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|
*> \brief \b DGESDD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGESDD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesdd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesdd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesdd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
* WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
* INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGESDD computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, optionally computing the left and right singular
*> vectors. If singular vectors are desired, it uses a
*> divide-and-conquer algorithm.
*>
*> The SVD is written
*>
*> A = U * SIGMA * transpose(V)
*>
*> where SIGMA is an M-by-N matrix which is zero except for its
*> min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
*> V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA
*> are the singular values of A; they are real and non-negative, and
*> are returned in descending order. The first min(m,n) columns of
*> U and V are the left and right singular vectors of A.
*>
*> Note that the routine returns VT = V**T, not V.
*>
*> The divide and conquer algorithm makes very mild assumptions about
*> floating point arithmetic. It will work on machines with a guard
*> digit in add/subtract, or on those binary machines without guard
*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> Specifies options for computing all or part of the matrix U:
*> = 'A': all M columns of U and all N rows of V**T are
*> returned in the arrays U and VT;
*> = 'S': the first min(M,N) columns of U and the first
*> min(M,N) rows of V**T are returned in the arrays U
*> and VT;
*> = 'O': If M >= N, the first N columns of U are overwritten
*> on the array A and all rows of V**T are returned in
*> the array VT;
*> otherwise, all columns of U are returned in the
*> array U and the first M rows of V**T are overwritten
*> in the array A;
*> = 'N': no columns of U or rows of V**T are computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A. N >= 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,
*> if JOBZ = 'O', A is overwritten with the first N columns
*> of U (the left singular vectors, stored
*> columnwise) if M >= N;
*> A is overwritten with the first M rows
*> of V**T (the right singular vectors, stored
*> rowwise) otherwise.
*> if JOBZ .ne. 'O', the contents of A are destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (min(M,N))
*> The singular values of A, sorted so that S(i) >= S(i+1).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,UCOL)
*> UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
*> UCOL = min(M,N) if JOBZ = 'S'.
*> If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
*> orthogonal matrix U;
*> if JOBZ = 'S', U contains the first min(M,N) columns of U
*> (the left singular vectors, stored columnwise);
*> if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= 1; if
*> JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
*> If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
*> N-by-N orthogonal matrix V**T;
*> if JOBZ = 'S', VT contains the first min(M,N) rows of
*> V**T (the right singular vectors, stored rowwise);
*> if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= 1;
*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*> if JOBZ = 'S', LDVT >= min(M,N).
*> \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.
*> If LWORK = -1, a workspace query is assumed. The optimal
*> size for the WORK array is calculated and stored in WORK(1),
*> and no other work except argument checking is performed.
*>
*> Let mx = max(M,N) and mn = min(M,N).
*> If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
*> If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
*> If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
*> If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
*> These are not tight minimums in all cases; see comments inside code.
*> For good performance, LWORK should generally be larger;
*> a query is recommended.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (8*min(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: DBDSDC did not converge, updating process failed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup doubleGEsing
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
$ WORK, LWORK, IWORK, INFO )
implicit none
*
* -- 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..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IL,
$ IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR, NWORK, WRKBL
INTEGER LWORK_DGEBRD_MN, LWORK_DGEBRD_MM,
$ LWORK_DGEBRD_NN, LWORK_DGELQF_MN,
$ LWORK_DGEQRF_MN,
$ LWORK_DORGBR_P_MM, LWORK_DORGBR_Q_NN,
$ LWORK_DORGLQ_MN, LWORK_DORGLQ_NN,
$ LWORK_DORGQR_MM, LWORK_DORGQR_MN,
$ LWORK_DORMBR_PRT_MM, LWORK_DORMBR_QLN_MM,
$ LWORK_DORMBR_PRT_MN, LWORK_DORMBR_QLN_MN,
$ LWORK_DORMBR_PRT_NN, LWORK_DORMBR_QLN_NN
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSDC, DGEBRD, DGELQF, DGEMM, DGEQRF, DLACPY,
$ DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE, LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
MINMN = MIN( M, N )
WNTQA = LSAME( JOBZ, 'A' )
WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
WNTQO = LSAME( JOBZ, 'O' )
WNTQN = LSAME( JOBZ, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDU.LT.1 .OR. ( WNTQAS .AND. LDU.LT.M ) .OR.
