 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dgesvdq()

 subroutine dgesvdq ( character JOBA, character JOBP, character JOBR, character JOBU, character JOBV, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * ) V, integer LDV, integer NUMRANK, integer, dimension( * ) IWORK, integer LIWORK, double precision, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer INFO )

DGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

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Purpose:
 DGESVDQ computes the singular value decomposition (SVD) of a real
M-by-N matrix A, where M >= N. The SVD of A is written as
[++]   [xx]   [x0]   [xx]
A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]
[++]   [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
Parameters
 [in] JOBA  JOBA is CHARACTER*1 Specifies the level of accuracy in the computed SVD = 'A' The requested accuracy corresponds to having the backward error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F, where EPS = DLAMCH('Epsilon'). This authorises DGESVDQ to truncate the computed triangular factor in a rank revealing QR factorization whenever the truncated part is below the threshold of the order of EPS * ||A||_F. This is aggressive truncation level. = 'M' Similarly as with 'A', but the truncation is more gentle: it is allowed only when there is a drop on the diagonal of the triangular factor in the QR factorization. This is medium truncation level. = 'H' High accuracy requested. No numerical rank determination based on the rank revealing QR factorization is attempted. = 'E' Same as 'H', and in addition the condition number of column scaled A is estimated and returned in RWORK(1). N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1) [in] JOBP  JOBP is CHARACTER*1 = 'P' The rows of A are ordered in decreasing order with respect to ||A(i,:)||_\infty. This enhances numerical accuracy at the cost of extra data movement. Recommended for numerical robustness. = 'N' No row pivoting. [in] JOBR  JOBR is CHARACTER*1 = 'T' After the initial pivoted QR factorization, DGESVD is applied to the transposed R**T of the computed triangular factor R. This involves some extra data movement (matrix transpositions). Useful for experiments, research and development. = 'N' The triangular factor R is given as input to DGESVD. This may be preferred as it involves less data movement. [in] JOBU  JOBU is CHARACTER*1 = 'A' All M left singular vectors are computed and returned in the matrix U. See the description of U. = 'S' or 'U' N = min(M,N) left singular vectors are computed and returned in the matrix U. See the description of U. = 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular vectors are computed and returned in the matrix U. = 'F' The N left singular vectors are returned in factored form as the product of the Q factor from the initial QR factorization and the N left singular vectors of (R**T , 0)**T. If row pivoting is used, then the necessary information on the row pivoting is stored in IWORK(N+1:N+M-1). = 'N' The left singular vectors are not computed. [in] JOBV  JOBV is CHARACTER*1 = 'A', 'V' All N right singular vectors are computed and returned in the matrix V. = 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular vectors are computed and returned in the matrix V. This option is allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal. = 'N' The right singular vectors are not computed. [in] M  M is INTEGER The number of rows of the input matrix A. M >= 0. [in] N  N is INTEGER The number of columns of the input matrix A. M >= N >= 0. [in,out] A  A is DOUBLE PRECISION array of dimensions LDA x N On entry, the input matrix A. On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains the Householder vectors as stored by DGEQP3. If JOBU = 'F', these Householder vectors together with WORK(1:N) can be used to restore the Q factors from the initial pivoted QR factorization of A. See the description of U. [in] LDA  LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M). [out] S  S is DOUBLE PRECISION array of dimension N. The singular values of A, ordered so that S(i) >= S(i+1). [out] U  U is DOUBLE PRECISION array, dimension LDU x M if JOBU = 'A'; see the description of LDU. In this case, on exit, U contains the M left singular vectors. LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this case, U contains the leading N or the leading NUMRANK left singular vectors. LDU x N if JOBU = 'F' ; see the description of LDU. In this case U contains N x N orthogonal matrix that can be used to form the left singular vectors. If JOBU = 'N', U is not referenced. [in] LDU  LDU is INTEGER. The leading dimension of the array U. If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M). If JOBU = 'F', LDU >= max(1,N). Otherwise, LDU >= 1. [out] V  V is DOUBLE PRECISION array, dimension LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' . If JOBV = 'A', or 'V', V contains the N-by-N orthogonal matrix V**T; If JOBV = 'R', V contains the first NUMRANK rows of V**T (the right singular vectors, stored rowwise, of the NUMRANK largest singular values). If JOBV = 'N' and JOBA = 'E', V is used as a workspace. If JOBV = 'N', and JOBA.NE.'E', V is not referenced. [in] LDV  LDV is INTEGER The leading dimension of the array V. If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N). Otherwise, LDV >= 1. [out] NUMRANK  NUMRANK is INTEGER NUMRANK is the numerical rank first determined after the rank revealing QR factorization, following the strategy specified by the value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK leading singular values and vectors are then requested in the call of DGESVD. The final value of NUMRANK might be further reduced if some singular values are computed as zeros. [out] IWORK  IWORK is INTEGER array, dimension (max(1, LIWORK)). On exit, IWORK(1:N) contains column pivoting permutation of the rank revealing QR factorization. If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence of row swaps used in row pivoting. These can be used to restore the left singular vectors in the case JOBU = 'F'. If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, IWORK(1) returns the minimal LIWORK. [in] LIWORK  LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= N + M - 1, if JOBP = 'P' and JOBA .NE. 'E'; LIWORK >= N if JOBP = 'N' and JOBA .NE. 'E'; LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E'; LIWORK >= N + N if JOBP = 'N' and JOBA = 'E'. If LIWORK = -1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA. [out] WORK  WORK is DOUBLE PRECISION array, dimension (max(2, LWORK)), used as a workspace. On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters needed to recover the Q factor from the QR factorization computed by DGEQP3. If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, WORK(1) returns the optimal LWORK, and WORK(2) returns the minimal LWORK. [in,out] LWORK  LWORK is INTEGER The dimension of the array WORK. It is determined as follows: Let LWQP3 = 3*N+1, LWCON = 3*N, and let LWORQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U' { MAX( M, 1 ), if JOBU = 'A' LWSVD = MAX( 5*N, 1 ) LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ), LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 ) Then the minimal value of LWORK is: = MAX( N + LWQP3, LWSVD ) if only the singular values are needed; = MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed, and a scaled condition estimate requested; = N + MAX( LWQP3, LWSVD, LWORQ ) if the singular values and the left singular vectors are requested; = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the singular values and the left singular vectors are requested, and also a scaled condition estimate requested; = N + MAX( LWQP3, LWSVD ) if the singular values and the right singular vectors are requested; = N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right singular vectors are requested, and also a scaled condition etimate requested; = N + MAX( LWQP3, LWSVD, LWORQ ) if the full SVD is requested with JOBV = 'R'; independent of JOBR; = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested, JOBV = 'R' and, also a scaled condition estimate requested; independent of JOBR; = MAX( N + MAX( LWQP3, LWSVD, LWORQ ), N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the full SVD is requested with JOBV = 'A' or 'V', and JOBR ='N' = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ), N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) ) if the full SVD is requested with JOBV = 'A' or 'V', and JOBR ='N', and also a scaled condition number estimate requested. = MAX( N + MAX( LWQP3, LWSVD, LWORQ ), N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the full SVD is requested with JOBV = 'A', 'V', and JOBR ='T' = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ), N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the full SVD is requested with JOBV = 'A' or 'V', and JOBR ='T', and also a scaled condition number estimate requested. Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ). If LWORK = -1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA. [out] RWORK  RWORK is DOUBLE PRECISION array, dimension (max(1, LRWORK)). On exit, 1. If JOBA = 'E', RWORK(1) contains an estimate of the condition number of column scaled A. If A = C * D where D is diagonal and C has unit columns in the Euclidean norm, then, assuming full column rank, N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1). Otherwise, RWORK(1) = -1. 2. RWORK(2) contains the number of singular values computed as exact zeros in DGESVD applied to the upper triangular or trapezoidal R (from the initial QR factorization). In case of early exit (no call to DGESVD, such as in the case of zero matrix) RWORK(2) = -1. If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, RWORK(1) returns the minimal LRWORK. [in] LRWORK  LRWORK is INTEGER. The dimension of the array RWORK. If JOBP ='P', then LRWORK >= MAX(2, M). Otherwise, LRWORK >= 2 If LRWORK = -1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA. [out] INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if DBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B (computed in DGESVD) did not converge to zero.
