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cgesdd.f
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*> \brief \b CGESDD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGESDD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgesdd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgesdd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgesdd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
* WORK, LWORK, RWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
* INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* REAL RWORK( * ), S( * )
* COMPLEX A( LDA, * ), U( LDU, * ), VT( LDVT, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGESDD computes the singular value decomposition (SVD) of a complex
*> M-by-N matrix A, optionally computing the left and/or right singular
*> vectors, by using divide-and-conquer method. 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 VT = V**H, not V.
*>
*> The divide and conquer algorithm makes very mild assumptions about
*> floating point arithmetic. It will work on machines with a guard
*> digit in add/subtract, or on those binary machines without guard
*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> Specifies options for computing all or part of the matrix U:
*> = 'A': all M columns of U and all N rows of V**H are
*> returned in the arrays U and VT;
*> = 'S': the first min(M,N) columns of U and the first
*> min(M,N) rows of V**H are returned in the arrays U
*> and VT;
*> = 'O': If M >= N, the first N columns of U are overwritten
*> in the array A and all rows of V**H are returned in
*> the array VT;
*> otherwise, all columns of U are returned in the
*> array U and the first M rows of V**H are overwritten
*> in the array A;
*> = 'N': no columns of U or rows of V**H are computed.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit,
*> if JOBZ = 'O', A is overwritten with the first N columns
*> of U (the left singular vectors, stored
*> columnwise) if M >= N;
*> A is overwritten with the first M rows
*> of V**H (the right singular vectors, stored
*> rowwise) otherwise.
*> if JOBZ .ne. 'O', the contents of A are destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is 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)
*> UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
*> UCOL = min(M,N) if JOBZ = 'S'.
*> If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
*> unitary matrix U;
*> if JOBZ = 'S', U contains the first min(M,N) columns of U
*> (the left singular vectors, stored columnwise);
*> if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= 1;
*> if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is COMPLEX array, dimension (LDVT,N)
*> If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
*> N-by-N unitary matrix V**H;
*> if JOBZ = 'S', VT contains the first min(M,N) rows of
*> V**H (the right singular vectors, stored rowwise);
*> if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= 1;
*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*> if JOBZ = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is 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 >= 1.
*> If LWORK = -1, a workspace query is assumed. The optimal
*> size for the WORK array is calculated and stored in WORK(1),
*> and no other work except argument checking is performed.
*>
*> Let mx = max(M,N) and mn = min(M,N).
*> If JOBZ = 'N', LWORK >= 2*mn + mx.
*> If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
*> If JOBZ = 'S', LWORK >= mn*mn + 3*mn.
*> If JOBZ = 'A', LWORK >= mn*mn + 2*mn + mx.
*> These are not tight minimums in all cases; see comments inside code.
*> For good performance, LWORK should generally be larger;
*> a query is recommended.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (MAX(1,LRWORK))
*> Let mx = max(M,N) and mn = min(M,N).
