+1 ), lda, tau, work, info )
498 *
499  IF( wantu ) THEN
500 *
501 * Update U(:,K+1:M) := U(:,K+1:M)*U1
502 *
503  CALL dorm2r( 'Right', 'No transpose', m, m-k, min( m-k, l ),
504  \$ a( k+1, n-l+1 ), lda, tau, u( 1, k+1 ), ldu,
505  \$ work, info )
506  END IF
507 *
508 * Clean up
509 *
510  DO 140 j = n - l + 1, n
511  DO 130 i = j - n + k + l + 1, m
512  a( i, j ) = zero
513  130 CONTINUE
514  140 CONTINUE
515 *
516  END IF
517 *
518  RETURN
519 *
520 * End of DGGSVP
521 *
subroutine dgeqr2(M, N, A, LDA, TAU, WORK, INFO)
DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm...
Definition: dgeqr2.f:123
subroutine dorg2r(M, N, K, A, LDA, TAU, WORK, INFO)
DORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf ...
Definition: dorg2r.f:116
subroutine dlapmt(FORWRD, M, N, X, LDX, K)
DLAPMT performs a forward or backward permutation of the columns of a matrix.
Definition: dlapmt.f:106
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dgeqpf(M, N, A, LDA, JPVT, TAU, WORK, INFO)
DGEQPF
Definition: dgeqpf.f:144
subroutine dorm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
DORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sge...
Definition: dorm2r.f:161
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dormr2(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
DORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sge...
Definition: dormr2.f:161
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: dlaset.f:112
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
subroutine dgerq2(M, N, A, LDA, TAU, WORK, INFO)
DGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm...
Definition: dgerq2.f:125
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