*> \brief <b> ZGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgesvdq.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZGESVDQ( JOBA, JOBP, JOBR, JOBU, JOBV, M, N, A, LDA,
* S, U, LDU, V, LDV, NUMRANK, IWORK, LIWORK,
* CWORK, LCWORK, RWORK, LRWORK, INFO )
*
* .. Scalar Arguments ..
* IMPLICIT NONE
* CHARACTER JOBA, JOBP, JOBR, JOBU, JOBV
* INTEGER M, N, LDA, LDU, LDV, NUMRANK, LIWORK, LCWORK, LRWORK,
* INFO
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( * )
* DOUBLE PRECISION S( * ), RWORK( * )
* INTEGER IWORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
* ZCGESVDQ computes the singular value decomposition (SVD) of a complex
*> M-by-N matrix A, where M >= N. The SVD of A is written as
*> [++] [xx] [x0] [xx]
*> A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
*> [++] [xx]
*> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
*> matrix, and V is an N-by-N unitary matrix. The diagonal elements
*> of SIGMA are the singular values of A. The columns of U and V are the
*> left and the right singular vectors of A, respectively.
*> \endverbatim
*
* Arguments
* =========
*
*> \param[in] JOBA
*> \verbatim
*> JOBA is CHARACTER*1
*> Specifies the level of accuracy in the computed SVD
*> = 'A' The requested accuracy corresponds to having the backward
*> error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,
*> where EPS = DLAMCH('Epsilon'). This authorises ZGESVDQ to
*> truncate the computed triangular factor in a rank revealing
*> QR factorization whenever the truncated part is below the
*> threshold of the order of EPS * ||A||_F. This is aggressive
*> truncation level.
*> = 'M' Similarly as with 'A', but the truncation is more gentle: it
*> is allowed only when there is a drop on the diagonal of the
*> triangular factor in the QR factorization. This is medium
*> truncation level.
*> = 'H' High accuracy requested. No numerical rank determination based
*> on the rank revealing QR factorization is attempted.
*> = 'E' Same as 'H', and in addition the condition number of column
*> scaled A is estimated and returned in RWORK(1).
*> N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)
*> \endverbatim
*>
*> \param[in] JOBP
*> \verbatim
*> JOBP is CHARACTER*1
*> = 'P' The rows of A are ordered in decreasing order with respect to
*> ||A(i,:)||_\infty. This enhances numerical accuracy at the cost
*> of extra data movement. Recommended for numerical robustness.
*> = 'N' No row pivoting.
*> \endverbatim
*>
*> \param[in] JOBR
*> \verbatim
*> JOBR is CHARACTER*1
*> = 'T' After the initial pivoted QR factorization, ZGESVD is applied to
*> the adjoint R**H of the computed triangular factor R. This involves
*> some extra data movement (matrix transpositions). Useful for
*> experiments, research and development.
*> = 'N' The triangular factor R is given as input to CGESVD. This may be
*> preferred as it involves less data movement.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> = 'A' All M left singular vectors are computed and returned in the
*> matrix U. See the description of U.
*> = 'S' or 'U' N = min(M,N) left singular vectors are computed and returned
*> in the matrix U. See the description of U.
*> = 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular
*> vectors are computed and returned in the matrix U.
*> = 'F' The N left singular vectors are returned in factored form as the
*> product of the Q factor from the initial QR factorization and the
*> N left singular vectors of (R**H , 0)**H. If row pivoting is used,
*> then the necessary information on the row pivoting is stored in
*> IWORK(N+1:N+M-1).
*> = 'N' The left singular vectors are not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> = 'A', 'V' All N right singular vectors are computed and returned in
*> the matrix V.
*> = 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular
*> vectors are computed and returned in the matrix V. This option is
*> allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
*> = 'N' The right singular vectors are not 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. M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array of dimensions LDA x N
*> On entry, the input matrix A.
*> On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
*> the Householder vectors as stored by ZGEQP3. If JOBU = 'F', these Householder
*> vectors together with CWORK(1:N) can be used to restore the Q factors from
*> the initial pivoted QR factorization of A. See the description of U.
*> \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 of dimension N.
*> The singular values of A, ordered so that S(i) >= S(i+1).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is COMPLEX*16 array, dimension
*> LDU x M if JOBU = 'A'; see the description of LDU. In this case,
*> on exit, U contains the M left singular vectors.
*> LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this
*> case, U contains the leading N or the leading NUMRANK left singular vectors.
*> LDU x N if JOBU = 'F' ; see the description of LDU. In this case U
*> contains N x N unitary matrix that can be used to form the left
*> singular vectors.
*> If JOBU = 'N', U is not referenced.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER.
*> The leading dimension of the array U.
