LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ sggsvp()

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

SGGSVP

Purpose:
``` This routine is deprecated and has been replaced by routine SGGSVP3.

SGGSVP 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
SGGSVD.```
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 REAL 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 REAL 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 REAL` [in] TOLB ``` TOLB is REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (N)` [out] WORK ` WORK is REAL 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 SGEQPF 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 253 of file sggsvp.f.

256*
257* -- LAPACK computational routine --
258* -- LAPACK is a software package provided by Univ. of Tennessee, --
259* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
260*
261* .. Scalar Arguments ..
262 CHARACTER JOBQ, JOBU, JOBV
263 INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
264 REAL TOLA, TOLB
265* ..
266* .. Array Arguments ..
267 INTEGER IWORK( * )
268 REAL A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
269 \$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
270* ..
271*
272* =====================================================================
273*
274* .. Parameters ..
275 REAL ZERO, ONE
276 parameter( zero = 0.0e+0, one = 1.0e+0 )
277* ..
278* .. Local Scalars ..
279 LOGICAL FORWRD, WANTQ, WANTU, WANTV
280 INTEGER I, J
281* ..
282* .. External Functions ..
283 LOGICAL LSAME
284 EXTERNAL lsame
285* ..
286* .. External Subroutines ..
287 EXTERNAL sgeqpf, sgeqr2, sgerq2, slacpy, slapmt, slaset,
289* ..
290* .. Intrinsic Functions ..
291 INTRINSIC abs, max, min
292* ..
293* .. Executable Statements ..
294*
295* Test the input parameters
296*
297 wantu = lsame( jobu, 'U' )
298 wantv = lsame( jobv, 'V' )
299 wantq = lsame( jobq, 'Q' )
300 forwrd = .true.
301*
302 info = 0
303 IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
304 info = -1
305 ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
306 info = -2
307 ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
308 info = -3
309 ELSE IF( m.LT.0 ) THEN
310 info = -4
311 ELSE IF( p.LT.0 ) THEN
312 info = -5
313 ELSE IF( n.LT.0 ) THEN
314 info = -6
315 ELSE IF( lda.LT.max( 1, m ) ) THEN
316 info = -8
317 ELSE IF( ldb.LT.max( 1, p ) ) THEN
318 info = -10
319 ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
320 info = -16
321 ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
322 info = -18
323 ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
324 info = -20
325 END IF
326 IF( info.NE.0 ) THEN
327 CALL xerbla( 'SGGSVP', -info )
328 RETURN
329 END IF
330*
331* QR with column pivoting of B: B*P = V*( S11 S12 )
332* ( 0 0 )
333*
334 DO 10 i = 1, n
335 iwork( i ) = 0
336 10 CONTINUE
337 CALL sgeqpf( p, n, b, ldb, iwork, tau, work, info )
338*
339* Update A := A*P
340*
341 CALL slapmt( forwrd, m, n, a, lda, iwork )
342*
343* Determine the effective rank of matrix B.
344*
345 l = 0
346 DO 20 i = 1, min( p, n )
347 IF( abs( b( i, i ) ).GT.tolb )
348 \$ l = l + 1
349 20 CONTINUE
350*
351 IF( wantv ) THEN
352*
353* Copy the details of V, and form V.
354*
355 CALL slaset( 'Full', p, p, zero, zero, v, ldv )
356 IF( p.GT.1 )
357 \$ CALL slacpy( 'Lower', p-1, n, b( 2, 1 ), ldb, v( 2, 1 ),
358 \$ ldv )
359 CALL sorg2r( p, p, min( p, n ), v, ldv, tau, work, info )
360 END IF
361*
362* Clean up B
363*
364 DO 40 j = 1, l - 1
365 DO 30 i = j + 1, l
366 b( i, j ) = zero
367 30 CONTINUE
368 40 CONTINUE
369 IF( p.GT.l )
370 \$ CALL slaset( 'Full', p-l, n, zero, zero, b( l+1, 1 ), ldb )
371*
372 IF( wantq ) THEN
373*
374* Set Q = I and Update Q := Q*P
375*
376 CALL slaset( 'Full', n, n, zero, one, q, ldq )
377 CALL slapmt( forwrd, n, n, q, ldq, iwork )
378 END IF
379*
380 IF( p.GE.l .AND. n.NE.l ) THEN
381*
382* RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z
383*
384 CALL sgerq2( l, n, b, ldb, tau, work, info )
385*
386* Update A := A*Z**T
387*