Further Details:
   1. The data movement (matrix transpose) is coded using simple nested
DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
Those DO-loops are easily identified in this source code - by the CONTINUE
statements labeled with 11**. In an optimized version of this code, the
nested DO loops should be replaced with calls to an optimized subroutine.
2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
column norm overflow. This is the minial precaution and it is left to the
SVD routine (CGESVD) to do its own preemptive scaling if potential over-
or underflows are detected. To avoid repeated scanning of the array A,
an optimal implementation would do all necessary scaling before calling
CGESVD and the scaling in CGESVD can be switched off.
3. Other comments related to code optimization are given in comments in the
code, enlosed in [[double brackets]].
Bugs, examples and comments
  Please report all bugs and send interesting examples and/or comments to
drmac@math.hr. Thank you.
References
   Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
44(1): 11:1-11:30 (2017)

SIGMA library, xGESVDQ section updated February 2016.
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Contributors:
 Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr

Definition at line 412 of file dgesvdq.f.

415 * .. Scalar Arguments ..
416  IMPLICIT NONE
417  CHARACTER JOBA, JOBP, JOBR, JOBU, JOBV
418  INTEGER M, N, LDA, LDU, LDV, NUMRANK, LIWORK, LWORK, LRWORK,
419  $INFO 420 * .. 421 * .. Array Arguments .. 422 DOUBLE PRECISION A( LDA, * ), U( LDU, * ), V( LDV, * ), WORK( * ) 423 DOUBLE PRECISION S( * ), RWORK( * ) 424 INTEGER IWORK( * ) 425 * 426 * ===================================================================== 427 * 428 * .. Parameters .. 429 DOUBLE PRECISION ZERO, ONE 430 parameter( zero = 0.0d0, one = 1.0d0 ) 431 * .. Local Scalars .. 432 INTEGER IERR, IWOFF, NR, N1, OPTRATIO, p, q 433 INTEGER LWCON, LWQP3, LWRK_DGELQF, LWRK_DGESVD, LWRK_DGESVD2, 434$ LWRK_DGEQP3, LWRK_DGEQRF, LWRK_DORMLQ, LWRK_DORMQR,
435  $LWRK_DORMQR2, LWLQF, LWQRF, LWSVD, LWSVD2, LWORQ, 436$ LWORQ2, LWORLQ, MINWRK, MINWRK2, OPTWRK, OPTWRK2,
437  $IMINWRK, RMINWRK 438 LOGICAL ACCLA, ACCLM, ACCLH, ASCALED, CONDA, DNTWU, DNTWV, 439$ LQUERY, LSVC0, LSVEC, ROWPRM, RSVEC, RTRANS, WNTUA,
440  $WNTUF, WNTUR, WNTUS, WNTVA, WNTVR 441 DOUBLE PRECISION BIG, EPSLN, RTMP, SCONDA, SFMIN 442 * .. Local Arrays 443 DOUBLE PRECISION RDUMMY(1) 444 * .. 445 * .. External Subroutines (BLAS, LAPACK) 446 EXTERNAL dgelqf, dgeqp3, dgeqrf, dgesvd, dlacpy, dlapmt, 448$ dormqr, xerbla
449 * ..
450 * .. External Functions (BLAS, LAPACK)
451  LOGICAL LSAME
452  INTEGER IDAMAX
453  DOUBLE PRECISION DLANGE, DNRM2, DLAMCH
454  EXTERNAL dlange, lsame, idamax, dnrm2, dlamch
455 * ..
456 * .. Intrinsic Functions ..
457 *
458  INTRINSIC abs, max, min, dble, sqrt
459 *
460 * Test the input arguments
461 *
462  wntus = lsame( jobu, 'S' ) .OR. lsame( jobu, 'U' )
463  wntur = lsame( jobu, 'R' )
464  wntua = lsame( jobu, 'A' )
465  wntuf = lsame( jobu, 'F' )
466  lsvc0 = wntus .OR. wntur .OR. wntua
467  lsvec = lsvc0 .OR. wntuf
468  dntwu = lsame( jobu, 'N' )
469 *
470  wntvr = lsame( jobv, 'R' )
471  wntva = lsame( jobv, 'A' ) .OR. lsame( jobv, 'V' )
472  rsvec = wntvr .OR. wntva
473  dntwv = lsame( jobv, 'N' )
474 *
475  accla = lsame( joba, 'A' )
476  acclm = lsame( joba, 'M' )
477  conda = lsame( joba, 'E' )
478  acclh = lsame( joba, 'H' ) .OR. conda
479 *
480  rowprm = lsame( jobp, 'P' )
481  rtrans = lsame( jobr, 'T' )
482 *
483  IF ( rowprm ) THEN
484  IF ( conda ) THEN
485  iminwrk = max( 1, n + m - 1 + n )
486  ELSE
487  iminwrk = max( 1, n + m - 1 )
488  END IF
489  rminwrk = max( 2, m )
490  ELSE
491  IF ( conda ) THEN
492  iminwrk = max( 1, n + n )
493  ELSE
494  iminwrk = max( 1, n )
495  END IF
496  rminwrk = 2
497  END IF
498  lquery = (liwork .EQ. -1 .OR. lwork .EQ. -1 .OR. lrwork .EQ. -1)
499  info = 0
500  IF ( .NOT. ( accla .OR. acclm .OR. acclh ) ) THEN
501  info = -1
502  ELSE IF ( .NOT.( rowprm .OR. lsame( jobp, 'N' ) ) ) THEN
503  info = -2
504  ELSE IF ( .NOT.( rtrans .OR. lsame( jobr, 'N' ) ) ) THEN
505  info = -3
506  ELSE IF ( .NOT.( lsvec .OR. dntwu ) ) THEN
507  info = -4
508  ELSE IF ( wntur .AND. wntva ) THEN
509  info = -5
510  ELSE IF ( .NOT.( rsvec .OR. dntwv )) THEN
511  info = -5
512  ELSE IF ( m.LT.0 ) THEN
513  info = -6
514  ELSE IF ( ( n.LT.0 ) .OR. ( n.GT.m ) ) THEN
515  info = -7
516  ELSE IF ( lda.LT.max( 1, m ) ) THEN
517  info = -9
518  ELSE IF ( ldu.LT.1 .OR. ( lsvc0 .AND. ldu.LT.m ) .OR.
519  $( wntuf .AND. ldu.LT.n ) ) THEN 520 info = -12 521 ELSE IF ( ldv.LT.1 .OR. ( rsvec .AND. ldv.LT.n ) .OR. 522$ ( conda .AND. ldv.LT.n ) ) THEN
523  info = -14
524  ELSE IF ( liwork .LT. iminwrk .AND. .NOT. lquery ) THEN
525  info = -17
526  END IF
527 *
528 *
529  IF ( info .EQ. 0 ) THEN
530 * .. compute the minimal and the optimal workspace lengths
531 * [[The expressions for computing the minimal and the optimal
532 * values of LWORK are written with a lot of redundancy and
533 * can be simplified. However, this detailed form is easier for
534 * maintenance and modifications of the code.]]