*> If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M).
*> If JOBU = 'F', LDU >= max(1,N).
*> Otherwise, LDU >= 1.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is COMPLEX*16 array, dimension
*> LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .
*> If JOBV = 'A', or 'V', V contains the N-by-N unitary matrix V**H;
*> If JOBV = 'R', V contains the first NUMRANK rows of V**H (the right
*> singular vectors, stored rowwise, of the NUMRANK largest singular values).
*> If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
*> If JOBV = 'N', and JOBA.NE.'E', V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N).
*> Otherwise, LDV >= 1.
*> \endverbatim
*>
*> \param[out] NUMRANK
*> \verbatim
*> NUMRANK is INTEGER
*> NUMRANK is the numerical rank first determined after the rank
*> revealing QR factorization, following the strategy specified by the
*> value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK
*> leading singular values and vectors are then requested in the call
*> of CGESVD. The final value of NUMRANK might be further reduced if
*> some singular values are computed as zeros.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (max(1, LIWORK)).
*> On exit, IWORK(1:N) contains column pivoting permutation of the
*> rank revealing QR factorization.
*> If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence
*> of row swaps used in row pivoting. These can be used to restore the
*> left singular vectors in the case JOBU = 'F'.
*
*> If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
*> IWORK(1) returns the minimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*> LIWORK is INTEGER
*> The dimension of the array IWORK.
*> LIWORK >= N + M - 1, if JOBP = 'P';
*> LIWORK >= N if JOBP = 'N'.
*>
*> If LIWORK = -1, then a workspace query is assumed; the routine
*> only calculates and returns the optimal and minimal sizes
*> for the CWORK, IWORK, and RWORK arrays, and no error
*> message related to LCWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] CWORK
*> \verbatim
*> CWORK is COMPLEX*12 array, dimension (max(2, LCWORK)), used as a workspace.
*> On exit, if, on entry, LCWORK.NE.-1, CWORK(1:N) contains parameters
*> needed to recover the Q factor from the QR factorization computed by
*> ZGEQP3.
*
*> If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
*> CWORK(1) returns the optimal LCWORK, and
*> CWORK(2) returns the minimal LCWORK.
*> \endverbatim
*>
*> \param[in,out] LCWORK
*> \verbatim
*> LCWORK is INTEGER
*> The dimension of the array CWORK. It is determined as follows:
*> Let LWQP3 = N+1, LWCON = 2*N, and let
*> LWUNQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U'
*> { MAX( M, 1 ), if JOBU = 'A'
*> LWSVD = MAX( 3*N, 1 )
*> LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
*> LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
*> Then the minimal value of LCWORK is:
*> = MAX( N + LWQP3, LWSVD ) if only the singular values are needed;
*> = MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
*> and a scaled condition estimate requested;
*>
*> = N + MAX( LWQP3, LWSVD, LWUNQ ) if the singular values and the left
*> singular vectors are requested;
*> = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the singular values and the left
*> singular vectors are requested, and also
*> a scaled condition estimate requested;
*>
*> = N + MAX( LWQP3, LWSVD ) if the singular values and the right
*> singular vectors are requested;
*> = N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
*> singular vectors are requested, and also
*> a scaled condition etimate requested;
*>
*> = N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
*> independent of JOBR;
*> = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
*> JOBV = 'R' and, also a scaled condition
*> estimate requested; independent of JOBR;
*> = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
*> N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
*> full SVD is requested with JOBV = 'A' or 'V', and
*> JOBR ='N'
*> = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
*> N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
*> if the full SVD is requested with JOBV = 'A' or 'V', and
*> JOBR ='N', and also a scaled condition number estimate
*> requested.
*> = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
*> N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
*> full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
*> = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
*> N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
*> if the full SVD is requested with JOBV = 'A', 'V' and
*> JOBR ='T', and also a scaled condition number estimate
*> requested.
*> Finally, LCWORK must be at least two: LCWORK = MAX( 2, LCWORK ).
*>
*> If LCWORK = -1, then a workspace query is assumed; the routine
*> only calculates and returns the optimal and minimal sizes
*> for the CWORK, IWORK, and RWORK arrays, and no error
*> message related to LCWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (max(1, LRWORK)).
*> On exit,
*> 1. If JOBA = 'E', RWORK(1) contains an estimate of the condition
*> number of column scaled A. If A = C * D where D is diagonal and C
*> has unit columns in the Euclidean norm, then, assuming full column rank,
*> N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).
*> Otherwise, RWORK(1) = -1.
*> 2. RWORK(2) contains the number of singular values computed as
*> exact zeros in ZGESVD applied to the upper triangular or trapeziodal
*> R (from the initial QR factorization). In case of early exit (no call to
*> ZGESVD, such as in the case of zero matrix) RWORK(2) = -1.
