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cgesvd.f
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*> \brief <b> CGESVD 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 CGESVD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgesvd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgesvd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgesvd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGESVD( 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 ..
* REAL RWORK( * ), S( * )
* COMPLEX A( LDA, * ), U( LDU, * ), VT( LDVT, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGESVD 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 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 REAL 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 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 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 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 REAL 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 CBDSQR 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 complexGEsing
*
* =====================================================================
SUBROUTINE CGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
$ WORK, LWORK, RWORK, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* April 2012
*
* .. Scalar Arguments ..
CHARACTER JOBU, JOBVT
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
REAL RWORK( * ), S( * )
COMPLEX A( LDA, * ), U( LDU, * ), VT( LDVT, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
$ CONE = ( 1.0E0, 0.0E0 ) )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. 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_CGEQRF, LWORK_CUNGQR_N, LWORK_CUNGQR_M,
$ LWORK_CGEBRD, LWORK_CUNGBR_P, LWORK_CUNGBR_Q,
$ LWORK_CGELQF, LWORK_CUNGLQ_N, LWORK_CUNGLQ_M
REAL ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
REAL DUM( 1 )
COMPLEX CDUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL CBDSQR, CGEBRD, CGELQF, CGEMM, CGEQRF, CLACPY,
$ CLASCL, CLASET, CUNGBR, CUNGLQ, CUNGQR, CUNMBR,
$ SLASCL, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL CLANGE, SLAMCH
EXTERNAL LSAME, ILAENV, CLANGE, SLAMCH
* ..
* .. 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, 'CGESVD', JOBU // JOBVT, M, N, 0, 0 )
* Compute space needed for CGEQRF
CALL CGEQRF( M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEQRF = INT( CDUM(1) )
* Compute space needed for CUNGQR
CALL CUNGQR( M, N, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
LWORK_CUNGQR_N = INT( CDUM(1) )
CALL CUNGQR( M, M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
LWORK_CUNGQR_M = INT( CDUM(1) )
* Compute space needed for CGEBRD
CALL CGEBRD( N, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD = INT( CDUM(1) )
* Compute space needed for CUNGBR
CALL CUNGBR( 'P', N, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_P = INT( CDUM(1) )
CALL CUNGBR( 'Q', N, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_Q = INT( CDUM(1) )
*
MNTHR = ILAENV( 6, 'CGESVD', JOBU // JOBVT, M, N, 0, 0 )
IF( M.GE.MNTHR ) THEN
IF( WNTUN ) THEN
*
* Path 1 (M much larger than N, JOBU='N')
*
MAXWRK = N + LWORK_CGEQRF
MAXWRK = MAX( MAXWRK, 2*N+LWORK_CGEBRD )
IF( WNTVO .OR. WNTVAS )
$ MAXWRK = MAX( MAXWRK, 2*N+LWORK_CUNGBR_P )
MINWRK = 3*N
ELSE IF( WNTUO .AND. WNTVN ) THEN
*
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
*
WRKBL = N + LWORK_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_N )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_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_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_N )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_Q )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_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_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_N )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_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_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_N )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_Q )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_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_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_N )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_Q )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_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_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_M )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_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_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_M )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_Q )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_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_CGEQRF
WRKBL = MAX( WRKBL, N+LWORK_CUNGQR_M )
WRKBL = MAX( WRKBL, 2*N+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_Q )
WRKBL = MAX( WRKBL, 2*N+LWORK_CUNGBR_P )
MAXWRK = N*N + WRKBL
MINWRK = 2*N + M
END IF
ELSE
*
* Path 10 (M at least N, but not much larger)
*
CALL CGEBRD( M, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD = INT( CDUM(1) )
MAXWRK = 2*N + LWORK_CGEBRD
IF( WNTUS .OR. WNTUO ) THEN
CALL CUNGBR( 'Q', M, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_Q = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*N+LWORK_CUNGBR_Q )
END IF
IF( WNTUA ) THEN
CALL CUNGBR( 'Q', M, M, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_Q = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*N+LWORK_CUNGBR_Q )
END IF
IF( .NOT.WNTVN ) THEN
MAXWRK = MAX( MAXWRK, 2*N+LWORK_CUNGBR_P )
END IF
MINWRK = 2*N + M
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
* Space needed for CBDSQR is BDSPAC = 5*M
*
MNTHR = ILAENV( 6, 'CGESVD', JOBU // JOBVT, M, N, 0, 0 )
* Compute space needed for CGELQF
CALL CGELQF( M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
LWORK_CGELQF = INT( CDUM(1) )
* Compute space needed for CUNGLQ
CALL CUNGLQ( N, N, M, CDUM(1), N, CDUM(1), CDUM(1), -1,
