3695 lines
139 KiB
Fortran
3695 lines
139 KiB
Fortran
*> \brief <b> ZGESVD computes the singular value decomposition (SVD) for GE matrices</b>
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download ZGESVD + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgesvd.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgesvd.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgesvd.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE ZGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
|
|
* WORK, LWORK, RWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* CHARACTER JOBU, JOBVT
|
|
* INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* DOUBLE PRECISION RWORK( * ), S( * )
|
|
* COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ),
|
|
* $ WORK( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> ZGESVD computes the singular value decomposition (SVD) of a complex
|
|
*> M-by-N matrix A, optionally computing the left and/or right singular
|
|
*> vectors. The SVD is written
|
|
*>
|
|
*> A = U * SIGMA * conjugate-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 unitary matrix, and
|
|
*> V is an N-by-N unitary 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 V**H, not V.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \param[in] JOBU
|
|
*> \verbatim
|
|
*> JOBU is CHARACTER*1
|
|
*> Specifies options for computing all or part of the matrix U:
|
|
*> = 'A': all M columns of U are returned in array U:
|
|
*> = 'S': the first min(m,n) columns of U (the left singular
|
|
*> vectors) are returned in the array U;
|
|
*> = 'O': the first min(m,n) columns of U (the left singular
|
|
*> vectors) are overwritten on the array A;
|
|
*> = 'N': no columns of U (no left singular vectors) are
|
|
*> computed.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] JOBVT
|
|
*> \verbatim
|
|
*> JOBVT is CHARACTER*1
|
|
*> Specifies options for computing all or part of the matrix
|
|
*> V**H:
|
|
*> = 'A': all N rows of V**H are returned in the array VT;
|
|
*> = 'S': the first min(m,n) rows of V**H (the right singular
|
|
*> vectors) are returned in the array VT;
|
|
*> = 'O': the first min(m,n) rows of V**H (the right singular
|
|
*> vectors) are overwritten on the array A;
|
|
*> = 'N': no rows of V**H (no right singular vectors) are
|
|
*> computed.
|
|
*>
|
|
*> JOBVT and JOBU cannot both be 'O'.
|
|
*> \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 COMPLEX*16 array, dimension (LDA,N)
|
|
*> On entry, the M-by-N matrix A.
|
|
*> On exit,
|
|
*> if JOBU = 'O', A is overwritten with the first min(m,n)
|
|
*> columns of U (the left singular vectors,
|
|
*> stored columnwise);
|
|
*> if JOBVT = 'O', A is overwritten with the first min(m,n)
|
|
*> rows of V**H (the right singular vectors,
|
|
*> stored rowwise);
|
|
*> if JOBU .ne. 'O' and JOBVT .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 COMPLEX*16 array, dimension (LDU,UCOL)
|
|
*> (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
|
|
*> If JOBU = 'A', U contains the M-by-M unitary matrix U;
|
|
*> if JOBU = 'S', U contains the first min(m,n) columns of U
|
|
*> (the left singular vectors, stored columnwise);
|
|
*> if JOBU = 'N' or 'O', U is not referenced.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDU
|
|
*> \verbatim
|
|
*> LDU is INTEGER
|
|
*> The leading dimension of the array U. LDU >= 1; if
|
|
*> JOBU = 'S' or 'A', LDU >= M.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] VT
|
|
*> \verbatim
|
|
*> VT is COMPLEX*16 array, dimension (LDVT,N)
|
|
*> If JOBVT = 'A', VT contains the N-by-N unitary matrix
|
|
*> V**H;
|
|
*> if JOBVT = 'S', VT contains the first min(m,n) rows of
|
|
*> V**H (the right singular vectors, stored rowwise);
|
|
*> if JOBVT = 'N' or 'O', VT is not referenced.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDVT
|
|
*> \verbatim
|
|
*> LDVT is INTEGER
|
|
*> The leading dimension of the array VT. LDVT >= 1; if
|
|
*> JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WORK
|
|
*> \verbatim
|
|
*> WORK is COMPLEX*16 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 >= MAX(1,2*MIN(M,N)+MAX(M,N)).
|
|
*> 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] RWORK
|
|
*> \verbatim
|
|
*> RWORK is DOUBLE PRECISION array, dimension (5*min(M,N))
|
|
*> On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the
|
|
*> unconverged superdiagonal elements of an upper bidiagonal
|
|
*> matrix B whose diagonal is in S (not necessarily sorted).
|
|
*> B satisfies A = U * B * VT, so it has the same singular
|
|
*> values as A, and singular vectors related by U and VT.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] INFO
|
|
*> \verbatim
|
|
*> INFO is INTEGER
|
|
*> = 0: successful exit.
|
|
*> < 0: if INFO = -i, the i-th argument had an illegal value.
|
|
*> > 0: if ZBDSQR did not converge, INFO specifies how many
|
|
*> superdiagonals of an intermediate bidiagonal form B
|
|
*> did not converge to zero. See the description of RWORK
|
|
*> above for details.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \date April 2012
|
|
*
|
|
*> \ingroup complex16GEsing
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE ZGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU,
|
|
$ VT, LDVT, WORK, LWORK, RWORK, INFO )
|
|
*
|
|
* -- LAPACK driver routine (version 3.4.1) --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
* April 2012
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER JOBU, JOBVT
|
|
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
|
|
* ..
|
|
* .. Array Arguments ..