535 *
536 * .. minimal workspace length for DGEQP3 of an M x N matrix
537  lwqp3 = 3 * n + 1
538 * .. minimal workspace length for DORMQR to build left singular vectors
539  IF ( wntus .OR. wntur ) THEN
540  lworq = max( n , 1 )
541  ELSE IF ( wntua ) THEN
542  lworq = max( m , 1 )
543  END IF
544 * .. minimal workspace length for DPOCON of an N x N matrix
545  lwcon = 3 * n
546 * .. DGESVD of an N x N matrix
547  lwsvd = max( 5 * n, 1 )
548  IF ( lquery ) THEN
549  CALL dgeqp3( m, n, a, lda, iwork, rdummy, rdummy, -1,
550  $ierr ) 551 lwrk_dgeqp3 = int( rdummy(1) ) 552 IF ( wntus .OR. wntur ) THEN 553 CALL dormqr( 'L', 'N', m, n, n, a, lda, rdummy, u, 554$ ldu, rdummy, -1, ierr )
555  lwrk_dormqr = int( rdummy(1) )
556  ELSE IF ( wntua ) THEN
557  CALL dormqr( 'L', 'N', m, m, n, a, lda, rdummy, u,
558  $ldu, rdummy, -1, ierr ) 559 lwrk_dormqr = int( rdummy(1) ) 560 ELSE 561 lwrk_dormqr = 0 562 END IF 563 END IF 564 minwrk = 2 565 optwrk = 2 566 IF ( .NOT. (lsvec .OR. rsvec )) THEN 567 * .. minimal and optimal sizes of the workspace if 568 * only the singular values are requested 569 IF ( conda ) THEN 570 minwrk = max( n+lwqp3, lwcon, lwsvd ) 571 ELSE 572 minwrk = max( n+lwqp3, lwsvd ) 573 END IF 574 IF ( lquery ) THEN 575 CALL dgesvd( 'N', 'N', n, n, a, lda, s, u, ldu, 576$ v, ldv, rdummy, -1, ierr )
577  lwrk_dgesvd = int( rdummy(1) )
578  IF ( conda ) THEN
579  optwrk = max( n+lwrk_dgeqp3, n+lwcon, lwrk_dgesvd )
580  ELSE
581  optwrk = max( n+lwrk_dgeqp3, lwrk_dgesvd )
582  END IF
583  END IF
584  ELSE IF ( lsvec .AND. (.NOT.rsvec) ) THEN
585 * .. minimal and optimal sizes of the workspace if the
586 * singular values and the left singular vectors are requested
587  IF ( conda ) THEN
588  minwrk = n + max( lwqp3, lwcon, lwsvd, lworq )
589  ELSE
590  minwrk = n + max( lwqp3, lwsvd, lworq )
591  END IF
592  IF ( lquery ) THEN
593  IF ( rtrans ) THEN
594  CALL dgesvd( 'N', 'O', n, n, a, lda, s, u, ldu,
595  $v, ldv, rdummy, -1, ierr ) 596 ELSE 597 CALL dgesvd( 'O', 'N', n, n, a, lda, s, u, ldu, 598$ v, ldv, rdummy, -1, ierr )
599  END IF
600  lwrk_dgesvd = int( rdummy(1) )
601  IF ( conda ) THEN
602  optwrk = n + max( lwrk_dgeqp3, lwcon, lwrk_dgesvd,
603  $lwrk_dormqr ) 604 ELSE 605 optwrk = n + max( lwrk_dgeqp3, lwrk_dgesvd, 606$ lwrk_dormqr )
607  END IF
608  END IF
609  ELSE IF ( rsvec .AND. (.NOT.lsvec) ) THEN
610 * .. minimal and optimal sizes of the workspace if the
611 * singular values and the right singular vectors are requested
612  IF ( conda ) THEN
613  minwrk = n + max( lwqp3, lwcon, lwsvd )
614  ELSE
615  minwrk = n + max( lwqp3, lwsvd )
616  END IF
617  IF ( lquery ) THEN
618  IF ( rtrans ) THEN
619  CALL dgesvd( 'O', 'N', n, n, a, lda, s, u, ldu,
620  $v, ldv, rdummy, -1, ierr ) 621 ELSE 622 CALL dgesvd( 'N', 'O', n, n, a, lda, s, u, ldu, 623$ v, ldv, rdummy, -1, ierr )
624  END IF
625  lwrk_dgesvd = int( rdummy(1) )
626  IF ( conda ) THEN
627  optwrk = n + max( lwrk_dgeqp3, lwcon, lwrk_dgesvd )
628  ELSE
629  optwrk = n + max( lwrk_dgeqp3, lwrk_dgesvd )
630  END IF
631  END IF
632  ELSE
633 * .. minimal and optimal sizes of the workspace if the
634 * full SVD is requested
635  IF ( rtrans ) THEN
636  minwrk = max( lwqp3, lwsvd, lworq )
637  IF ( conda ) minwrk = max( minwrk, lwcon )
638  minwrk = minwrk + n
639  IF ( wntva ) THEN
640 * .. minimal workspace length for N x N/2 DGEQRF
641  lwqrf = max( n/2, 1 )
642 * .. minimal workspace length for N/2 x N/2 DGESVD
643  lwsvd2 = max( 5 * (n/2), 1 )
644  lworq2 = max( n, 1 )
645  minwrk2 = max( lwqp3, n/2+lwqrf, n/2+lwsvd2,
646  $n/2+lworq2, lworq ) 647 IF ( conda ) minwrk2 = max( minwrk2, lwcon ) 648 minwrk2 = n + minwrk2 649 minwrk = max( minwrk, minwrk2 ) 650 END IF 651 ELSE 652 minwrk = max( lwqp3, lwsvd, lworq ) 653 IF ( conda ) minwrk = max( minwrk, lwcon ) 654 minwrk = minwrk + n 655 IF ( wntva ) THEN 656 * .. minimal workspace length for N/2 x N DGELQF 657 lwlqf = max( n/2, 1 ) 658 lwsvd2 = max( 5 * (n/2), 1 ) 659 lworlq = max( n , 1 ) 660 minwrk2 = max( lwqp3, n/2+lwlqf, n/2+lwsvd2, 661$ n/2+lworlq, lworq )
662  IF ( conda ) minwrk2 = max( minwrk2, lwcon )
663  minwrk2 = n + minwrk2
664  minwrk = max( minwrk, minwrk2 )
665  END IF
666  END IF
667  IF ( lquery ) THEN
668  IF ( rtrans ) THEN
669  CALL dgesvd( 'O', 'A', n, n, a, lda, s, u, ldu,
670  $v, ldv, rdummy, -1, ierr ) 671 lwrk_dgesvd = int( rdummy(1) ) 672 optwrk = max(lwrk_dgeqp3,lwrk_dgesvd,lwrk_dormqr) 673 IF ( conda ) optwrk = max( optwrk, lwcon ) 674 optwrk = n + optwrk 675 IF ( wntva ) THEN 676 CALL dgeqrf(n,n/2,u,ldu,rdummy,rdummy,-1,ierr) 677 lwrk_dgeqrf = int( rdummy(1) ) 678 CALL dgesvd( 'S', 'O', n/2,n/2, v,ldv, s, u,ldu, 679$ v, ldv, rdummy, -1, ierr )