*> If JOBZ = 'N', LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
*> else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
*> else LRWORK >= max( 5*mn*mn + 5*mn,
*> 2*mx*mn + 2*mn*mn + mn ).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (8*min(M,N))
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: The updating process of SBDSDC did not converge.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup complexGEsing
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE CGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
$ WORK, LWORK, RWORK, IWORK, INFO )
implicit none
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL RWORK( * ), S( * )
COMPLEX A( LDA, * ), U( LDU, * ), VT( LDVT, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
INTEGER BLK, CHUNK, I, IE, IERR, IL, IR, IRU, IRVT,
$ ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR1, MNTHR2, NRWORK, NWORK, WRKBL
INTEGER LWORK_CGEBRD_MN, LWORK_CGEBRD_MM,
$ LWORK_CGEBRD_NN, LWORK_CGELQF_MN,
$ LWORK_CGEQRF_MN,
$ LWORK_CUNGBR_P_MN, LWORK_CUNGBR_P_NN,
$ LWORK_CUNGBR_Q_MN, LWORK_CUNGBR_Q_MM,
$ LWORK_CUNGLQ_MN, LWORK_CUNGLQ_NN,
$ LWORK_CUNGQR_MM, LWORK_CUNGQR_MN,
$ LWORK_CUNMBR_PRC_MM, LWORK_CUNMBR_QLN_MM,
$ LWORK_CUNMBR_PRC_MN, LWORK_CUNMBR_QLN_MN,
$ LWORK_CUNMBR_PRC_NN, LWORK_CUNMBR_QLN_NN
REAL ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
REAL DUM( 1 )
COMPLEX CDUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL CGEBRD, CGELQF, CGEMM, CGEQRF, CLACP2, CLACPY,
$ CLACRM, CLARCM, CLASCL, CLASET, CUNGBR, CUNGLQ,
$ CUNGQR, CUNMBR, SBDSDC, SLASCL, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, CLANGE
EXTERNAL LSAME, SLAMCH, CLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
MINMN = MIN( M, N )
MNTHR1 = INT( MINMN*17.0E0 / 9.0E0 )
MNTHR2 = INT( MINMN*5.0E0 / 3.0E0 )
WNTQA = LSAME( JOBZ, 'A' )
WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
WNTQO = LSAME( JOBZ, 'O' )
WNTQN = LSAME( JOBZ, 'N' )
LQUERY = ( LWORK.EQ.-1 )
MINWRK = 1
MAXWRK = 1
*
IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDU.LT.1 .OR. ( WNTQAS .AND. LDU.LT.M ) .OR.
$ ( WNTQO .AND. M.LT.N .AND. LDU.LT.M ) ) THEN
INFO = -8
ELSE IF( LDVT.LT.1 .OR. ( WNTQA .AND. LDVT.LT.N ) .OR.
$ ( WNTQS .AND. LDVT.LT.MINMN ) .OR.
$ ( WNTQO .AND. M.GE.N .AND. LDVT.LT.N ) ) THEN
INFO = -10
END IF
*
* Compute workspace
* Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace allocated at that point in the code,
* as well as the preferred amount for good performance.
* 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
*
* There is no complex work space needed for bidiagonal SVD
* The real work space needed for bidiagonal SVD (sbdsdc) is
* BDSPAC = 3*N*N + 4*N for singular values and vectors;
* BDSPAC = 4*N for singular values only;
* not including e, RU, and RVT matrices.
*
* Compute space preferred for each routine
CALL CGEBRD( M, N, CDUM(1), M, DUM(1), DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD_MN = INT( CDUM(1) )
*
CALL CGEBRD( N, N, CDUM(1), N, DUM(1), DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD_NN = INT( CDUM(1) )
*
CALL CGEQRF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEQRF_MN = INT( CDUM(1) )
*
CALL CUNGBR( 'P', N, N, N, CDUM(1), N, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGBR_P_NN = INT( CDUM(1) )
*
CALL CUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGBR_Q_MM = INT( CDUM(1) )
*
CALL CUNGBR( 'Q', M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGBR_Q_MN = INT( CDUM(1) )
*
CALL CUNGQR( M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGQR_MM = INT( CDUM(1) )