|
|
DOUBLE PRECISION RWORK( * ), S( * )
|
|
COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ),
|
|
$ WORK( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
COMPLEX*16 CZERO, CONE
|
|
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
|
|
$ CONE = ( 1.0D0, 0.0D0 ) )
|
|
DOUBLE PRECISION ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL LQUERY, WNTUA, WNTUAS, WNTUN, WNTUO, WNTUS,
|
|
$ WNTVA, WNTVAS, WNTVN, WNTVO, WNTVS
|
|
INTEGER BLK, CHUNK, I, IE, IERR, IR, IRWORK, ISCL,
|
|
$ ITAU, ITAUP, ITAUQ, IU, IWORK, LDWRKR, LDWRKU,
|
|
$ MAXWRK, MINMN, MINWRK, MNTHR, NCU, NCVT, NRU,
|
|
$ NRVT, WRKBL
|
|
INTEGER LWORK_ZGEQRF, LWORK_ZUNGQR_N, LWORK_ZUNGQR_M,
|
|
$ LWORK_ZGEBRD, LWORK_ZUNGBR_P, LWORK_ZUNGBR_Q,
|
|
$ LWORK_ZGELQF, LWORK_ZUNGLQ_N, LWORK_ZUNGLQ_M
|
|
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
|
|
* ..
|
|
* .. Local Arrays ..
|
|
DOUBLE PRECISION DUM( 1 )
|
|
COMPLEX*16 CDUM( 1 )
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL DLASCL, XERBLA, ZBDSQR, ZGEBRD, ZGELQF, ZGEMM,
|
|
$ ZGEQRF, ZLACPY, ZLASCL, ZLASET, ZUNGBR, ZUNGLQ,
|
|
$ ZUNGQR, ZUNMBR
|
|
* ..
|
|
* .. External Functions ..
|
|
LOGICAL LSAME
|
|
INTEGER ILAENV
|
|
DOUBLE PRECISION DLAMCH, ZLANGE
|
|
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, MIN, SQRT
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input arguments
|
|
*
|
|
INFO = 0
|
|
MINMN = MIN( M, N )
|
|
WNTUA = LSAME( JOBU, 'A' )
|
|
WNTUS = LSAME( JOBU, 'S' )
|
|
WNTUAS = WNTUA .OR. WNTUS
|
|
WNTUO = LSAME( JOBU, 'O' )
|
|
WNTUN = LSAME( JOBU, 'N' )
|
|
WNTVA = LSAME( JOBVT, 'A' )
|
|
WNTVS = LSAME( JOBVT, 'S' )
|
|
WNTVAS = WNTVA .OR. WNTVS
|
|
WNTVO = LSAME( JOBVT, 'O' )
|
|
WNTVN = LSAME( JOBVT, 'N' )
|
|
LQUERY = ( LWORK.EQ.-1 )
|
|
*
|
|
IF( .NOT.( WNTUA .OR. WNTUS .OR. WNTUO .OR. WNTUN ) ) THEN
|
|
INFO = -1
|
|
ELSE IF( .NOT.( WNTVA .OR. WNTVS .OR. WNTVO .OR. WNTVN ) .OR.
|
|
$ ( WNTVO .AND. WNTUO ) ) THEN
|
|
INFO = -2
|
|
ELSE IF( M.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -4
|
|
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
|
|
INFO = -6
|
|
ELSE IF( LDU.LT.1 .OR. ( WNTUAS .AND. LDU.LT.M ) ) THEN
|
|
INFO = -9
|
|
ELSE IF( LDVT.LT.1 .OR. ( WNTVA .AND. LDVT.LT.N ) .OR.
|
|
$ ( WNTVS .AND. LDVT.LT.MINMN ) ) THEN
|
|
INFO = -11
|
|
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.
|
|
* CWorkspace refers to complex workspace, and RWorkspace to
|
|
* real workspace. 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( M.GE.N .AND. MINMN.GT.0 ) THEN
|
|
*
|
|
* Space needed for ZBDSQR is BDSPAC = 5*N
|
|
*
|
|
MNTHR = ILAENV( 6, 'ZGESVD', JOBU // JOBVT, M, N, 0, 0 )
|
|
* Compute space needed for ZGEQRF
|
|
CALL ZGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZGEQRF=DUM(1)
|
|
* Compute space needed for ZUNGQR
|
|
CALL ZUNGQR( M, N, N, A, LDA, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZUNGQR_N=DUM(1)
|
|
CALL ZUNGQR( M, M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZUNGQR_M=DUM(1)
|
|
* Compute space needed for ZGEBRD
|
|
CALL ZGEBRD( N, N, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZGEBRD=DUM(1)
|
|
* Compute space needed for ZUNGBR
|
|
CALL ZUNGBR( 'P', N, N, N, A, LDA, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_P=DUM(1)
|
|
CALL ZUNGBR( 'Q', N, N, N, A, LDA, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_Q=DUM(1)
|
|
*
|
|
IF( M.GE.MNTHR ) THEN
|
|
IF( WNTUN ) THEN
|
|
*
|
|
* Path 1 (M much larger than N, JOBU='N')
|
|
*
|
|
MAXWRK = N + LWORK_ZGEQRF
|
|
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZGEBRD )
|
|
IF( WNTVO .OR. WNTVAS )
|
|
$ MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_P )
|
|
MINWRK = 3*N
|
|
ELSE IF( WNTUO .AND. WNTVN ) THEN
|
|
*
|
|
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_N )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
MAXWRK = MAX( N*N+WRKBL, N*N+M*N )
|
|
MINWRK = 2*N + M
|
|
ELSE IF( WNTUO .AND. WNTVAS ) THEN
|
|
*
|
|
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or
|
|
* 'A')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_N )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_P )
|
|
MAXWRK = MAX( N*N+WRKBL, N*N+M*N )
|
|
MINWRK = 2*N + M
|
|
ELSE IF( WNTUS .AND. WNTVN ) THEN
|
|
*
|
|
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_N )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
MAXWRK = N*N + WRKBL
|
|
MINWRK = 2*N + M
|
|
ELSE IF( WNTUS .AND. WNTVO ) THEN
|
|
*
|
|
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_N )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_P )
|
|
MAXWRK = 2*N*N + WRKBL
|
|
MINWRK = 2*N + M
|
|
ELSE IF( WNTUS .AND. WNTVAS ) THEN
|
|
*
|
|
* Path 6 (M much larger than N, JOBU='S', JOBVT='S' or
|
|
* 'A')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_N )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_P )
|
|
MAXWRK = N*N + WRKBL
|
|
MINWRK = 2*N + M
|
|
ELSE IF( WNTUA .AND. WNTVN ) THEN
|
|
*
|
|
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_M )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
MAXWRK = N*N + WRKBL
|
|
MINWRK = 2*N + M
|
|
ELSE IF( WNTUA .AND. WNTVO ) THEN
|
|
*
|
|
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_M )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_P )
|
|
MAXWRK = 2*N*N + WRKBL
|
|
MINWRK = 2*N + M
|
|
ELSE IF( WNTUA .AND. WNTVAS ) THEN
|
|
*
|
|
* Path 9 (M much larger than N, JOBU='A', JOBVT='S' or
|
|
* 'A')
|
|
*
|
|
WRKBL = N + LWORK_ZGEQRF
|
|
WRKBL = MAX( WRKBL, N+LWORK_ZUNGQR_M )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_Q )
|
|
WRKBL = MAX( WRKBL, 2*N+LWORK_ZUNGBR_P )
|
|
MAXWRK = N*N + WRKBL
|
|
MINWRK = 2*N + M
|
|
END IF
|
|
ELSE
|
|
*
|
|
* Path 10 (M at least N, but not much larger)
|
|
*
|
|
CALL ZGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZGEBRD=DUM(1)
|
|
MAXWRK = 2*N + LWORK_ZGEBRD
|
|
IF( WNTUS .OR. WNTUO ) THEN
|
|
CALL ZUNGBR( 'Q', M, N, N, A, LDA, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_Q=DUM(1)
|
|
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_Q )
|
|
END IF
|
|
IF( WNTUA ) THEN
|
|
CALL ZUNGBR( 'Q', M, M, N, A, LDA, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_Q=DUM(1)
|
|
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_Q )
|
|
END IF
|
|
IF( .NOT.WNTVN ) THEN
|
|
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_P )
|
|
MINWRK = 2*N + M
|
|
END IF
|
|
END IF
|
|
ELSE IF( MINMN.GT.0 ) THEN
|
|
*
|
|
* Space needed for ZBDSQR is BDSPAC = 5*M
|
|
*
|
|
MNTHR = ILAENV( 6, 'ZGESVD', JOBU // JOBVT, M, N, 0, 0 )
|
|
* Compute space needed for ZGELQF