680  lwrk_dgesvd2 = int( rdummy(1) )
681  CALL dormqr( 'R', 'C', n, n, n/2, u, ldu, rdummy,
682  $v, ldv, rdummy, -1, ierr ) 683 lwrk_dormqr2 = int( rdummy(1) ) 684 optwrk2 = max( lwrk_dgeqp3, n/2+lwrk_dgeqrf, 685$ n/2+lwrk_dgesvd2, n/2+lwrk_dormqr2 )
686  IF ( conda ) optwrk2 = max( optwrk2, lwcon )
687  optwrk2 = n + optwrk2
688  optwrk = max( optwrk, optwrk2 )
689  END IF
690  ELSE
691  CALL dgesvd( 'S', 'O', n, n, a, lda, s, u, ldu,
692  $v, ldv, rdummy, -1, ierr ) 693 lwrk_dgesvd = int( rdummy(1) ) 694 optwrk = max(lwrk_dgeqp3,lwrk_dgesvd,lwrk_dormqr) 695 IF ( conda ) optwrk = max( optwrk, lwcon ) 696 optwrk = n + optwrk 697 IF ( wntva ) THEN 698 CALL dgelqf(n/2,n,u,ldu,rdummy,rdummy,-1,ierr) 699 lwrk_dgelqf = int( rdummy(1) ) 700 CALL dgesvd( 'S','O', n/2,n/2, v, ldv, s, u, ldu, 701$ v, ldv, rdummy, -1, ierr )
702  lwrk_dgesvd2 = int( rdummy(1) )
703  CALL dormlq( 'R', 'N', n, n, n/2, u, ldu, rdummy,
704  $v, ldv, rdummy,-1,ierr ) 705 lwrk_dormlq = int( rdummy(1) ) 706 optwrk2 = max( lwrk_dgeqp3, n/2+lwrk_dgelqf, 707$ n/2+lwrk_dgesvd2, n/2+lwrk_dormlq )
708  IF ( conda ) optwrk2 = max( optwrk2, lwcon )
709  optwrk2 = n + optwrk2
710  optwrk = max( optwrk, optwrk2 )
711  END IF
712  END IF
713  END IF
714  END IF
715 *
716  minwrk = max( 2, minwrk )
717  optwrk = max( 2, optwrk )
718  IF ( lwork .LT. minwrk .AND. (.NOT.lquery) ) info = -19
719 *
720  END IF
721 *
722  IF (info .EQ. 0 .AND. lrwork .LT. rminwrk .AND. .NOT. lquery) THEN
723  info = -21
724  END IF
725  IF( info.NE.0 ) THEN
726  CALL xerbla( 'DGESVDQ', -info )
727  RETURN
728  ELSE IF ( lquery ) THEN
729 *
730 * Return optimal workspace
731 *
732  iwork(1) = iminwrk
733  work(1) = optwrk
734  work(2) = minwrk
735  rwork(1) = rminwrk
736  RETURN
737  END IF
738 *
739 * Quick return if the matrix is void.
740 *
741  IF( ( m.EQ.0 ) .OR. ( n.EQ.0 ) ) THEN
742 * .. all output is void.
743  RETURN
744  END IF
745 *
746  big = dlamch('O')
747  ascaled = .false.
748  iwoff = 1
749  IF ( rowprm ) THEN
750  iwoff = m
751 * .. reordering the rows in decreasing sequence in the
752 * ell-infinity norm - this enhances numerical robustness in
753 * the case of differently scaled rows.
754  DO 1904 p = 1, m
755 * RWORK(p) = ABS( A(p,ICAMAX(N,A(p,1),LDA)) )
756 * [[DLANGE will return NaN if an entry of the p-th row is Nan]]
757  rwork(p) = dlange( 'M', 1, n, a(p,1), lda, rdummy )
758 * .. check for NaN's and Inf's
759  IF ( ( rwork(p) .NE. rwork(p) ) .OR.
760  $( (rwork(p)*zero) .NE. zero ) ) THEN 761 info = -8 762 CALL xerbla( 'DGESVDQ', -info ) 763 RETURN 764 END IF 765 1904 CONTINUE 766 DO 1952 p = 1, m - 1 767 q = idamax( m-p+1, rwork(p), 1 ) + p - 1 768 iwork(n+p) = q 769 IF ( p .NE. q ) THEN 770 rtmp = rwork(p) 771 rwork(p) = rwork(q) 772 rwork(q) = rtmp 773 END IF 774 1952 CONTINUE 775 * 776 IF ( rwork(1) .EQ. zero ) THEN 777 * Quick return: A is the M x N zero matrix. 778 numrank = 0 779 CALL dlaset( 'G', n, 1, zero, zero, s, n ) 780 IF ( wntus ) CALL dlaset('G', m, n, zero, one, u, ldu) 781 IF ( wntua ) CALL dlaset('G', m, m, zero, one, u, ldu) 782 IF ( wntva ) CALL dlaset('G', n, n, zero, one, v, ldv) 783 IF ( wntuf ) THEN 784 CALL dlaset( 'G', n, 1, zero, zero, work, n ) 785 CALL dlaset( 'G', m, n, zero, one, u, ldu ) 786 END IF 787 DO 5001 p = 1, n 788 iwork(p) = p 789 5001 CONTINUE 790 IF ( rowprm ) THEN 791 DO 5002 p = n + 1, n + m - 1 792 iwork(p) = p - n 793 5002 CONTINUE 794 END IF 795 IF ( conda ) rwork(1) = -1 796 rwork(2) = -1 797 RETURN 798 END IF 799 * 800 IF ( rwork(1) .GT. big / sqrt(dble(m)) ) THEN 801 * .. to prevent overflow in the QR factorization, scale the 802 * matrix by 1/sqrt(M) if too large entry detected 803 CALL dlascl('G',0,0,sqrt(dble(m)),one, m,n, a,lda, ierr) 804 ascaled = .true. 805 END IF 806 CALL dlaswp( n, a, lda, 1, m-1, iwork(n+1), 1 ) 807 END IF 808 * 809 * .. At this stage, preemptive scaling is done only to avoid column 810 * norms overflows during the QR factorization. The SVD procedure should 811 * have its own scaling to save the singular values from overflows and 812 * underflows. That depends on the SVD procedure. 813 * 814 IF ( .NOT.rowprm ) THEN 815 rtmp = dlange( 'M', m, n, a, lda, rdummy ) 816 IF ( ( rtmp .NE. rtmp ) .OR. 817$ ( (rtmp*zero) .NE. zero ) ) THEN
818  info = -8
819  CALL xerbla( 'DGESVDQ', -info )
820  RETURN
821  END IF
822  IF ( rtmp .GT. big / sqrt(dble(m)) ) THEN
823 * .. to prevent overflow in the QR factorization, scale the
824 * matrix by 1/sqrt(M) if too large entry detected
825  CALL dlascl('G',0,0, sqrt(dble(m)),one, m,n, a,lda, ierr)
826  ascaled = .true.