$ ( WNTQO .AND. M.LT.N .AND. LDU.LT.M ) ) THEN
INFO = -8
ELSE IF( LDVT.LT.1 .OR. ( WNTQA .AND. LDVT.LT.N ) .OR.
$ ( WNTQS .AND. LDVT.LT.MINMN ) .OR.
$ ( WNTQO .AND. M.GE.N .AND. LDVT.LT.N ) ) THEN
INFO = -10
END IF
*
* Compute workspace
* Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace allocated 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
BDSPAC = 0
MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSDC
*
IF( WNTQN ) THEN
* dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
* keep 7*N for backwards compatability.
BDSPAC = 7*N
ELSE
BDSPAC = 3*N*N + 4*N
END IF
*
* Compute space preferred for each routine
CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_MN = INT( DUM(1) )
*
CALL DGEBRD( N, N, DUM(1), N, DUM(1), DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_NN = INT( DUM(1) )
*
CALL DGEQRF( M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
LWORK_DGEQRF_MN = INT( DUM(1) )
*
CALL DORGBR( 'Q', N, N, N, DUM(1), N, DUM(1), DUM(1), -1,
$ IERR )
LWORK_DORGBR_Q_NN = INT( DUM(1) )
*
CALL DORGQR( M, M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
LWORK_DORGQR_MM = INT( DUM(1) )
*
CALL DORGQR( M, N, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
LWORK_DORGQR_MN = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, DUM(1), N,
$ DUM(1), DUM(1), N, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, DUM(1), N,
$ DUM(1), DUM(1), N, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_NN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_MN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
IF( M.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
* Path 1 (M >> N, JOBZ='N')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
MAXWRK = MAX( WRKBL, BDSPAC + N )
MINWRK = BDSPAC + N
ELSE IF( WNTQO ) THEN
*
* Path 2 (M >> N, JOBZ='O')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + 2*N*N
MINWRK = BDSPAC + 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
* Path 3 (M >> N, JOBZ='S')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
MINWRK = BDSPAC + N*N + 3*N
ELSE IF( WNTQA ) THEN
*
* Path 4 (M >> N, JOBZ='A')
*
WRKBL = N + LWORK_DGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MM )
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
MINWRK = N*N + MAX( 3*N + BDSPAC, N + M )
END IF
ELSE
*
* Path 5 (M >= N, but not much larger)
*
WRKBL = 3*N + LWORK_DGEBRD_MN
IF( WNTQN ) THEN
* Path 5n (M >= N, jobz='N')
MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQO ) THEN
* Path 5o (M >= N, jobz='O')
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + M*N
MINWRK = 3*N + MAX( M, N*N + BDSPAC )
ELSE IF( WNTQS ) THEN
* Path 5s (M >= N, jobz='S')
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQA ) THEN
* Path 5a (M >= N, jobz='A')
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
END IF
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSDC
*
IF( WNTQN ) THEN
* dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
* keep 7*N for backwards compatability.