|
|
CALL ZGELQF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZGELQF=DUM(1)
|
|
* Compute space needed for ZUNGLQ
|
|
CALL ZUNGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZUNGLQ_N=DUM(1)
|
|
CALL ZUNGLQ( M, N, M, A, LDA, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZUNGLQ_M=DUM(1)
|
|
* Compute space needed for ZGEBRD
|
|
CALL ZGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZGEBRD=DUM(1)
|
|
* Compute space needed for ZUNGBR P
|
|
CALL ZUNGBR( 'P', M, M, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_P=DUM(1)
|
|
* Compute space needed for ZUNGBR Q
|
|
CALL ZUNGBR( 'Q', M, M, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_Q=DUM(1)
|
|
IF( N.GE.MNTHR ) THEN
|
|
IF( WNTVN ) THEN
|
|
*
|
|
* Path 1t(N much larger than M, JOBVT='N')
|
|
*
|
|
MAXWRK = M + LWORK_ZGELQF
|
|
MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZGEBRD )
|
|
IF( WNTUO .OR. WNTUAS )
|
|
$ MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZUNGBR_Q )
|
|
MINWRK = 3*M
|
|
ELSE IF( WNTVO .AND. WNTUN ) THEN
|
|
*
|
|
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_M )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
MAXWRK = MAX( M*M+WRKBL, M*M+M*N )
|
|
MINWRK = 2*M + N
|
|
ELSE IF( WNTVO .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 3t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='O')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_M )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_Q )
|
|
MAXWRK = MAX( M*M+WRKBL, M*M+M*N )
|
|
MINWRK = 2*M + N
|
|
ELSE IF( WNTVS .AND. WNTUN ) THEN
|
|
*
|
|
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_M )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = 2*M + N
|
|
ELSE IF( WNTVS .AND. WNTUO ) THEN
|
|
*
|
|
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_M )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_Q )
|
|
MAXWRK = 2*M*M + WRKBL
|
|
MINWRK = 2*M + N
|
|
ELSE IF( WNTVS .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 6t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='S')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_M )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_Q )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = 2*M + N
|
|
ELSE IF( WNTVA .AND. WNTUN ) THEN
|
|
*
|
|
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_N )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = 2*M + N
|
|
ELSE IF( WNTVA .AND. WNTUO ) THEN
|
|
*
|
|
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_N )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_Q )
|
|
MAXWRK = 2*M*M + WRKBL
|
|
MINWRK = 2*M + N
|
|
ELSE IF( WNTVA .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 9t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='A')
|
|
*
|
|
WRKBL = M + LWORK_ZGELQF
|
|
WRKBL = MAX( WRKBL, M+LWORK_ZUNGLQ_N )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZGEBRD )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_P )
|
|
WRKBL = MAX( WRKBL, 2*M+LWORK_ZUNGBR_Q )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = 2*M + N
|
|
END IF
|
|
ELSE
|
|
*
|
|
* Path 10t(N greater than M, but not much larger)
|
|
*
|
|
CALL ZGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, IERR )
|
|
LWORK_ZGEBRD=DUM(1)
|
|
MAXWRK = 2*M + LWORK_ZGEBRD
|
|
IF( WNTVS .OR. WNTVO ) THEN
|
|
* Compute space needed for ZUNGBR P
|
|
CALL ZUNGBR( 'P', M, N, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_P=DUM(1)
|
|
MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZUNGBR_P )
|
|
END IF
|
|
IF( WNTVA ) THEN
|
|
CALL ZUNGBR( 'P', N, N, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_ZUNGBR_P=DUM(1)
|
|
MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZUNGBR_P )
|
|
END IF
|
|
IF( .NOT.WNTUN ) THEN
|
|
MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZUNGBR_Q )
|
|
MINWRK = 2*M + N
|
|
END IF
|
|
END IF
|
|
END IF
|
|
MAXWRK = MAX( MAXWRK, MINWRK )
|
|
WORK( 1 ) = MAXWRK
|
|
*
|
|
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
|
|
INFO = -13
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'ZGESVD', -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 = ZLANGE( 'M', M, N, A, LDA, DUM )
|
|
ISCL = 0
|
|
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
|
ISCL = 1
|
|
CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
|
|
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
|
ISCL = 1
|
|
CALL ZLASCL( '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( WNTUN ) THEN
|
|
*
|
|
* Path 1 (M much larger than N, JOBU='N')
|
|
* No left singular vectors to be computed
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: need 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Zero out below R
|
|
*
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO, A( 2, 1 ),
|
|
$ LDA )
|
|
IE = 1
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
NCVT = 0
|
|
IF( WNTVO .OR. WNTVAS ) THEN
|
|
*
|
|
* If right singular vectors desired, generate P'.
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
NCVT = N
|
|
END IF
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in A if desired
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, NCVT, 0, 0, S, RWORK( IE ), A, LDA,
|
|
$ CDUM, 1, CDUM, 1, RWORK( IRWORK ), INFO )
|
|
*
|
|
* If right singular vectors desired in VT, copy them there
|
|
*
|
|
IF( WNTVAS )
|
|
$ CALL ZLACPY( 'F', N, N, A, LDA, VT, LDVT )
|
|
*
|
|
ELSE IF( WNTUO .AND. WNTVN ) THEN
|
|
*
|
|
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
|
|
* N left singular vectors to be overwritten on A and
|
|
* no right singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.N*N+3*N ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N )+LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N, WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N )+N*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N, WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is LDWRKU by N, WORK(IR) is N by N
|
|
*
|
|
LDWRKU = ( LWORK-N*N ) / N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IR) and zero out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ WORK( IR+1 ), LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IR)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing R
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
|
|
* (RWorkspace: need 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR)
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, 0, N, 0, S, RWORK( IE ), CDUM, 1,
|
|
$ WORK( IR ), LDWRKR, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