827  END IF
828  END IF
829 *
830 * .. QR factorization with column pivoting
831 *
832 * A * P = Q * [ R ]
833 * [ 0 ]
834 *
835  DO 1963 p = 1, n
836 * .. all columns are free columns
837  iwork(p) = 0
838  1963 CONTINUE
839  CALL dgeqp3( m, n, a, lda, iwork, work, work(n+1), lwork-n,
840  $ierr ) 841 * 842 * If the user requested accuracy level allows truncation in the 843 * computed upper triangular factor, the matrix R is examined and, 844 * if possible, replaced with its leading upper trapezoidal part. 845 * 846 epsln = dlamch('E') 847 sfmin = dlamch('S') 848 * SMALL = SFMIN / EPSLN 849 nr = n 850 * 851 IF ( accla ) THEN 852 * 853 * Standard absolute error bound suffices. All sigma_i with 854 * sigma_i < N*EPS*||A||_F are flushed to zero. This is an 855 * aggressive enforcement of lower numerical rank by introducing a 856 * backward error of the order of N*EPS*||A||_F. 857 nr = 1 858 rtmp = sqrt(dble(n))*epsln 859 DO 3001 p = 2, n 860 IF ( abs(a(p,p)) .LT. (rtmp*abs(a(1,1))) ) GO TO 3002 861 nr = nr + 1 862 3001 CONTINUE 863 3002 CONTINUE 864 * 865 ELSEIF ( acclm ) THEN 866 * .. similarly as above, only slightly more gentle (less aggressive). 867 * Sudden drop on the diagonal of R is used as the criterion for being 868 * close-to-rank-deficient. The threshold is set to EPSLN=DLAMCH('E'). 869 * [[This can be made more flexible by replacing this hard-coded value 870 * with a user specified threshold.]] Also, the values that underflow 871 * will be truncated. 872 nr = 1 873 DO 3401 p = 2, n 874 IF ( ( abs(a(p,p)) .LT. (epsln*abs(a(p-1,p-1))) ) .OR. 875$ ( abs(a(p,p)) .LT. sfmin ) ) GO TO 3402
876  nr = nr + 1
877  3401 CONTINUE
878  3402 CONTINUE
879 *
880  ELSE
881 * .. RRQR not authorized to determine numerical rank except in the
882 * obvious case of zero pivots.
883 * .. inspect R for exact zeros on the diagonal;
884 * R(i,i)=0 => R(i:N,i:N)=0.
885  nr = 1
886  DO 3501 p = 2, n
887  IF ( abs(a(p,p)) .EQ. zero ) GO TO 3502
888  nr = nr + 1
889  3501 CONTINUE
890  3502 CONTINUE
891 *
892  IF ( conda ) THEN
893 * Estimate the scaled condition number of A. Use the fact that it is
894 * the same as the scaled condition number of R.
895 * .. V is used as workspace
896  CALL dlacpy( 'U', n, n, a, lda, v, ldv )
897 * Only the leading NR x NR submatrix of the triangular factor
898 * is considered. Only if NR=N will this give a reliable error
899 * bound. However, even for NR < N, this can be used on an
900 * expert level and obtain useful information in the sense of
901 * perturbation theory.
902  DO 3053 p = 1, nr
903  rtmp = dnrm2( p, v(1,p), 1 )
904  CALL dscal( p, one/rtmp, v(1,p), 1 )
905  3053 CONTINUE
906  IF ( .NOT. ( lsvec .OR. rsvec ) ) THEN
907  CALL dpocon( 'U', nr, v, ldv, one, rtmp,
908  $work, iwork(n+iwoff), ierr ) 909 ELSE 910 CALL dpocon( 'U', nr, v, ldv, one, rtmp, 911$ work(n+1), iwork(n+iwoff), ierr )
912  END IF
913  sconda = one / sqrt(rtmp)
914 * For NR=N, SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1),
915 * N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
916 * See the reference  for more details.
917  END IF
918 *
919  ENDIF
920 *
921  IF ( wntur ) THEN
922  n1 = nr
923  ELSE IF ( wntus .OR. wntuf) THEN
924  n1 = n
925  ELSE IF ( wntua ) THEN
926  n1 = m
927  END IF
928 *
929  IF ( .NOT. ( rsvec .OR. lsvec ) ) THEN
930 *.......................................................................
931 * .. only the singular values are requested
932 *.......................................................................
933  IF ( rtrans ) THEN
934 *
935 * .. compute the singular values of R**T = [A](1:NR,1:N)**T
936 * .. set the lower triangle of [A] to [A](1:NR,1:N)**T and
937 * the upper triangle of [A] to zero.
938  DO 1146 p = 1, min( n, nr )
939  DO 1147 q = p + 1, n
940  a(q,p) = a(p,q)
941  IF ( q .LE. nr ) a(p,q) = zero
942  1147 CONTINUE
943  1146 CONTINUE
944 *
945  CALL dgesvd( 'N', 'N', n, nr, a, lda, s, u, ldu,
946  $v, ldv, work, lwork, info ) 947 * 948 ELSE 949 * 950 * .. compute the singular values of R = [A](1:NR,1:N) 951 * 952 IF ( nr .GT. 1 ) 953$ CALL dlaset( 'L', nr-1,nr-1, zero,zero, a(2,1), lda )
954  CALL dgesvd( 'N', 'N', nr, n, a, lda, s, u, ldu,
955  $v, ldv, work, lwork, info ) 956 * 957 END IF 958 * 959 ELSE IF ( lsvec .AND. ( .NOT. rsvec) ) THEN 960 *....................................................................... 961 * .. the singular values and the left singular vectors requested 962 *......................................................................."""""""" 963 IF ( rtrans ) THEN 964 * .. apply DGESVD to R**T 965 * .. copy R**T into [U] and overwrite [U] with the right singular 966 * vectors of R 967 DO 1192 p = 1, nr 968 DO 1193 q = p, n 969 u(q,p) = a(p,q) 970 1193 CONTINUE 971 1192 CONTINUE 972 IF ( nr .GT. 1 ) 973$ CALL dlaset( 'U', nr-1,nr-1, zero,zero, u(1,2), ldu )
974 * .. the left singular vectors not computed, the NR right singular
975 * vectors overwrite [U](1:NR,1:NR) as transposed. These
976 * will be pre-multiplied by Q to build the left singular vectors of A.
