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dgelss.f
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*> \brief <b> DGELSS solves overdetermined or underdetermined systems for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGELSS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelss.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelss.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelss.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
* WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGELSS computes the minimum norm solution to a real linear least
*> squares problem:
*>
*> Minimize 2-norm(| b - A*x |).
*>
*> using the singular value decomposition (SVD) of A. A is an M-by-N
*> matrix which may be rank-deficient.
*>
*> Several right hand side vectors b and solution vectors x can be
*> handled in a single call; they are stored as the columns of the
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix
*> X.
*>
*> The effective rank of A is determined by treating as zero those
*> singular values which are less than RCOND times the largest singular
*> value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the first min(m,n) rows of A are overwritten with
*> its right singular vectors, stored rowwise.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> On entry, the M-by-NRHS right hand side matrix B.
*> On exit, B is overwritten by the N-by-NRHS solution
*> matrix X. If m >= n and RANK = n, the residual
*> sum-of-squares for the solution in the i-th column is given
*> by the sum of squares of elements n+1:m in that column.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,max(M,N)).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (min(M,N))
*> The singular values of A in decreasing order.
*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> RCOND is used to determine the effective rank of A.
*> Singular values S(i) <= RCOND*S(1) are treated as zero.
*> If RCOND < 0, machine precision is used instead.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The effective rank of A, i.e., the number of singular values
*> which are greater than RCOND*S(1).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1, and also:
*> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: the algorithm for computing the SVD failed to converge;
*> if INFO = i, i off-diagonal elements of an intermediate
*> bidiagonal form did not converge to zero.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleGEsolve
*
* =====================================================================
SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
$ WORK, LWORK, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL,
$ ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN,
$ MAXWRK, MINMN, MINWRK, MM, MNTHR
INTEGER LWORK_DGEQRF, LWORK_DORMQR, LWORK_DGEBRD,
$ LWORK_DORMBR, LWORK_DORGBR, LWORK_DORMLQ,
$ LWORK_DGELQF
DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSQR, DCOPY, DGEBRD, DGELQF, DGEMM, DGEMV,
$ DGEQRF, DLABAD, DLACPY, DLASCL, DLASET, DORGBR,
$ DORMBR, DORMLQ, DORMQR, DRSCL, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
MINMN = MIN( M, N )
MAXMN = MAX( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
INFO = -7
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.)
*
IF( INFO.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
IF( MINMN.GT.0 ) THEN
MM = M
MNTHR = ILAENV( 6, 'DGELSS', ' ', M, N, NRHS, -1 )
IF( M.GE.N .AND. M.GE.MNTHR ) THEN
*
* Path 1a - overdetermined, with many more rows than
* columns
*
* Compute space needed for DGEQRF
CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, INFO )
LWORK_DGEQRF=DUM(1)
* Compute space needed for DORMQR
CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, DUM(1), B,
$ LDB, DUM(1), -1, INFO )
LWORK_DORMQR=DUM(1)
MM = N
MAXWRK = MAX( MAXWRK, N + LWORK_DGEQRF )
MAXWRK = MAX( MAXWRK, N + LWORK_DORMQR )
END IF
IF( M.GE.N ) THEN
*
* Path 1 - overdetermined or exactly determined
*
* Compute workspace needed for DBDSQR
*
BDSPAC = MAX( 1, 5*N )
* Compute space needed for DGEBRD
CALL DGEBRD( MM, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, INFO )
LWORK_DGEBRD=DUM(1)
* Compute space needed for DORMBR
CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, DUM(1),
$ B, LDB, DUM(1), -1, INFO )
LWORK_DORMBR=DUM(1)
* Compute space needed for DORGBR
CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, INFO )
LWORK_DORGBR=DUM(1)
* Compute total workspace needed
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORMBR )
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, BDSPAC )
MAXWRK = MAX( MAXWRK, N*NRHS )
MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC )
MAXWRK = MAX( MINWRK, MAXWRK )
END IF
IF( N.GT.M ) THEN
*
* Compute workspace needed for DBDSQR
*
BDSPAC = MAX( 1, 5*M )
MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC )
IF( N.GE.MNTHR ) THEN
*
* Path 2a - underdetermined, with many more columns
* than rows
*
* Compute space needed for DGELQF
CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1),
$ -1, INFO )
LWORK_DGELQF=DUM(1)
* Compute space needed for DGEBRD
CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, INFO )
LWORK_DGEBRD=DUM(1)
* Compute space needed for DORMBR
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
$ DUM(1), B, LDB, DUM(1), -1, INFO )
LWORK_DORMBR=DUM(1)
* Compute space needed for DORGBR
CALL DORGBR( 'P', M, M, M, A, LDA, DUM(1),
$ DUM(1), -1, INFO )
LWORK_DORGBR=DUM(1)
* Compute space needed for DORMLQ
CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, DUM(1),
$ B, LDB, DUM(1), -1, INFO )
LWORK_DORMLQ=DUM(1)
* Compute total workspace needed
MAXWRK = M + LWORK_DGELQF
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DGEBRD )
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORMBR )
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC )
IF( NRHS.GT.1 ) THEN
MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
ELSE
MAXWRK = MAX( MAXWRK, M*M + 2*M )
END IF
MAXWRK = MAX( MAXWRK, M + LWORK_DORMLQ )
ELSE
*
* Path 2 - underdetermined
*
* Compute space needed for DGEBRD
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, INFO )
LWORK_DGEBRD=DUM(1)
* Compute space needed for DORMBR
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, A, LDA,
$ DUM(1), B, LDB, DUM(1), -1, INFO )
LWORK_DORMBR=DUM(1)
* Compute space needed for DORGBR
CALL DORGBR( 'P', M, N, M, A, LDA, DUM(1),
$ DUM(1), -1, INFO )
LWORK_DORGBR=DUM(1)
MAXWRK = 3*M + LWORK_DGEBRD
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORMBR )
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, BDSPAC )
MAXWRK = MAX( MAXWRK, N*NRHS )
END IF
END IF
MAXWRK = MAX( MINWRK, MAXWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
$ INFO = -12
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGELSS', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RANK = 0
RETURN
END IF
*
* Get machine parameters
*
EPS = DLAMCH( 'P' )
SFMIN = DLAMCH( 'S' )
SMLNUM = SFMIN / EPS
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
IASCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
IASCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
IASCL = 2
ELSE IF( ANRM.EQ.ZERO ) THEN
*
* Matrix all zero. Return zero solution.
*
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, MINMN )
RANK = 0
GO TO 70
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
IBSCL = 0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
IBSCL = 2
END IF
*
* Overdetermined case
*
IF( M.GE.N ) THEN
*
* Path 1 - overdetermined or exactly determined
*
MM = M
IF( M.GE.MNTHR ) THEN
*
* Path 1a - overdetermined, with many more rows than columns
*
MM = N
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
*
* Multiply B by transpose(Q)
* (Workspace: need N+NRHS, prefer N+NRHS*NB)
*
CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
* Zero out below R
*
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
END IF
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in A
* (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
*
CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors of R
* (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
* Generate right bidiagonalizing vectors of R in A
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
IWORK = IE + N
*
* Perform bidiagonal QR iteration
* multiply B by transpose of left singular vectors
* compute right singular vectors in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
$ 1, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70
*
* Multiply B by reciprocals of singular values
*
THR = MAX( RCOND*S( 1 ), SFMIN )
IF( RCOND.LT.ZERO )
$ THR = MAX( EPS*S( 1 ), SFMIN )
RANK = 0
DO 10 I = 1, N
IF( S( I ).GT.THR ) THEN
CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
RANK = RANK + 1
ELSE
CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
END IF
10 CONTINUE
*
* Multiply B by right singular vectors
* (Workspace: need N, prefer N*NRHS)
*
IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB, ZERO,
$ WORK, LDB )
CALL DLACPY( 'G', N, NRHS, WORK, LDB, B, LDB )
ELSE IF( NRHS.GT.1 ) THEN
CHUNK = LWORK / N
DO 20 I = 1, NRHS, CHUNK
BL = MIN( NRHS-I+1, CHUNK )
CALL DGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1, I ),
$ LDB, ZERO, WORK, N )