*
CALL CUNGQR( M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGQR_MN = INT( CDUM(1) )
*
CALL CUNMBR( 'P', 'R', 'C', N, N, N, CDUM(1), N, CDUM(1),
$ CDUM(1), N, CDUM(1), -1, IERR )
LWORK_CUNMBR_PRC_NN = INT( CDUM(1) )
*
CALL CUNMBR( 'Q', 'L', 'N', M, M, N, CDUM(1), M, CDUM(1),
$ CDUM(1), M, CDUM(1), -1, IERR )
LWORK_CUNMBR_QLN_MM = INT( CDUM(1) )
*
CALL CUNMBR( 'Q', 'L', 'N', M, N, N, CDUM(1), M, CDUM(1),
$ CDUM(1), M, CDUM(1), -1, IERR )
LWORK_CUNMBR_QLN_MN = INT( CDUM(1) )
*
CALL CUNMBR( 'Q', 'L', 'N', N, N, N, CDUM(1), N, CDUM(1),
$ CDUM(1), N, CDUM(1), -1, IERR )
LWORK_CUNMBR_QLN_NN = INT( CDUM(1) )
*
IF( M.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
* Path 1 (M >> N, JOBZ='N')
*
MAXWRK = N + LWORK_CGEQRF_MN
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CGEBRD_NN )
MINWRK = 3*N
ELSE IF( WNTQO ) THEN
*
* Path 2 (M >> N, JOBZ='O')
*
WRKBL = N + LWORK_CGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_CUNGQR_MN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CGEBRD_NN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_QLN_NN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_PRC_NN )
MAXWRK = M*N + N*N + WRKBL
MINWRK = 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
* Path 3 (M >> N, JOBZ='S')
*
WRKBL = N + LWORK_CGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_CUNGQR_MN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CGEBRD_NN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_QLN_NN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
MINWRK = N*N + 3*N
ELSE IF( WNTQA ) THEN
*
* Path 4 (M >> N, JOBZ='A')
*
WRKBL = N + LWORK_CGEQRF_MN
WRKBL = MAX( WRKBL, N + LWORK_CUNGQR_MM )
WRKBL = MAX( WRKBL, 2*N + LWORK_CGEBRD_NN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_QLN_NN )
WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
MINWRK = N*N + MAX( 3*N, N + M )
END IF
ELSE IF( M.GE.MNTHR2 ) THEN
*
* Path 5 (M >> N, but not as much as MNTHR1)
*
MAXWRK = 2*N + LWORK_CGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
* Path 5o (M >> N, JOBZ='O')
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_P_NN )
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_Q_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
* Path 5s (M >> N, JOBZ='S')
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_P_NN )
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_Q_MN )
ELSE IF( WNTQA ) THEN
* Path 5a (M >> N, JOBZ='A')
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_P_NN )
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_Q_MM )
END IF
ELSE
*
* Path 6 (M >= N, but not much larger)
*
MAXWRK = 2*N + LWORK_CGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
* Path 6o (M >= N, JOBZ='O')
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_PRC_NN )
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_QLN_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
* Path 6s (M >= N, JOBZ='S')
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_QLN_MN )
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_PRC_NN )
ELSE IF( WNTQA ) THEN
* Path 6a (M >= N, JOBZ='A')
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_QLN_MM )
MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_PRC_NN )
END IF
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
* There is no complex work space needed for bidiagonal SVD
* The real work space needed for bidiagonal SVD (sbdsdc) is
* BDSPAC = 3*M*M + 4*M for singular values and vectors;
* BDSPAC = 4*M for singular values only;
* not including e, RU, and RVT matrices.