977  CALL dgesvd( 'N', 'O', n, nr, u, ldu, s, u, ldu,
978  $u, ldu, work(n+1), lwork-n, info ) 979 * 980 DO 1119 p = 1, nr 981 DO 1120 q = p + 1, nr 982 rtmp = u(q,p) 983 u(q,p) = u(p,q) 984 u(p,q) = rtmp 985 1120 CONTINUE 986 1119 CONTINUE 987 * 988 ELSE 989 * .. apply DGESVD to R 990 * .. copy R into [U] and overwrite [U] with the left singular vectors 991 CALL dlacpy( 'U', nr, n, a, lda, u, ldu ) 992 IF ( nr .GT. 1 ) 993$ CALL dlaset( 'L', nr-1, nr-1, zero, zero, u(2,1), ldu )
994 * .. the right singular vectors not computed, the NR left singular
995 * vectors overwrite [U](1:NR,1:NR)
996  CALL dgesvd( 'O', 'N', nr, n, u, ldu, s, u, ldu,
997  $v, ldv, work(n+1), lwork-n, info ) 998 * .. now [U](1:NR,1:NR) contains the NR left singular vectors of 999 * R. These will be pre-multiplied by Q to build the left singular 1000 * vectors of A. 1001 END IF 1002 * 1003 * .. assemble the left singular vector matrix U of dimensions 1004 * (M x NR) or (M x N) or (M x M). 1005 IF ( ( nr .LT. m ) .AND. ( .NOT.wntuf ) ) THEN 1006 CALL dlaset('A', m-nr, nr, zero, zero, u(nr+1,1), ldu) 1007 IF ( nr .LT. n1 ) THEN 1008 CALL dlaset( 'A',nr,n1-nr,zero,zero,u(1,nr+1), ldu ) 1009 CALL dlaset( 'A',m-nr,n1-nr,zero,one, 1010$ u(nr+1,nr+1), ldu )
1011  END IF
1012  END IF
1013 *
1014 * The Q matrix from the first QRF is built into the left singular
1015 * vectors matrix U.
1016 *
1017  IF ( .NOT.wntuf )
1018  $CALL dormqr( 'L', 'N', m, n1, n, a, lda, work, u, 1019$ ldu, work(n+1), lwork-n, ierr )
1020  IF ( rowprm .AND. .NOT.wntuf )
1021  $CALL dlaswp( n1, u, ldu, 1, m-1, iwork(n+1), -1 ) 1022 * 1023 ELSE IF ( rsvec .AND. ( .NOT. lsvec ) ) THEN 1024 *....................................................................... 1025 * .. the singular values and the right singular vectors requested 1026 *....................................................................... 1027 IF ( rtrans ) THEN 1028 * .. apply DGESVD to R**T 1029 * .. copy R**T into V and overwrite V with the left singular vectors 1030 DO 1165 p = 1, nr 1031 DO 1166 q = p, n 1032 v(q,p) = (a(p,q)) 1033 1166 CONTINUE 1034 1165 CONTINUE 1035 IF ( nr .GT. 1 ) 1036$ CALL dlaset( 'U', nr-1,nr-1, zero,zero, v(1,2), ldv )
1037 * .. the left singular vectors of R**T overwrite V, the right singular
1038 * vectors not computed
1039  IF ( wntvr .OR. ( nr .EQ. n ) ) THEN
1040  CALL dgesvd( 'O', 'N', n, nr, v, ldv, s, u, ldu,
1041  $u, ldu, work(n+1), lwork-n, info ) 1042 * 1043 DO 1121 p = 1, nr 1044 DO 1122 q = p + 1, nr 1045 rtmp = v(q,p) 1046 v(q,p) = v(p,q) 1047 v(p,q) = rtmp 1048 1122 CONTINUE 1049 1121 CONTINUE 1050 * 1051 IF ( nr .LT. n ) THEN 1052 DO 1103 p = 1, nr 1053 DO 1104 q = nr + 1, n 1054 v(p,q) = v(q,p) 1055 1104 CONTINUE 1056 1103 CONTINUE 1057 END IF 1058 CALL dlapmt( .false., nr, n, v, ldv, iwork ) 1059 ELSE 1060 * .. need all N right singular vectors and NR < N 1061 * [!] This is simple implementation that augments [V](1:N,1:NR) 1062 * by padding a zero block. In the case NR << N, a more efficient 1063 * way is to first use the QR factorization. For more details 1064 * how to implement this, see the " FULL SVD " branch. 1065 CALL dlaset('G', n, n-nr, zero, zero, v(1,nr+1), ldv) 1066 CALL dgesvd( 'O', 'N', n, n, v, ldv, s, u, ldu, 1067$ u, ldu, work(n+1), lwork-n, info )
1068 *
1069  DO 1123 p = 1, n
1070  DO 1124 q = p + 1, n
1071  rtmp = v(q,p)
1072  v(q,p) = v(p,q)
1073  v(p,q) = rtmp
1074  1124 CONTINUE
1075  1123 CONTINUE
1076  CALL dlapmt( .false., n, n, v, ldv, iwork )
1077  END IF
1078 *
1079  ELSE
1080 * .. aply DGESVD to R
1081 * .. copy R into V and overwrite V with the right singular vectors
1082  CALL dlacpy( 'U', nr, n, a, lda, v, ldv )
1083  IF ( nr .GT. 1 )
1084  $CALL dlaset( 'L', nr-1, nr-1, zero, zero, v(2,1), ldv ) 1085 * .. the right singular vectors overwrite V, the NR left singular 1086 * vectors stored in U(1:NR,1:NR) 1087 IF ( wntvr .OR. ( nr .EQ. n ) ) THEN 1088 CALL dgesvd( 'N', 'O', nr, n, v, ldv, s, u, ldu, 1089$ v, ldv, work(n+1), lwork-n, info )
1090  CALL dlapmt( .false., nr, n, v, ldv, iwork )
1091 * .. now [V](1:NR,1:N) contains V(1:N,1:NR)**T
1092  ELSE
1093 * .. need all N right singular vectors and NR < N
1094 * [!] This is simple implementation that augments [V](1:NR,1:N)
1095 * by padding a zero block. In the case NR << N, a more efficient
1096 * way is to first use the LQ factorization. For more details
1097 * how to implement this, see the " FULL SVD " branch.
1098  CALL dlaset('G', n-nr, n, zero,zero, v(nr+1,1), ldv)
1099  CALL dgesvd( 'N', 'O', n, n, v, ldv, s, u, ldu,
1100  $v, ldv, work(n+1), lwork-n, info ) 1101 CALL dlapmt( .false., n, n, v, ldv, iwork ) 1102 END IF 1103 * .. now [V] contains the transposed matrix of the right singular 1104 * vectors of A. 1105 END IF 1106 * 1107 ELSE 1108 *....................................................................... 1109 * .. FULL SVD requested 1110 *....................................................................... 1111 IF ( rtrans ) THEN 1112 * 1113 * .. apply DGESVD to R**T [[this option is left for R&D&T]] 1114 * 1115 IF ( wntvr .OR. ( nr .EQ. n ) ) THEN 1116 * .. copy R**T into [V] and overwrite [V] with the left singular 1117 * vectors of R**T 1118 DO 1168 p = 1, nr 1119 DO 1169 q = p, n 1120 v(q,p) = a(p,q) 1121 1169 CONTINUE 1122 1168 CONTINUE 1123 IF ( nr .GT. 1 ) 1124$ CALL dlaset( 'U', nr-1,nr-1, zero,zero, v(1,2), ldv )
1125 *
1126 * .. the left singular vectors of R**T overwrite [V], the NR right
1127 * singular vectors of R**T stored in [U](1:NR,1:NR) as transposed
1128  CALL dgesvd( 'O', 'A', n, nr, v, ldv, s, v, ldv,
1129  $u, ldu, work(n+1), lwork-n, info ) 1130 * .. assemble V 1131 DO 1115 p = 1, nr 1132 DO 1116 q = p + 1, nr 1133 rtmp = v(q,p) 1134 v(q,p) = v(p,q) 1135 v(p,q) = rtmp 1136 1116 CONTINUE 1137 1115 CONTINUE 1138 IF ( nr .LT. n ) THEN 1139 DO 1101 p = 1, nr 1140 DO 1102 q = nr+1, n 1141 v(p,q) = v(q,p) 1142 1102 CONTINUE 1143 1101 CONTINUE 1144 END IF 1145 CALL dlapmt( .false., nr, n, v, ldv, iwork ) 1146 * 1147 DO 1117 p = 1, nr 1148 DO 1118 q = p + 1, nr 1149 rtmp = u(q,p) 1150 u(q,p) = u(p,q) 1151 u(p,q) = rtmp 1152 1118 CONTINUE 1153 1117 CONTINUE 1154 * 1155 IF ( ( nr .LT. m ) .AND. .NOT.(wntuf)) THEN 1156 CALL dlaset('A', m-nr,nr, zero,zero, u(nr+1,1), ldu) 1157 IF ( nr .LT. n1 ) THEN 1158 CALL dlaset('A',nr,n1-nr,zero,zero,u(1,nr+1),ldu) 1159 CALL dlaset( 'A',m-nr,n1-nr,zero,one, 1160$ u(nr+1,nr+1), ldu )
1161  END IF
1162  END IF
1163 *
1164  ELSE
1165 * .. need all N right singular vectors and NR < N
1166 * .. copy R**T into [V] and overwrite [V] with the left singular
1167 * vectors of R**T
1168 * [[The optimal ratio N/NR for using QRF instead of padding
1169 * with zeros. Here hard coded to 2; it must be at least
1170 * two due to work space constraints.]]