IU = ITAUQ
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IR), storing result in WORK(IU) and copying to A
|
|
* (CWorkspace: need N*N+N, prefer N*N+M*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
DO 10 I = 1, M, LDWRKU
|
|
CHUNK = MIN( M-I+1, LDWRKU )
|
|
CALL ZGEMM( 'N', 'N', CHUNK, N, N, CONE, A( I, 1 ),
|
|
$ LDA, WORK( IR ), LDWRKR, CZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL ZLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
|
|
$ A( I, 1 ), LDA )
|
|
10 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
IE = 1
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize A
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
|
|
* (RWorkspace: N)
|
|
*
|
|
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing A
|
|
* (CWorkspace: need 3*N, prefer 2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in A
|
|
* (CWorkspace: need 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, 0, M, 0, S, RWORK( IE ), CDUM, 1,
|
|
$ A, LDA, CDUM, 1, RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUO .AND. WNTVAS ) THEN
|
|
*
|
|
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or 'A')
|
|
* N left singular vectors to be overwritten on A and
|
|
* N right singular vectors to be computed in VT
|
|
*
|
|
IF( LWORK.GE.N*N+3*N ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N )+LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N )+N*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is LDWRKU by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = ( LWORK-N*N ) / N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to VT, zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT, copying result to WORK(IR)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, VT, LDVT, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', N, N, VT, LDVT, WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate left vectors bidiagonalizing R in WORK(IR)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing R in VT
|
|
* (CWorkspace: need N*N+3*N-1, prefer N*N+2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR) and computing right
|
|
* singular vectors of R in VT
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, N, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, WORK( IR ), LDWRKR, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
IU = ITAUQ
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IR), storing result in WORK(IU) and copying to A
|
|
* (CWorkspace: need N*N+N, prefer N*N+M*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
DO 20 I = 1, M, LDWRKU
|
|
CHUNK = MIN( M-I+1, LDWRKU )
|
|
CALL ZGEMM( 'N', 'N', CHUNK, N, N, CONE, A( I, 1 ),
|
|
$ LDA, WORK( IR ), LDWRKR, CZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL ZLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
|
|
$ A( I, 1 ), LDA )
|
|
20 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to VT, zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: N)
|
|
*
|
|
CALL ZGEBRD( N, N, VT, LDVT, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in A by left vectors bidiagonalizing R
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
|
|
$ WORK( ITAUQ ), A, LDA, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing R in VT
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in A and computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, M, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, A, LDA, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUS ) THEN
|
|
*
|
|
IF( WNTVN ) THEN
|
|
*
|
|
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
|
|
* N left singular vectors to be computed in U and
|
|
* no right singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.N*N+3*N ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IR) is LDA by N
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is N by N
|
|
*
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IR), zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ WORK( IR+1 ), LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IR)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, WORK( IR ), LDWRKR, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing R in WORK(IR)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR)
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, 0, N, 0, S, RWORK( IE ), CDUM,
|
|
$ 1, WORK( IR ), LDWRKR, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IR), storing result in U
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, N, CONE, A, LDA,
|
|
$ WORK( IR ), LDWRKR, CZERO, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left vectors bidiagonalizing R
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, 0, M, 0, S, RWORK( IE ), CDUM,
|
|
$ 1, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVO ) THEN
|
|
*
|
|
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
|
|
* N left singular vectors to be computed in U and
|
|
* N right singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*N*N+3*N ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = N
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (CWorkspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (CWorkspace: need 2*N*N+3*N,
|
|
* prefer 2*N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IR)
|
|
* (CWorkspace: need 2*N*N+3*N-1,
|
|
* prefer 2*N*N+2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in WORK(IR)
|
|
* (CWorkspace: need 2*N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, N, 0, S, RWORK( IE ),
|
|
$ WORK( IR ), LDWRKR, WORK( IU ),
|
|
$ LDWRKU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IU), storing result in U
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, N, CONE, A, LDA,
|
|
$ WORK( IU ), LDWRKU, CZERO, U, LDU )
|
|
*
|
|
* Copy right singular vectors of R to A
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left vectors bidiagonalizing R
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing R in A
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in A
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, M, 0, S, RWORK( IE ), A,
|
|
$ LDA, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVAS ) THEN
|
|
*
|
|
* Path 6 (M much larger than N, JOBU='S', JOBVT='S'
|
|
* or 'A')
|
|
* N left singular vectors to be computed in U and
|
|
* N right singular vectors to be computed in VT
|
|
*
|
|
IF( LWORK.GE.N*N+3*N ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N
|
|
*
|
|
LDWRKU = N
|
|
END IF
|
|
ITAU = IU + LDWRKU*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to VT
