 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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``` This routine is deprecated and has been replaced by routine SGGSVP3.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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