1171 * OPTRATIO = ILAENV(6, 'DGESVD', 'S' // 'O', NR,N,0,0)
1172 * OPTRATIO = MAX( OPTRATIO, 2 )
1173  optratio = 2
1174  IF ( optratio*nr .GT. n ) THEN
1175  DO 1198 p = 1, nr
1176  DO 1199 q = p, n
1177  v(q,p) = a(p,q)
1178  1199 CONTINUE
1179  1198 CONTINUE
1180  IF ( nr .GT. 1 )
1181  $CALL dlaset('U',nr-1,nr-1, zero,zero, v(1,2),ldv) 1182 * 1183 CALL dlaset('A',n,n-nr,zero,zero,v(1,nr+1),ldv) 1184 CALL dgesvd( 'O', 'A', n, n, v, ldv, s, v, ldv, 1185$ u, ldu, work(n+1), lwork-n, info )
1186 *
1187  DO 1113 p = 1, n
1188  DO 1114 q = p + 1, n
1189  rtmp = v(q,p)
1190  v(q,p) = v(p,q)
1191  v(p,q) = rtmp
1192  1114 CONTINUE
1193  1113 CONTINUE
1194  CALL dlapmt( .false., n, n, v, ldv, iwork )
1195 * .. assemble the left singular vector matrix U of dimensions
1196 * (M x N1), i.e. (M x N) or (M x M).
1197 *
1198  DO 1111 p = 1, n
1199  DO 1112 q = p + 1, n
1200  rtmp = u(q,p)
1201  u(q,p) = u(p,q)
1202  u(p,q) = rtmp
1203  1112 CONTINUE
1204  1111 CONTINUE
1205 *
1206  IF ( ( n .LT. m ) .AND. .NOT.(wntuf)) THEN
1207  CALL dlaset('A',m-n,n,zero,zero,u(n+1,1),ldu)
1208  IF ( n .LT. n1 ) THEN
1209  CALL dlaset('A',n,n1-n,zero,zero,u(1,n+1),ldu)
1210  CALL dlaset('A',m-n,n1-n,zero,one,
1211  $u(n+1,n+1), ldu ) 1212 END IF 1213 END IF 1214 ELSE 1215 * .. copy R**T into [U] and overwrite [U] with the right 1216 * singular vectors of R 1217 DO 1196 p = 1, nr 1218 DO 1197 q = p, n 1219 u(q,nr+p) = a(p,q) 1220 1197 CONTINUE 1221 1196 CONTINUE 1222 IF ( nr .GT. 1 ) 1223$ CALL dlaset('U',nr-1,nr-1,zero,zero,u(1,nr+2),ldu)
1224  CALL dgeqrf( n, nr, u(1,nr+1), ldu, work(n+1),
1225  $work(n+nr+1), lwork-n-nr, ierr ) 1226 DO 1143 p = 1, nr 1227 DO 1144 q = 1, n 1228 v(q,p) = u(p,nr+q) 1229 1144 CONTINUE 1230 1143 CONTINUE 1231 CALL dlaset('U',nr-1,nr-1,zero,zero,v(1,2),ldv) 1232 CALL dgesvd( 'S', 'O', nr, nr, v, ldv, s, u, ldu, 1233$ v,ldv, work(n+nr+1),lwork-n-nr, info )
1234  CALL dlaset('A',n-nr,nr,zero,zero,v(nr+1,1),ldv)
1235  CALL dlaset('A',nr,n-nr,zero,zero,v(1,nr+1),ldv)
1236  CALL dlaset('A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv)
1237  CALL dormqr('R','C', n, n, nr, u(1,nr+1), ldu,
1238  $work(n+1),v,ldv,work(n+nr+1),lwork-n-nr,ierr) 1239 CALL dlapmt( .false., n, n, v, ldv, iwork ) 1240 * .. assemble the left singular vector matrix U of dimensions 1241 * (M x NR) or (M x N) or (M x M). 1242 IF ( ( nr .LT. m ) .AND. .NOT.(wntuf)) THEN 1243 CALL dlaset('A',m-nr,nr,zero,zero,u(nr+1,1),ldu) 1244 IF ( nr .LT. n1 ) THEN 1245 CALL dlaset('A',nr,n1-nr,zero,zero,u(1,nr+1),ldu) 1246 CALL dlaset( 'A',m-nr,n1-nr,zero,one, 1247$ u(nr+1,nr+1),ldu)
1248  END IF
1249  END IF
1250  END IF
1251  END IF
1252 *
1253  ELSE
1254 *
1255 * .. apply DGESVD to R [[this is the recommended option]]
1256 *
1257  IF ( wntvr .OR. ( nr .EQ. n ) ) THEN
1258 * .. copy R into [V] and overwrite V with the right singular vectors
1259  CALL dlacpy( 'U', nr, n, a, lda, v, ldv )
1260  IF ( nr .GT. 1 )
1261  $CALL dlaset( 'L', nr-1,nr-1, zero,zero, v(2,1), ldv ) 1262 * .. the right singular vectors of R overwrite [V], the NR left 1263 * singular vectors of R stored in [U](1:NR,1:NR) 1264 CALL dgesvd( 'S', 'O', nr, n, v, ldv, s, u, ldu, 1265$ v, ldv, work(n+1), lwork-n, info )
1266  CALL dlapmt( .false., nr, n, v, ldv, iwork )
1267 * .. now [V](1:NR,1:N) contains V(1:N,1:NR)**T
1268 * .. assemble the left singular vector matrix U of dimensions
1269 * (M x NR) or (M x N) or (M x M).