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
|
|
$ LDVT )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (CWorkspace: need N*N+3*N-1,
|
|
* prefer N*N+2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in VT
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, N, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, WORK( IU ), LDWRKU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IU), storing result in U
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, N, CONE, A, LDA,
|
|
$ WORK( IU ), LDWRKU, CZERO, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, N, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to VT, zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, VT, LDVT, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in VT
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, M, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, U, LDU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUA ) THEN
|
|
*
|
|
IF( WNTVN ) THEN
|
|
*
|
|
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
|
|
* M left singular vectors to be computed in U and
|
|
* no right singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( N+M, 3*N ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IR) is LDA by N
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is N by N
|
|
*
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Copy R to WORK(IR), zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ WORK( IR+1 ), LDWRKR )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need N*N+N+M, prefer N*N+N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IR)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, WORK( IR ), LDWRKR, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IR)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR)
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, 0, N, 0, S, RWORK( IE ), CDUM,
|
|
$ 1, WORK( IR ), LDWRKR, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply Q in U by left singular vectors of R in
|
|
* WORK(IR), storing result in A
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, N, CONE, U, LDU,
|
|
$ WORK( IR ), LDWRKR, CZERO, A, LDA )
|
|
*
|
|
* Copy left singular vectors of A from A to U
|
|
*
|
|
CALL ZLACPY( 'F', M, N, A, LDA, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need N+M, prefer N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in A
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, 0, M, 0, S, RWORK( IE ), CDUM,
|
|
$ 1, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVO ) THEN
|
|
*
|
|
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
|
|
* M left singular vectors to be computed in U and
|
|
* N right singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*N*N+MAX( N+M, 3*N ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = N
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need 2*N*N+N+M, prefer 2*N*N+N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (CWorkspace: need 2*N*N+3*N,
|
|
* prefer 2*N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IR)
|
|
* (CWorkspace: need 2*N*N+3*N-1,
|
|
* prefer 2*N*N+2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in WORK(IR)
|
|
* (CWorkspace: need 2*N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, N, 0, S, RWORK( IE ),
|
|
$ WORK( IR ), LDWRKR, WORK( IU ),
|
|
$ LDWRKU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
* Multiply Q in U by left singular vectors of R in
|
|
* WORK(IU), storing result in A
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, N, CONE, U, LDU,
|
|
$ WORK( IU ), LDWRKU, CZERO, A, LDA )
|
|
*
|
|
* Copy left singular vectors of A from A to U
|
|
*
|
|
CALL ZLACPY( 'F', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Copy right singular vectors of R from WORK(IR) to A
|
|
*
|
|
CALL ZLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need N+M, prefer N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in A
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in A
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in A
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, M, 0, S, RWORK( IE ), A,
|
|
$ LDA, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVAS ) THEN
|
|
*
|
|
* Path 9 (M much larger than N, JOBU='A', JOBVT='S'
|
|
* or 'A')
|
|
* M left singular vectors to be computed in U and
|
|
* N right singular vectors to be computed in VT
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( N+M, 3*N ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N
|
|
*
|
|
LDWRKU = N
|
|
END IF
|
|
ITAU = IU + LDWRKU*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need N*N+N+M, prefer N*N+N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to VT
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
|
|
$ LDVT )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (CWorkspace: need N*N+3*N-1,
|
|
* prefer N*N+2*N+(N-1)*NB)
|
|
* (RWorkspace: need 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in VT
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, N, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, WORK( IU ), LDWRKU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply Q in U by left singular vectors of R in
|
|
* WORK(IU), storing result in A
|
|
* (CWorkspace: need N*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, N, CONE, U, LDU,
|
|
$ WORK( IU ), LDWRKU, CZERO, A, LDA )
|
|
*
|
|
* Copy left singular vectors of A from A to U
|
|
*
|
|
CALL ZLACPY( 'F', M, N, A, LDA, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (CWorkspace: need 2*N, prefer N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (CWorkspace: need N+M, prefer N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R from A to VT, zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT
|
|
* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( N, N, VT, LDVT, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in VT
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, N, M, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, U, LDU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* M .LT. MNTHR
|
|
*
|
|
* Path 10 (M at least N, but not much larger)
|
|
* Reduce to bidiagonal form without QR decomposition
|
|
*
|
|
IE = 1
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize A
|
|
* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
IF( WNTUAS ) THEN
|
|
*
|
|
* If left singular vectors desired in U, copy result to U
|
|
* and generate left bidiagonalizing vectors in U
|
|
* (CWorkspace: need 2*N+NCU, prefer 2*N+NCU*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