$ IERR )
LWORK_CUNGLQ_N = INT( CDUM(1) )
CALL CUNGLQ( M, N, M, A, LDA, CDUM(1), CDUM(1), -1, IERR )
LWORK_CUNGLQ_M = INT( CDUM(1) )
* Compute space needed for CGEBRD
CALL CGEBRD( M, M, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD = INT( CDUM(1) )
* Compute space needed for CUNGBR P
CALL CUNGBR( 'P', M, M, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_P = INT( CDUM(1) )
* Compute space needed for CUNGBR Q
CALL CUNGBR( 'Q', M, M, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_Q = INT( CDUM(1) )
IF( N.GE.MNTHR ) THEN
IF( WNTVN ) THEN
*
* Path 1t(N much larger than M, JOBVT='N')
*
MAXWRK = M + LWORK_CGELQF
MAXWRK = MAX( MAXWRK, 2*M+LWORK_CGEBRD )
IF( WNTUO .OR. WNTUAS )
$ MAXWRK = MAX( MAXWRK, 2*M+LWORK_CUNGBR_Q )
MINWRK = 3*M
ELSE IF( WNTVO .AND. WNTUN ) THEN
*
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
*
WRKBL = M + LWORK_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_M )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_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_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_M )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_P )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_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_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_M )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_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_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_M )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_P )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_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_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_M )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_P )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_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_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_N )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_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_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_N )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_P )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_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_CGELQF
WRKBL = MAX( WRKBL, M+LWORK_CUNGLQ_N )
WRKBL = MAX( WRKBL, 2*M+LWORK_CGEBRD )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_P )
WRKBL = MAX( WRKBL, 2*M+LWORK_CUNGBR_Q )
MAXWRK = M*M + WRKBL
MINWRK = 2*M + N
END IF
ELSE
*
* Path 10t(N greater than M, but not much larger)
*
CALL CGEBRD( M, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD = INT( CDUM(1) )
MAXWRK = 2*M + LWORK_CGEBRD
IF( WNTVS .OR. WNTVO ) THEN
* Compute space needed for CUNGBR P
CALL CUNGBR( 'P', M, N, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_P = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*M+LWORK_CUNGBR_P )
END IF
IF( WNTVA ) THEN
CALL CUNGBR( 'P', N, N, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
LWORK_CUNGBR_P = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*M+LWORK_CUNGBR_P )
END IF
IF( .NOT.WNTUN ) THEN
MAXWRK = MAX( MAXWRK, 2*M+LWORK_CUNGBR_Q )
END IF
MINWRK = 2*M + N
END IF
END IF
MAXWRK = MAX( MINWRK, MAXWRK )
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGESVD', -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 = SLAMCH( 'P' )
SMLNUM = SQRT( SLAMCH( 'S' ) ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = CLANGE( 'M', M, N, A, LDA, DUM )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ISCL = 1
CALL CLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
ELSE IF( ANRM.GT.BIGNUM ) THEN
ISCL = 1
CALL CLASCL( '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 CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Zero out below R
*
IF( N .GT. 1 ) THEN
CALL CLASET( 'L', N-1, N-1, CZERO, CZERO, A( 2, 1 ),
$ LDA )
END IF
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 CGEBRD( 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 CUNGBR( '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 CBDSQR( '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 CLACPY( '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 CGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IR) and zero out below it
*
CALL CLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
CALL CLASET( '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 CUNGQR( 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 CGEBRD( 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 CUNGBR( '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 CBDSQR( '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 CGEMM( 'N', 'N', CHUNK, N, N, CONE, A( I, 1 ),
$ LDA, WORK( IR ), LDWRKR, CZERO,
$ WORK( IU ), LDWRKU )
CALL CLACPY( '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 CGEBRD( 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 CUNGBR( '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 CBDSQR( '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 CGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL CLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL CLASET( '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 CUNGQR( 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 CGEBRD( N, N, VT, LDVT, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL CLACPY( '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 CUNGBR( '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 CUNGBR( '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 CBDSQR( '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 CGEMM( 'N', 'N', CHUNK, N, N, CONE, A( I, 1 ),
$ LDA, WORK( IR ), LDWRKR, CZERO,
$ WORK( IU ), LDWRKU )
CALL CLACPY( '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 CGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL CLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL CLASET( '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 CUNGQR( 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)