BDSPAC = 7*M
ELSE
BDSPAC = 3*M*M + 4*M
END IF
*
* Compute space preferred for each routine
CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_MN = INT( DUM(1) )
*
CALL DGEBRD( M, M, A, M, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
LWORK_DGEBRD_MM = INT( DUM(1) )
*
CALL DGELQF( M, N, A, M, DUM(1), DUM(1), -1, IERR )
LWORK_DGELQF_MN = INT( DUM(1) )
*
CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
LWORK_DORGLQ_NN = INT( DUM(1) )
*
CALL DORGLQ( M, N, M, A, M, DUM(1), DUM(1), -1, IERR )
LWORK_DORGLQ_MN = INT( DUM(1) )
*
CALL DORGBR( 'P', M, M, M, A, N, DUM(1), DUM(1), -1, IERR )
LWORK_DORGBR_P_MM = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', M, M, M, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_MM = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', M, N, M, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_MN = INT( DUM(1) )
*
CALL DORMBR( 'P', 'R', 'T', N, N, M, DUM(1), N,
$ DUM(1), DUM(1), N, DUM(1), -1, IERR )
LWORK_DORMBR_PRT_NN = INT( DUM(1) )
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, DUM(1), M,
$ DUM(1), DUM(1), M, DUM(1), -1, IERR )
LWORK_DORMBR_QLN_MM = INT( DUM(1) )
*
IF( N.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
* Path 1t (N >> M, JOBZ='N')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
MAXWRK = MAX( WRKBL, BDSPAC + M )
MINWRK = BDSPAC + M
ELSE IF( WNTQO ) THEN
*
* Path 2t (N >> M, JOBZ='O')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_MN )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + 2*M*M
MINWRK = BDSPAC + 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
* Path 3t (N >> M, JOBZ='S')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_MN )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
MINWRK = BDSPAC + M*M + 3*M
ELSE IF( WNTQA ) THEN
*
* Path 4t (N >> M, JOBZ='A')
*
WRKBL = M + LWORK_DGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_NN )
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
MINWRK = M*M + MAX( 3*M + BDSPAC, M + N )
END IF
ELSE
*
* Path 5t (N > M, but not much larger)
*
WRKBL = 3*M + LWORK_DGEBRD_MN
IF( WNTQN ) THEN
* Path 5tn (N > M, jobz='N')
MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQO ) THEN
* Path 5to (N > M, jobz='O')
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*N
MINWRK = 3*M + MAX( N, M*M + BDSPAC )
ELSE IF( WNTQS ) THEN
* Path 5ts (N > M, jobz='S')
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQA ) THEN
* Path 5ta (N > M, jobz='A')
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_NN )
MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
END IF
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGESDD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
ELSE IF( ANRM.GT.BIGNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
END IF
*
IF( M.GE.N ) THEN
*
* A has at least as many rows as columns. If A has sufficiently
* more rows than columns, first reduce using the QR
* decomposition (if sufficient workspace available)
*
IF( M.GE.MNTHR ) THEN
*
IF( WNTQN ) THEN
*
* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
* Workspace: need N [tau] + N [work]
* Workspace: prefer N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Zero out below R
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in A
* Workspace: need 3*N [e, tauq, taup] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
NWORK = IE + N
*
* Perform bidiagonal SVD, computing singular values only
* Workspace: need N [e] + BDSPAC
*
CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
* Path 2 (M >> N, JOBZ = 'O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
IR = 1
*
* WORK(IR) is LDWRKR by N
*
IF( LWORK .GE. LDA*N + N*N + 3*N + BDSPAC ) THEN
LDWRKR = LDA
ELSE
LDWRKR = ( LWORK - N*N - 3*N - BDSPAC ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* WORK(IU) is N by N
*
IU = NWORK
NWORK = IU + N*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite WORK(IU) by left singular vectors of R
* and VT by right singular vectors of R
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U]
* Workspace: prefer M*N [R] + 3*N [e, tauq, taup] + N*N [U]
*
DO 10 I = 1, M, LDWRKR
CHUNK = MIN( M - I + 1, LDWRKR )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IU ), N, ZERO, WORK( IR ),
$ LDWRKR )
CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
$ A( I, 1 ), LDA )
10 CONTINUE
*
ELSE IF( WNTQS ) THEN
*
* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IR = 1
*
* WORK(IR) is N by N
*
LDWRKR = N
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
* Workspace: need N*N [R] + N [tau] + N [work]
* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagoal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of R and VT
* by right singular vectors of R
* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
* Workspace: need N*N [R]
*
CALL DLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, WORK( IR ),
$ LDWRKR, ZERO, U, LDU )
*
ELSE IF( WNTQA ) THEN
*
* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IU = 1
*
* WORK(IU) is N by N
*
LDWRKU = N
ITAU = IU + LDWRKU*N
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* Workspace: need N*N [U] + N [tau] + N [work]