IF( WNTUS )
|
|
$ NCU = N
|
|
IF( WNTUA )
|
|
$ NCU = M
|
|
CALL ZUNGBR( 'Q', M, NCU, N, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVAS ) THEN
|
|
*
|
|
* If right singular vectors desired in VT, copy result to
|
|
* VT and generate right bidiagonalizing vectors in VT
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTUO ) THEN
|
|
*
|
|
* If left singular vectors desired in A, generate left
|
|
* bidiagonalizing vectors in A
|
|
* (CWorkspace: need 3*N, prefer 2*N+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVO ) THEN
|
|
*
|
|
* If right singular vectors desired in A, generate right
|
|
* bidiagonalizing vectors in A
|
|
* (CWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IRWORK = IE + N
|
|
IF( WNTUAS .OR. WNTUO )
|
|
$ NRU = M
|
|
IF( WNTUN )
|
|
$ NRU = 0
|
|
IF( WNTVAS .OR. WNTVO )
|
|
$ NCVT = N
|
|
IF( WNTVN )
|
|
$ NCVT = 0
|
|
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, NCVT, NRU, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in A
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, NCVT, NRU, 0, S, RWORK( IE ), A,
|
|
$ LDA, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
ELSE
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in A and computing right singular
|
|
* vectors in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', N, NCVT, NRU, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, A, LDA, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
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( WNTVN ) THEN
|
|
*
|
|
* Path 1t(N much larger than M, JOBVT='N')
|
|
* No right singular vectors to be computed
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Zero out above L
|
|
*
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO, A( 1, 2 ),
|
|
$ LDA )
|
|
IE = 1
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
IF( WNTUO .OR. WNTUAS ) THEN
|
|
*
|
|
* If left singular vectors desired, generate Q
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IRWORK = IE + M
|
|
NRU = 0
|
|
IF( WNTUO .OR. WNTUAS )
|
|
$ NRU = M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left singular
|
|
* vectors of A in A if desired
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, 0, NRU, 0, S, RWORK( IE ), CDUM, 1,
|
|
$ A, LDA, CDUM, 1, RWORK( IRWORK ), INFO )
|
|
*
|
|
* If left singular vectors desired in U, copy them there
|
|
*
|
|
IF( WNTUAS )
|
|
$ CALL ZLACPY( 'F', M, M, A, LDA, U, LDU )
|
|
*
|
|
ELSE IF( WNTVO .AND. WNTUN ) THEN
|
|
*
|
|
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
|
|
* M right singular vectors to be overwritten on A and
|
|
* no left singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.M*M+3*M ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N )+LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N )+M*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
CHUNK = ( LWORK-M*M ) / M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IR) and zero out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IR ), LDWRKR )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ WORK( IR+LDWRKR ), LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IR)
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IR ), LDWRKR, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing L
|
|
* (CWorkspace: need M*M+3*M-1, prefer M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, 0, 0, S, RWORK( IE ),
|
|
$ WORK( IR ), LDWRKR, CDUM, 1, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
IU = ITAUQ
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by Q
|
|
* in A, storing result in WORK(IU) and copying to A
|
|
* (CWorkspace: need M*M+M, prefer M*M+M*N)
|
|
* (RWorkspace: 0)
|
|
*
|
|
DO 30 I = 1, N, CHUNK
|
|
BLK = MIN( N-I+1, CHUNK )
|
|
CALL ZGEMM( 'N', 'N', M, BLK, M, CONE, WORK( IR ),
|
|
$ LDWRKR, A( 1, I ), LDA, CZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL ZLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
|
|
$ A( 1, I ), LDA )
|
|
30 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
IE = 1
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize A
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing A
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in A
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'L', M, N, 0, 0, S, RWORK( IE ), A, LDA,
|
|
$ CDUM, 1, CDUM, 1, RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVO .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 3t(N much larger than M, JOBU='S' or 'A', JOBVT='O')
|
|
* M right singular vectors to be overwritten on A and
|
|
* M left singular vectors to be computed in U
|
|
*
|
|
IF( LWORK.GE.M*M+3*M ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N )+LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N )+M*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
CHUNK = ( LWORK-M*M ) / M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing about above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO, U( 1, 2 ),
|
|
$ LDU )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U, copying result to WORK(IR)
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, U, LDU, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, M, U, LDU, WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate right vectors bidiagonalizing L in WORK(IR)
|
|
* (CWorkspace: need M*M+3*M-1, prefer M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing L in U
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in U, and computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, M, 0, S, RWORK( IE ),
|
|
$ WORK( IR ), LDWRKR, U, LDU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
IU = ITAUQ
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by Q
|
|
* in A, storing result in WORK(IU) and copying to A
|
|
* (CWorkspace: need M*M+M, prefer M*M+M*N))
|
|
* (RWorkspace: 0)
|
|
*
|
|
DO 40 I = 1, N, CHUNK
|
|
BLK = MIN( N-I+1, CHUNK )
|
|
CALL ZGEMM( 'N', 'N', M, BLK, M, CONE, WORK( IR ),
|
|
$ LDWRKR, A( 1, I ), LDA, CZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL ZLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
|
|
$ A( 1, I ), LDA )
|
|
40 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO, U( 1, 2 ),
|
|
$ LDU )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, U, LDU, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right vectors bidiagonalizing L by Q in A
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'C', M, N, M, U, LDU,
|
|
$ WORK( ITAUP ), A, LDA, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing L in U