388 CALL sormr2( 'Right', 'Transpose', m, n, l, b, ldb, tau, a,
389 \$ lda, work, info )
390*
391 IF( wantq ) THEN
392*
393* Update Q := Q*Z**T
394*
395 CALL sormr2( 'Right', 'Transpose', n, n, l, b, ldb, tau, q,
396 \$ ldq, work, info )
397 END IF
398*
399* Clean up B
400*
401 CALL slaset( 'Full', l, n-l, zero, zero, b, ldb )
402 DO 60 j = n - l + 1, n
403 DO 50 i = j - n + l + 1, l
404 b( i, j ) = zero
405 50 CONTINUE
406 60 CONTINUE
407*
408 END IF
409*
410* Let N-L L
411* A = ( A11 A12 ) M,
412*
413* then the following does the complete QR decomposition of A11:
414*
415* A11 = U*( 0 T12 )*P1**T
416* ( 0 0 )
417*
418 DO 70 i = 1, n - l
419 iwork( i ) = 0
420 70 CONTINUE
421 CALL sgeqpf( m, n-l, a, lda, iwork, tau, work, info )
422*
423* Determine the effective rank of A11
424*
425 k = 0
426 DO 80 i = 1, min( m, n-l )
427 IF( abs( a( i, i ) ).GT.tola )
428 \$ k = k + 1
429 80 CONTINUE
430*
431* Update A12 := U**T*A12, where A12 = A( 1:M, N-L+1:N )
432*
433 CALL sorm2r( 'Left', 'Transpose', m, l, min( m, n-l ), a, lda,
434 \$ tau, a( 1, n-l+1 ), lda, work, info )
435*
436 IF( wantu ) THEN
437*
438* Copy the details of U, and form U
439*
440 CALL slaset( 'Full', m, m, zero, zero, u, ldu )
441 IF( m.GT.1 )
442 \$ CALL slacpy( 'Lower', m-1, n-l, a( 2, 1 ), lda, u( 2, 1 ),
443 \$ ldu )
444 CALL sorg2r( m, m, min( m, n-l ), u, ldu, tau, work, info )
445 END IF
446*
447 IF( wantq ) THEN
448*
449* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
450*
451 CALL slapmt( forwrd, n, n-l, q, ldq, iwork )
452 END IF
453*
454* Clean up A: set the strictly lower triangular part of
455* A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
456*
457 DO 100 j = 1, k - 1
458 DO 90 i = j + 1, k
459 a( i, j ) = zero
460 90 CONTINUE
461 100 CONTINUE
462 IF( m.GT.k )
463 \$ CALL slaset( 'Full', m-k, n-l, zero, zero, a( k+1, 1 ), lda )
464*
465 IF( n-l.GT.k ) THEN
466*
467* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
468*
469 CALL sgerq2( k, n-l, a, lda, tau, work, info )
470*
471 IF( wantq ) THEN
472*
473* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1**T
474*
475 CALL sormr2( 'Right', 'Transpose', n, n-l, k, a, lda, tau,
476 \$ q, ldq, work, info )
477 END IF
478*
479* Clean up A
480*
481 CALL slaset( 'Full', k, n-l-k, zero, zero, a, lda )
482 DO 120 j = n - l - k + 1, n - l
483 DO 110 i = j - n + l + k + 1, k
484 a( i, j ) = zero
485 110 CONTINUE
486 120 CONTINUE
487*
488 END IF
489*
490 IF( m.GT.k ) THEN
491*
492* QR factorization of A( K+1:M,N-L+1:N )
493*
494 CALL sgeqr2( m-k, l, a( k+1, n-l+1 ), lda, tau, work, info )
495*
496 IF( wantu ) THEN
497*
498* Update U(:,K+1:M) := U(:,K+1:M)*U1
499*
500 CALL sorm2r( 'Right', 'No transpose', m, m-k, min( m-k, l ),
501 \$ a( k+1, n-l+1 ), lda, tau, u( 1, k+1 ), ldu,
502 \$ work, info )
503 END IF
504*
505* Clean up
506*
507 DO 140 j = n - l + 1, n
508 DO 130 i = j - n + k + l + 1, m
509 a( i, j ) = zero
510 130 CONTINUE
511 140 CONTINUE
512*
513 END IF
514*
515 RETURN
516*
517* End of SGGSVP
518*
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sgerq2(M, N, A, LDA, TAU, WORK, INFO)
SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
Definition: sgerq2.f:123
subroutine sgeqpf(M, N, A, LDA, JPVT, TAU, WORK, INFO)
SGEQPF
Definition: sgeqpf.f:142
subroutine sgeqr2(M, N, A, LDA, TAU, WORK, INFO)
SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Definition: sgeqr2.f:130
subroutine slapmt(FORWRD, M, N, X, LDX, K)
SLAPMT performs a forward or backward permutation of the columns of a matrix.
Definition: slapmt.f:104
subroutine sormr2(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORMR2 multiplies a general matrix by the orthogonal matrix from a RQ factorization determined by sge...
Definition: sormr2.f:159
subroutine sorm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sge...
Definition: sorm2r.f:159
subroutine sorg2r(M, N, K, A, LDA, TAU, WORK, INFO)
SORG2R generates all or part of the orthogonal matrix Q from a QR factorization determined by sgeqrf ...
Definition: sorg2r.f:114
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