* Workspace: prefer N*N [U] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* Workspace: need N*N [U] + N [tau] + M [work]
* Workspace: prefer N*N [U] + N [tau] + M*NB [work]
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce R in A, zeroing out other entries
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in A
* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
* Workspace: need N*N [U] + 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite WORK(IU) by left singular vectors of R and VT
* by right singular vectors of R
* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
* Workspace: need N*N [U]
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, WORK( IU ),
$ LDWRKU, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
END IF
*
ELSE
*
* M .LT. MNTHR
*
* Path 5 (M >= N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize A
* Workspace: need 3*N [e, tauq, taup] + M [work]
* Workspace: prefer 3*N [e, tauq, taup] + (M+N)*NB [work]
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
* Path 5n (M >= N, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
* Path 5o (M >= N, JOBZ='O')
IU = NWORK
IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
* WORK( IU ) is M by N
*
LDWRKU = M
NWORK = IU + LDWRKU*N
CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IU ),
$ LDWRKU )
* IR is unused; silence compile warnings
IR = -1
ELSE
*
* WORK( IU ) is N by N
*
LDWRKU = N
NWORK = IU + LDWRKU*N
*
* WORK(IR) is LDWRKR by N
*
IR = NWORK
LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
$ LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ),
$ IWORK, INFO )
*
* Overwrite VT by right singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
* Path 5o-fast
* Overwrite WORK(IU) by left singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + M*N [U] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + M*N [U] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy left singular vectors of A from WORK(IU) to A
*
CALL DLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
ELSE
*
* Path 5o-slow
* Generate Q in A
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of
* bidiagonal matrix in WORK(IU), storing result in
* WORK(IR) and copying to A
* Workspace: need 3*N [e, tauq, taup] + N*N [U] + NB*N [R]
* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + M*N [R]
*
DO 20 I = 1, M, LDWRKR
CHUNK = MIN( M - I + 1, LDWRKR )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IU ), LDWRKU, ZERO,
$ WORK( IR ), LDWRKR )
CALL DLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
$ A( I, 1 ), LDA )
20 CONTINUE
END IF
*
ELSE IF( WNTQS ) THEN
*
* Path 5s (M >= N, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, N, ZERO, ZERO, U, LDU )
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + N [work]
* Workspace: prefer 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
* Path 5a (M >= N, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, M, ZERO, ZERO, U, LDU )
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Set the right corner of U to identity matrix
*
IF( M.GT.N ) THEN
CALL DLASET( 'F', M - N, M - N, ZERO, ONE, U(N+1,N+1),
$ LDU )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*N [e, tauq, taup] + M [work]
* Workspace: prefer 3*N [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
END IF
*
END IF
*
ELSE
*
* A has more columns than rows. If A has sufficiently more
* columns than rows, first reduce using the LQ decomposition (if
* sufficient workspace available)
*
IF( N.GE.MNTHR ) THEN
*
IF( WNTQN ) THEN
*
* Path 1t (N >> M, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + M
*
* Compute A=L*Q
* Workspace: need M [tau] + M [work]
* Workspace: prefer M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Zero out above L
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA )
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in A
* Workspace: need 3*M [e, tauq, taup] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
NWORK = IE + M
*
* Perform bidiagonal SVD, computing singular values only
* Workspace: need M [e] + BDSPAC
*
CALL DBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
* Path 2t (N >> M, JOBZ='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
IVT = 1
*
* WORK(IVT) is M by M
* WORK(IL) is M by M; it is later resized to M by chunk for gemm
*
IL = IVT + M*M
IF( LWORK .GE. M*N + M*M + 3*M + BDSPAC ) THEN
LDWRKL = M
CHUNK = N
ELSE
LDWRKL = M
CHUNK = ( LWORK - M*M ) / M
END IF
ITAU = IL + LDWRKL*M
NWORK = ITAU + M
*
* Compute A=L*Q
* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing about above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
$ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U, and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), M, DUM, IDUM, WORK( NWORK ),
$ IWORK, INFO )
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUP ), WORK( IVT ), M,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by Q
* in A, storing result in WORK(IL) and copying to A
* Workspace: need M*M [VT] + M*M [L]
* Workspace: prefer M*M [VT] + M*N [L]
* At this point, L is resized as M by chunk.