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in A
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, N, M, 0, S, RWORK( IE ), A, LDA,
|
|
$ U, LDU, CDUM, 1, RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVS ) THEN
|
|
*
|
|
IF( WNTUN ) THEN
|
|
*
|
|
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
|
|
* M right singular vectors to be computed in VT and
|
|
* no left singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.M*M+3*M ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IR) is LDA by M
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is M by M
|
|
*
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IR), zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ WORK( IR+LDWRKR ), LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IR)
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IR ), LDWRKR, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing L in
|
|
* WORK(IR)
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, 0, 0, S, RWORK( IE ),
|
|
$ WORK( IR ), LDWRKR, CDUM, 1, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by
|
|
* Q in A, storing result in VT
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IR ),
|
|
$ LDWRKR, A, LDA, CZERO, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy result to VT
|
|
*
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ A( 1, 2 ), LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right vectors bidiagonalizing L by Q in VT
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'C', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, N, 0, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, CDUM, 1, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUO ) THEN
|
|
*
|
|
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
|
|
* M right singular vectors to be computed in VT and
|
|
* M left singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*M*M+3*M ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out below it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (CWorkspace: need 2*M*M+3*M,
|
|
* prefer 2*M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need 2*M*M+3*M-1,
|
|
* prefer 2*M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IR)
|
|
* (CWorkspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in WORK(IR) and computing
|
|
* right singular vectors of L in WORK(IU)
|
|
* (CWorkspace: need 2*M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, M, 0, S, RWORK( IE ),
|
|
$ WORK( IU ), LDWRKU, WORK( IR ),
|
|
$ LDWRKR, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in A, storing result in VT
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IU ),
|
|
$ LDWRKU, A, LDA, CZERO, VT, LDVT )
|
|
*
|
|
* Copy left singular vectors of L to A
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ A( 1, 2 ), LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right vectors bidiagonalizing L by Q in VT
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'C', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors of L in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in A and computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, N, M, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, A, LDA, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUAS ) THEN
|
|
*
|
|
* Path 6t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='S')
|
|
* M right singular vectors to be computed in VT and
|
|
* M left singular vectors to be computed in U
|
|
*
|
|
IF( LWORK.GE.M*M+3*M ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is LDA by M
|
|
*
|
|
LDWRKU = M
|
|
END IF
|
|
ITAU = IU + LDWRKU*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to U
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
|
|
$ LDU )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need M*M+3*M-1,
|
|
* prefer M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in U and computing right
|
|
* singular vectors of L in WORK(IU)
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, M, 0, S, RWORK( IE ),
|
|
$ WORK( IU ), LDWRKU, U, LDU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in A, storing result in VT
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IU ),
|
|
$ LDWRKU, A, LDA, CZERO, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ U( 1, 2 ), LDU )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, U, LDU, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in U by Q
|
|
* in VT
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'C', M, N, M, U, LDU,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, N, M, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, U, LDU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVA ) THEN
|
|
*
|
|
IF( WNTUN ) THEN
|
|
*
|
|
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
|
|
* N right singular vectors to be computed in VT and
|
|
* no left singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( N+M, 3*M ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IR) is LDA by M
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is M by M
|
|
*
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Copy L to WORK(IR), zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ WORK( IR+LDWRKR ), LDWRKR )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need M*M+M+N, prefer M*M+M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IR)
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IR ), LDWRKR, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IR)
|
|
* (CWorkspace: need M*M+3*M-1,
|
|
* prefer M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, 0, 0, S, RWORK( IE ),
|
|
$ WORK( IR ), LDWRKR, CDUM, 1, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by
|
|
* Q in VT, storing result in A
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IR ),
|
|
$ LDWRKR, VT, LDVT, CZERO, A, LDA )
|
|
*
|
|
* Copy right singular vectors of A from A to VT
|
|
*
|
|
CALL ZLACPY( 'F', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need M+N, prefer M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ A( 1, 2 ), LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in A by Q
|
|
* in VT
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'C', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, N, 0, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, CDUM, 1, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUO ) THEN
|
|
*
|
|
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
|
|
* N right singular vectors to be computed in VT and
|
|