*
* Compute space preferred for each routine
CALL CGEBRD( M, N, CDUM(1), M, DUM(1), DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD_MN = INT( CDUM(1) )
*
CALL CGEBRD( M, M, CDUM(1), M, DUM(1), DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_CGEBRD_MM = INT( CDUM(1) )
*
CALL CGELQF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
LWORK_CGELQF_MN = INT( CDUM(1) )
*
CALL CUNGBR( 'P', M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGBR_P_MN = INT( CDUM(1) )
*
CALL CUNGBR( 'P', N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGBR_P_NN = INT( CDUM(1) )
*
CALL CUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGBR_Q_MM = INT( CDUM(1) )
*
CALL CUNGLQ( M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGLQ_MN = INT( CDUM(1) )
*
CALL CUNGLQ( N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
$ -1, IERR )
LWORK_CUNGLQ_NN = INT( CDUM(1) )
*
CALL CUNMBR( 'P', 'R', 'C', M, M, M, CDUM(1), M, CDUM(1),
$ CDUM(1), M, CDUM(1), -1, IERR )
LWORK_CUNMBR_PRC_MM = INT( CDUM(1) )
*
CALL CUNMBR( 'P', 'R', 'C', M, N, M, CDUM(1), M, CDUM(1),
$ CDUM(1), M, CDUM(1), -1, IERR )
LWORK_CUNMBR_PRC_MN = INT( CDUM(1) )
*
CALL CUNMBR( 'P', 'R', 'C', N, N, M, CDUM(1), N, CDUM(1),
$ CDUM(1), N, CDUM(1), -1, IERR )
LWORK_CUNMBR_PRC_NN = INT( CDUM(1) )
*
CALL CUNMBR( 'Q', 'L', 'N', M, M, M, CDUM(1), M, CDUM(1),
$ CDUM(1), M, CDUM(1), -1, IERR )
LWORK_CUNMBR_QLN_MM = INT( CDUM(1) )
*
IF( N.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
* Path 1t (N >> M, JOBZ='N')
*
MAXWRK = M + LWORK_CGELQF_MN
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CGEBRD_MM )
MINWRK = 3*M
ELSE IF( WNTQO ) THEN
*
* Path 2t (N >> M, JOBZ='O')
*
WRKBL = M + LWORK_CGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_CUNGLQ_MN )
WRKBL = MAX( WRKBL, 2*M + LWORK_CGEBRD_MM )
WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_QLN_MM )
WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_PRC_MM )
MAXWRK = M*N + M*M + WRKBL
MINWRK = 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
* Path 3t (N >> M, JOBZ='S')
*
WRKBL = M + LWORK_CGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_CUNGLQ_MN )
WRKBL = MAX( WRKBL, 2*M + LWORK_CGEBRD_MM )
WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_QLN_MM )
WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
MINWRK = M*M + 3*M
ELSE IF( WNTQA ) THEN
*
* Path 4t (N >> M, JOBZ='A')
*
WRKBL = M + LWORK_CGELQF_MN
WRKBL = MAX( WRKBL, M + LWORK_CUNGLQ_NN )
WRKBL = MAX( WRKBL, 2*M + LWORK_CGEBRD_MM )
WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_QLN_MM )
WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
MINWRK = M*M + MAX( 3*M, M + N )
END IF
ELSE IF( N.GE.MNTHR2 ) THEN
*
* Path 5t (N >> M, but not as much as MNTHR1)
*
MAXWRK = 2*M + LWORK_CGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
* Path 5to (N >> M, JOBZ='O')
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_Q_MM )
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_P_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
* Path 5ts (N >> M, JOBZ='S')
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_Q_MM )
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_P_MN )
ELSE IF( WNTQA ) THEN
* Path 5ta (N >> M, JOBZ='A')
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_Q_MM )
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_P_NN )
END IF
ELSE
*
* Path 6t (N > M, but not much larger)
*
MAXWRK = 2*M + LWORK_CGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
* Path 6to (N > M, JOBZ='O')
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_QLN_MM )
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_PRC_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
* Path 6ts (N > M, JOBZ='S')
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_QLN_MM )
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_PRC_MN )
ELSE IF( WNTQA ) THEN
* Path 6ta (N > M, JOBZ='A')
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_QLN_MM )
MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_PRC_NN )
END IF
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = MAXWRK
IF( LWORK.LT.MINWRK .AND. .NOT. LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGESDD', -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.MNTHR1 ) THEN
*
IF( WNTQN ) THEN
*
* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
* CWorkspace: need N [tau] + N [work]
* CWorkspace: prefer N [tau] + N*NB [work]
* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Zero out below R
*
CALL CLASET( 'L', N-1, N-1, CZERO, CZERO, A( 2, 1 ),