*
DO 30 I = 1, N, CHUNK
BLK = MIN( N - I + 1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), M,
$ A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL )
CALL DLACPY( 'F', M, BLK, WORK( IL ), LDWRKL,
$ A( 1, I ), LDA )
30 CONTINUE
*
ELSE IF( WNTQS ) THEN
*
* Path 3t (N >> M, JOBZ='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IL = 1
*
* WORK(IL) is M by M
*
LDWRKL = M
ITAU = IL + LDWRKL*M
NWORK = ITAU + M
*
* Compute A=L*Q
* Workspace: need M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
$ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
* Workspace: need M*M [L] + M [tau] + M [work]
* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU).
* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of L and VT
* by right singular vectors of L
* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IL) by
* Q in A, storing result in VT
* Workspace: need M*M [L]
*
CALL DLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ), LDWRKL,
$ A, LDA, ZERO, VT, LDVT )
*
ELSE IF( WNTQA ) THEN
*
* Path 4t (N >> M, JOBZ='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IVT = 1
*
* WORK(IVT) is M by M
*
LDWKVT = M
ITAU = IVT + LDWKVT*M
NWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* Workspace: need M*M [VT] + M [tau] + M [work]
* Workspace: prefer M*M [VT] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* Workspace: need M*M [VT] + M [tau] + N [work]
* Workspace: prefer M*M [VT] + M [tau] + N*NB [work]
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce L in A, zeroing out other entries
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in A
* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + M [work]
* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
$ WORK( NWORK ), IWORK, INFO )
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
* Workspace: need M*M [VT] + 3*M [e, tauq, taup]+ M [work]
* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup]+ M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by
* Q in VT, storing result in A
* Workspace: need M*M [VT]
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ), LDWKVT,
$ VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
END IF
*
ELSE
*
* N .LT. MNTHR
*
* Path 5t (N > M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize A
* Workspace: need 3*M [e, tauq, taup] + N [work]
* Workspace: prefer 3*M [e, tauq, taup] + (M+N)*NB [work]
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
* Path 5tn (N > M, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
* Path 5to (N > M, JOBZ='O')
LDWKVT = M
IVT = NWORK
IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
* WORK( IVT ) is M by N
*
CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ),
$ LDWKVT )
NWORK = IVT + LDWKVT*N
* IL is unused; silence compile warnings
IL = -1
ELSE
*
* WORK( IVT ) is M by M
*
NWORK = IVT + LDWKVT*M
IL = NWORK
*
* WORK(IL) is M by CHUNK
*
CHUNK = ( LWORK - M*M - 3*M ) / M
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + BDSPAC
*
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
$ WORK( NWORK ), IWORK, INFO )
*
* Overwrite U by left singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
*
IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
* Path 5to-fast
* Overwrite WORK(IVT) by left singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + M*N [VT] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*N [VT] + M*NB [work]
*
CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy right singular vectors of A from WORK(IVT) to A
*
CALL DLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
ELSE
*
* Path 5to-slow
* Generate P**T in A
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by right singular vectors of
* bidiagonal matrix in WORK(IVT), storing result in
* WORK(IL) and copying to A
* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M*NB [L]
* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*N [L]
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N - I + 1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ),
$ LDWKVT, A( 1, I ), LDA, ZERO,
$ WORK( IL ), M )
CALL DLACPY( 'F', M, BLK, WORK( IL ), M, A( 1, I ),
$ LDA )
40 CONTINUE
END IF
ELSE IF( WNTQS ) THEN
*
* Path 5ts (N > M, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, N, ZERO, ZERO, VT, LDVT )
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + M [work]
* Workspace: prefer 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
* Path 5ta (N > M, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', N, N, ZERO, ZERO, VT, LDVT )
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
$ INFO )
*
* Set the right corner of VT to identity matrix
*
IF( N.GT.M ) THEN
CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT(M+1,M+1),
$ LDVT )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
* Workspace: need 3*M [e, tauq, taup] + N [work]
* Workspace: prefer 3*M [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK - NWORK + 1, IERR )
END IF
*
END IF
*
END IF
*
* Undo scaling if necessary
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
IF( ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
END IF
*
* Return optimal workspace in WORK(1)
*
WORK( 1 ) = MAXWRK
*
RETURN
*
* End of DGESDD
*
END
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