* M left singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*M*M+MAX( N+M, 3*M ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need 2*M*M+M+N, prefer 2*M*M+M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (CWorkspace: need 2*M*M+3*M,
|
|
* prefer 2*M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need 2*M*M+3*M-1,
|
|
* prefer 2*M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IR)
|
|
* (CWorkspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in WORK(IR) and computing
|
|
* right singular vectors of L in WORK(IU)
|
|
* (CWorkspace: need 2*M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, M, 0, S, RWORK( IE ),
|
|
$ WORK( IU ), LDWRKU, WORK( IR ),
|
|
$ LDWRKR, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in VT, storing result in A
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IU ),
|
|
$ LDWRKU, VT, LDVT, CZERO, A, LDA )
|
|
*
|
|
* Copy right singular vectors of A from A to VT
|
|
*
|
|
CALL ZLACPY( 'F', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Copy left singular vectors of A from WORK(IR) to A
|
|
*
|
|
CALL ZLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need M+N, prefer M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ A( 1, 2 ), LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in A by Q
|
|
* in VT
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'C', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in A and computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, N, M, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, A, LDA, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUAS ) THEN
|
|
*
|
|
* Path 9t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='A')
|
|
* N right singular vectors to be computed in VT and
|
|
* M left singular vectors to be computed in U
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( N+M, 3*M ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by M
|
|
*
|
|
LDWRKU = M
|
|
END IF
|
|
ITAU = IU + LDWRKU*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need M*M+M+N, prefer M*M+M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to U
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
|
|
$ LDU )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in U and computing right
|
|
* singular vectors of L in WORK(IU)
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, M, M, 0, S, RWORK( IE ),
|
|
$ WORK( IU ), LDWRKU, U, LDU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in VT, storing result in A
|
|
* (CWorkspace: need M*M)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IU ),
|
|
$ LDWRKU, VT, LDVT, CZERO, A, LDA )
|
|
*
|
|
* Copy right singular vectors of A from A to VT
|
|
*
|
|
CALL ZLACPY( 'F', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (CWorkspace: need 2*M, prefer M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (CWorkspace: need M+N, prefer M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing out above it
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO,
|
|
$ U( 1, 2 ), LDU )
|
|
IE = 1
|
|
ITAUQ = ITAU
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U
|
|
* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
|
|
* (RWorkspace: need M)
|
|
*
|
|
CALL ZGEBRD( M, M, U, LDU, S, RWORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in U by Q
|
|
* in VT
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNMBR( 'P', 'L', 'C', M, N, M, U, LDU,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IRWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'U', M, N, M, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, U, LDU, CDUM, 1,
|
|
$ RWORK( IRWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* N .LT. MNTHR
|
|
*
|
|
* Path 10t(N greater than M, but not much larger)
|
|
* Reduce to bidiagonal form without LQ decomposition
|
|
*
|
|
IE = 1
|
|
ITAUQ = 1
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize A
|
|
* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
|
|
* (RWorkspace: M)
|
|
*
|
|
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
IF( WNTUAS ) THEN
|
|
*
|
|
* If left singular vectors desired in U, copy result to U
|
|
* and generate left bidiagonalizing vectors in U
|
|
* (CWorkspace: need 3*M-1, prefer 2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVAS ) THEN
|
|
*
|
|
* If right singular vectors desired in VT, copy result to
|
|
* VT and generate right bidiagonalizing vectors in VT
|
|
* (CWorkspace: need 2*M+NRVT, prefer 2*M+NRVT*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
IF( WNTVA )
|
|
$ NRVT = N
|
|
IF( WNTVS )
|
|
$ NRVT = M
|
|
CALL ZUNGBR( 'P', NRVT, N, M, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTUO ) THEN
|
|
*
|
|
* If left singular vectors desired in A, generate left
|
|
* bidiagonalizing vectors in A
|
|
* (CWorkspace: need 3*M-1, prefer 2*M+(M-1)*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'Q', M, M, N, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVO ) THEN
|
|
*
|
|
* If right singular vectors desired in A, generate right
|
|
* bidiagonalizing vectors in A
|
|
* (CWorkspace: need 3*M, prefer 2*M+M*NB)
|
|
* (RWorkspace: 0)
|
|
*
|
|
CALL ZUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IRWORK = IE + M
|
|
IF( WNTUAS .OR. WNTUO )
|
|
$ NRU = M
|
|
IF( WNTUN )
|
|
$ NRU = 0
|
|
IF( WNTVAS .OR. WNTVO )
|
|
$ NCVT = N
|
|
IF( WNTVN )
|
|
$ NCVT = 0
|
|
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'L', M, NCVT, NRU, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in A
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'L', M, NCVT, NRU, 0, S, RWORK( IE ), A,
|
|
$ LDA, U, LDU, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
ELSE
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in A and computing right singular
|
|
* vectors in VT
|
|
* (CWorkspace: 0)
|
|
* (RWorkspace: need BDSPAC)
|
|
*
|
|
CALL ZBDSQR( 'L', M, NCVT, NRU, 0, S, RWORK( IE ), VT,
|
|
$ LDVT, A, LDA, CDUM, 1, RWORK( IRWORK ),
|
|
$ INFO )
|
|
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( INFO.NE.0 .AND. ANRM.GT.BIGNUM )
|
|
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN-1, 1,
|
|
$ RWORK( IE ), MINMN, IERR )
|
|
IF( ANRM.LT.SMLNUM )
|
|
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ IERR )
|
|
IF( INFO.NE.0 .AND. ANRM.LT.SMLNUM )
|
|
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN-1, 1,
|
|
$ RWORK( IE ), MINMN, IERR )
|
|
END IF
|
|
*
|
|
* Return optimal workspace in WORK(1)
|
|
*
|
|
WORK( 1 ) = MAXWRK
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZGESVD
|
|
*
|
|
END
|