CALL DLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB )
20 CONTINUE
ELSE
CALL DGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
CALL DCOPY( N, WORK, 1, B, 1 )
END IF
*
ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
$ MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN
*
* Path 2a - underdetermined, with many more columns than rows
* and sufficient workspace for an efficient algorithm
*
LDWORK = M
IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
$ M*LDA+M+M*NRHS ) )LDWORK = LDA
ITAU = 1
IWORK = M + 1
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
IL = IWORK
*
* Copy L to WORK(IL), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
$ LDWORK )
IE = IL + LDWORK*M
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
* (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors of L
* (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
$ WORK( ITAUQ ), B, LDB, WORK( IWORK ),
$ LWORK-IWORK+1, INFO )
*
* Generate right bidiagonalizing vectors of R in WORK(IL)
* (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
IWORK = IE + M
*
* Perform bidiagonal QR iteration,
* computing right singular vectors of L in WORK(IL) and
* multiplying B by transpose of left singular vectors
* (Workspace: need M*M+M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ),
$ LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70
*
* Multiply B by reciprocals of singular values
*
THR = MAX( RCOND*S( 1 ), SFMIN )
IF( RCOND.LT.ZERO )
$ THR = MAX( EPS*S( 1 ), SFMIN )
RANK = 0
DO 30 I = 1, M
IF( S( I ).GT.THR ) THEN
CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
RANK = RANK + 1
ELSE
CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
END IF
30 CONTINUE
IWORK = IE
*
* Multiply B by right singular vectors of L in WORK(IL)
* (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS)
*
IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN
CALL DGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ), LDWORK,
$ B, LDB, ZERO, WORK( IWORK ), LDB )
CALL DLACPY( 'G', M, NRHS, WORK( IWORK ), LDB, B, LDB )
ELSE IF( NRHS.GT.1 ) THEN
CHUNK = ( LWORK-IWORK+1 ) / M
DO 40 I = 1, NRHS, CHUNK
BL = MIN( NRHS-I+1, CHUNK )
CALL DGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ), LDWORK,
$ B( 1, I ), LDB, ZERO, WORK( IWORK ), M )
CALL DLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ),
$ LDB )
40 CONTINUE
ELSE
CALL DGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1, 1 ),
$ 1, ZERO, WORK( IWORK ), 1 )
CALL DCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 )
END IF
*
* Zero out below first M rows of B
*
CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
IWORK = ITAU + M
*
* Multiply transpose(Q) by B
* (Workspace: need M+NRHS, prefer M+NRHS*NB)
*
CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
ELSE
*
* Path 2 - remaining underdetermined cases
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize A
* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ INFO )
*
* Multiply B by transpose of left bidiagonalizing vectors
* (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
*
CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
*
* Generate right bidiagonalizing vectors in A
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
IWORK = IE + M
*
* Perform bidiagonal QR iteration,
* computing right singular vectors of A in A and
* multiplying B by transpose of left singular vectors
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
$ 1, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70
*
* Multiply B by reciprocals of singular values
*
THR = MAX( RCOND*S( 1 ), SFMIN )
IF( RCOND.LT.ZERO )
$ THR = MAX( EPS*S( 1 ), SFMIN )
RANK = 0
DO 50 I = 1, M
IF( S( I ).GT.THR ) THEN
CALL DRSCL( NRHS, S( I ), B( I, 1 ), LDB )
RANK = RANK + 1
ELSE
CALL DLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
END IF
50 CONTINUE
*
* Multiply B by right singular vectors of A
* (Workspace: need N, prefer N*NRHS)
*
IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
CALL DGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB, ZERO,
$ WORK, LDB )
CALL DLACPY( 'F', N, NRHS, WORK, LDB, B, LDB )
ELSE IF( NRHS.GT.1 ) THEN
CHUNK = LWORK / N
DO 60 I = 1, NRHS, CHUNK
BL = MIN( NRHS-I+1, CHUNK )
CALL DGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1, I ),
$ LDB, ZERO, WORK, N )
CALL DLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB )
60 CONTINUE
ELSE
CALL DGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
CALL DCOPY( N, WORK, 1, B, 1 )
END IF
END IF
*
* Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
END IF
*
70 CONTINUE
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGELSS
*
END