1270  IF ( ( nr .LT. m ) .AND. .NOT.(wntuf)) THEN
1271  CALL dlaset('A', m-nr,nr, zero,zero, u(nr+1,1), ldu)
1272  IF ( nr .LT. n1 ) THEN
1273  CALL dlaset('A',nr,n1-nr,zero,zero,u(1,nr+1),ldu)
1274  CALL dlaset( 'A',m-nr,n1-nr,zero,one,
1275  $u(nr+1,nr+1), ldu ) 1276 END IF 1277 END IF 1278 * 1279 ELSE 1280 * .. need all N right singular vectors and NR < N 1281 * .. the requested number of the left singular vectors 1282 * is then N1 (N or M) 1283 * [[The optimal ratio N/NR for using LQ instead of padding 1284 * with zeros. Here hard coded to 2; it must be at least 1285 * two due to work space constraints.]] 1286 * OPTRATIO = ILAENV(6, 'DGESVD', 'S' // 'O', NR,N,0,0) 1287 * OPTRATIO = MAX( OPTRATIO, 2 ) 1288 optratio = 2 1289 IF ( optratio * nr .GT. n ) THEN 1290 CALL dlacpy( 'U', nr, n, a, lda, v, ldv ) 1291 IF ( nr .GT. 1 ) 1292$ CALL dlaset('L', nr-1,nr-1, zero,zero, v(2,1),ldv)
1293 * .. the right singular vectors of R overwrite [V], the NR left
1294 * singular vectors of R stored in [U](1:NR,1:NR)
1295  CALL dlaset('A', n-nr,n, zero,zero, v(nr+1,1),ldv)
1296  CALL dgesvd( 'S', 'O', n, n, v, ldv, s, u, ldu,
1297  $v, ldv, work(n+1), lwork-n, info ) 1298 CALL dlapmt( .false., n, n, v, ldv, iwork ) 1299 * .. now [V] contains the transposed matrix of the right 1300 * singular vectors of A. The leading N left singular vectors 1301 * are in [U](1:N,1:N) 1302 * .. assemble the left singular vector matrix U of dimensions 1303 * (M x N1), i.e. (M x N) or (M x M). 1304 IF ( ( n .LT. m ) .AND. .NOT.(wntuf)) THEN 1305 CALL dlaset('A',m-n,n,zero,zero,u(n+1,1),ldu) 1306 IF ( n .LT. n1 ) THEN 1307 CALL dlaset('A',n,n1-n,zero,zero,u(1,n+1),ldu) 1308 CALL dlaset( 'A',m-n,n1-n,zero,one, 1309$ u(n+1,n+1), ldu )
1310  END IF
1311  END IF
1312  ELSE
1313  CALL dlacpy( 'U', nr, n, a, lda, u(nr+1,1), ldu )
1314  IF ( nr .GT. 1 )
1315  $CALL dlaset('L',nr-1,nr-1,zero,zero,u(nr+2,1),ldu) 1316 CALL dgelqf( nr, n, u(nr+1,1), ldu, work(n+1), 1317$ work(n+nr+1), lwork-n-nr, ierr )
1318  CALL dlacpy('L',nr,nr,u(nr+1,1),ldu,v,ldv)
1319  IF ( nr .GT. 1 )
1320  $CALL dlaset('U',nr-1,nr-1,zero,zero,v(1,2),ldv) 1321 CALL dgesvd( 'S', 'O', nr, nr, v, ldv, s, u, ldu, 1322$ v, ldv, work(n+nr+1), lwork-n-nr, info )
1323  CALL dlaset('A',n-nr,nr,zero,zero,v(nr+1,1),ldv)
1324  CALL dlaset('A',nr,n-nr,zero,zero,v(1,nr+1),ldv)
1325  CALL dlaset('A',n-nr,n-nr,zero,one,v(nr+1,nr+1),ldv)
1326  CALL dormlq('R','N',n,n,nr,u(nr+1,1),ldu,work(n+1),
1327  $v, ldv, work(n+nr+1),lwork-n-nr,ierr) 1328 CALL dlapmt( .false., n, n, v, ldv, iwork ) 1329 * .. assemble the left singular vector matrix U of dimensions 1330 * (M x NR) or (M x N) or (M x M). 1331 IF ( ( nr .LT. m ) .AND. .NOT.(wntuf)) THEN 1332 CALL dlaset('A',m-nr,nr,zero,zero,u(nr+1,1),ldu) 1333 IF ( nr .LT. n1 ) THEN 1334 CALL dlaset('A',nr,n1-nr,zero,zero,u(1,nr+1),ldu) 1335 CALL dlaset( 'A',m-nr,n1-nr,zero,one, 1336$ u(nr+1,nr+1), ldu )
1337  END IF
1338  END IF
1339  END IF
1340  END IF
1341 * .. end of the "R**T or R" branch
1342  END IF
1343 *
1344 * The Q matrix from the first QRF is built into the left singular
1345 * vectors matrix U.
1346 *
1347  IF ( .NOT. wntuf )
1348  $CALL dormqr( 'L', 'N', m, n1, n, a, lda, work, u, 1349$ ldu, work(n+1), lwork-n, ierr )
1350  IF ( rowprm .AND. .NOT.wntuf )
1351  $CALL dlaswp( n1, u, ldu, 1, m-1, iwork(n+1), -1 ) 1352 * 1353 * ... end of the "full SVD" branch 1354 END IF 1355 * 1356 * Check whether some singular values are returned as zeros, e.g. 1357 * due to underflow, and update the numerical rank. 1358 p = nr 1359 DO 4001 q = p, 1, -1 1360 IF ( s(q) .GT. zero ) GO TO 4002 1361 nr = nr - 1 1362 4001 CONTINUE 1363 4002 CONTINUE 1364 * 1365 * .. if numerical rank deficiency is detected, the truncated 1366 * singular values are set to zero. 1367 IF ( nr .LT. n ) CALL dlaset( 'G', n-nr,1, zero,zero, s(nr+1), n ) 1368 * .. undo scaling; this may cause overflow in the largest singular 1369 * values. 1370 IF ( ascaled ) 1371$ CALL dlascl( 'G',0,0, one,sqrt(dble(m)), nr,1, s, n, ierr )
1372  IF ( conda ) rwork(1) = sconda
1373  rwork(2) = p - nr
1374 * .. p-NR is the number of singular values that are computed as
1375 * exact zeros in DGESVD() applied to the (possibly truncated)
1376 * full row rank triangular (trapezoidal) factor of A.
1377  numrank = nr
1378 *
1379  RETURN
1380 *
1381 * End of DGESVDQ
1382 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: dlascl.f:143
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
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:110
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dgeqp3(M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO)
DGEQP3
Definition: dgeqp3.f:151
subroutine dgeqrf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGEQRF
Definition: dgeqrf.f:146
subroutine dgelqf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
DGELQF
Definition: dgelqf.f:143
subroutine dgesvd(JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO)
DGESVD computes the singular value decomposition (SVD) for GE matrices
Definition: dgesvd.f:211
subroutine dlaswp(N, A, LDA, K1, K2, IPIV, INCX)
DLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: dlaswp.f:115
subroutine dlapmt(FORWRD, M, N, X, LDX, K)
DLAPMT performs a forward or backward permutation of the columns of a matrix.
Definition: dlapmt.f:104
subroutine dormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMQR
Definition: dormqr.f:167
subroutine dormlq(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
DORMLQ
Definition: dormlq.f:167
subroutine dpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
DPOCON
Definition: dpocon.f:121
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition: dnrm2.f90:89
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