$ LDA )
IE = 1
ITAUQ = 1
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in A
* CWorkspace: need 2*N [tauq, taup] + N [work]
* CWorkspace: prefer 2*N [tauq, taup] + 2*N*NB [work]
* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
NRWORK = IE + N
*
* Perform bidiagonal SVD, compute singular values only
* CWorkspace: need 0
* RWorkspace: need N [e] + BDSPAC
*
CALL SBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
* Path 2 (M >> N, JOBZ='O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
IU = 1
*
* WORK(IU) is N by N
*
LDWRKU = N
IR = IU + LDWRKU*N
IF( LWORK .GE. M*N + N*N + 3*N ) THEN
*
* WORK(IR) is M by N
*
LDWRKR = M
ELSE
LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
* CWorkspace: need N*N [U] + N*N [R] + N [tau] + N [work]
* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Copy R to WORK( IR ), zeroing 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 [U] + N*N [R] + N [tau] + N [work]
* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
* RWorkspace: need 0
*
CALL CUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
IE = 1
ITAUQ = ITAU
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of R in WORK(IRU) and computing right singular vectors
* of R in WORK(IRVT)
* CWorkspace: need 0
* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
NRWORK = IRVT + N*N
CALL SBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
$ RWORK( NRWORK ), IWORK, INFO )
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by the left singular vectors of R
* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
CALL CUNMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by the right singular vectors of R
* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
* CWorkspace: need N*N [U] + N*N [R]
* CWorkspace: prefer N*N [U] + M*N [R]
* RWorkspace: need 0
*
DO 10 I = 1, M, LDWRKR
CHUNK = MIN( M-I+1, LDWRKR )
CALL CGEMM( 'N', 'N', CHUNK, N, N, CONE, A( I, 1 ),
$ LDA, WORK( IU ), LDWRKU, CZERO,
$ WORK( IR ), LDWRKR )
CALL CLACPY( 'F', CHUNK, N, WORK( IR ), LDWRKR,
$ A( I, 1 ), LDA )
10 CONTINUE
*
ELSE IF( WNTQS ) THEN
*
* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IR = 1
*
* WORK(IR) is N by N
*
LDWRKR = N
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
* CWorkspace: need N*N [R] + N [tau] + N [work]
* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Copy R to WORK(IR), zeroing 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 [R] + N [tau] + N [work]
* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
* RWorkspace: need 0
*
CALL CUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
IE = 1
ITAUQ = ITAU
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
* CWorkspace: need 0
* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
NRWORK = IRVT + N*N
CALL SBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
$ RWORK( NRWORK ), IWORK, INFO )
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of R
* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
* CWorkspace: need N*N [R]
* RWorkspace: need 0
*
CALL CLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL CGEMM( 'N', 'N', M, N, N, CONE, A, LDA, WORK( IR ),
$ LDWRKR, CZERO, U, LDU )
*
ELSE IF( WNTQA ) THEN
*
* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IU = 1
*
* WORK(IU) is N by N
*
LDWRKU = N
ITAU = IU + LDWRKU*N
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* CWorkspace: need N*N [U] + N [tau] + N [work]
* CWorkspace: prefer N*N [U] + N [tau] + N*NB [work]
* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
CALL CLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* CWorkspace: need N*N [U] + N [tau] + M [work]
* CWorkspace: prefer N*N [U] + N [tau] + M*NB [work]
* RWorkspace: need 0
*
CALL CUNGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Produce R in A, zeroing out below it
*
CALL CLASET( 'L', N-1, N-1, CZERO, CZERO, A( 2, 1 ),
$ LDA )
IE = 1
ITAUQ = ITAU
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in A
* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + 2*N*NB [work]
* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IRU = IE + N
IRVT = IRU + N*N
NRWORK = IRVT + N*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
* CWorkspace: need 0
* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
$ RWORK( NRWORK ), IWORK, INFO )
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by left singular vectors of R
* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
CALL CUNMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
* CWorkspace: need N*N [U]
* RWorkspace: need 0
*