*
*> If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
*> RWORK(1) returns the minimal LRWORK.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*> LRWORK is INTEGER.
*> The dimension of the array RWORK.
*> If JOBP ='P', then LRWORK >= MAX(2, M, 5*N);
*> Otherwise, LRWORK >= MAX(2, 5*N).
*
*> If LRWORK = -1, then a workspace query is assumed; the routine
*> only calculates and returns the optimal and minimal sizes
*> for the CWORK, IWORK, and RWORK arrays, and no error
*> message related to LCWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if ZBDSQR did not converge, INFO specifies how many superdiagonals
*> of an intermediate bidiagonal form B (computed in ZGESVD) did not
*> converge to zero.
*> \endverbatim
*
*> \par Further Details:
* ========================
*>
*> \verbatim
*>
*> 1. The data movement (matrix transpose) is coded using simple nested
*> DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
*> Those DO-loops are easily identified in this source code - by the CONTINUE
*> statements labeled with 11**. In an optimized version of this code, the
*> nested DO loops should be replaced with calls to an optimized subroutine.
*> 2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
*> column norm overflow. This is the minial precaution and it is left to the
*> SVD routine (CGESVD) to do its own preemptive scaling if potential over-
*> or underflows are detected. To avoid repeated scanning of the array A,
*> an optimal implementation would do all necessary scaling before calling
*> CGESVD and the scaling in CGESVD can be switched off.
*> 3. Other comments related to code optimization are given in comments in the
*> code, enlosed in [[double brackets]].
*> \endverbatim
*
*> \par Bugs, examples and comments
* ===========================
*
*> \verbatim
*> Please report all bugs and send interesting examples and/or comments to
*> drmac@math.hr. Thank you.
*> \endverbatim
*
*> \par References
* ===============
*
*> \verbatim
*> [1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
*> Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
*> 44(1): 11:1-11:30 (2017)
*>
*> SIGMA library, xGESVDQ section updated February 2016.
*> Developed and coded by Zlatko Drmac, Department of Mathematics
*> University of Zagreb, Croatia, drmac@math.hr
*> \endverbatim
*
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*> Developed and coded by Zlatko Drmac, Department of Mathematics
*> University of Zagreb, Croatia, drmac@math.hr
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2018
*
*> \ingroup complex16GEsing
*
* =====================================================================
SUBROUTINE ZGESVDQ( JOBA, JOBP, JOBR, JOBU, JOBV, M, N, A, LDA,
$ S, U, LDU, V, LDV, NUMRANK, IWORK, LIWORK,
$ CWORK, LCWORK, RWORK, LRWORK, INFO )
* .. Scalar Arguments ..
IMPLICIT NONE
CHARACTER JOBA, JOBP, JOBR, JOBU, JOBV
INTEGER M, N, LDA, LDU, LDV, NUMRANK, LIWORK, LCWORK, LRWORK,
$ INFO
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( * )
DOUBLE PRECISION S( * ), RWORK( * )
INTEGER IWORK( * )
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = (0.0D0,0.0D0), CONE = (1.0D0,0.0D0) )
* ..
* .. Local Scalars ..
INTEGER IERR, NR, N1, OPTRATIO, p, q
INTEGER LWCON, LWQP3, LWRK_ZGELQF, LWRK_ZGESVD, LWRK_ZGESVD2,
$ LWRK_ZGEQP3, LWRK_ZGEQRF, LWRK_ZUNMLQ, LWRK_ZUNMQR,
$ LWRK_ZUNMQR2, LWLQF, LWQRF, LWSVD, LWSVD2, LWUNQ,
$ LWUNQ2, LWUNLQ, MINWRK, MINWRK2, OPTWRK, OPTWRK2,
$ IMINWRK, RMINWRK
LOGICAL ACCLA, ACCLM, ACCLH, ASCALED, CONDA, DNTWU, DNTWV,
$ LQUERY, LSVC0, LSVEC, ROWPRM, RSVEC, RTRANS, WNTUA,
$ WNTUF, WNTUR, WNTUS, WNTVA, WNTVR
DOUBLE PRECISION BIG, EPSLN, RTMP, SCONDA, SFMIN
COMPLEX*16 CTMP
* ..
* .. Local Arrays
COMPLEX*16 CDUMMY(1)
DOUBLE PRECISION RDUMMY(1)
* ..
* .. External Subroutines (BLAS, LAPACK)
EXTERNAL ZGELQF, ZGEQP3, ZGEQRF, ZGESVD, ZLACPY, ZLAPMT,
$ ZLASCL, ZLASET, ZLASWP, ZDSCAL, DLASET, DLASCL,
$ ZPOCON, ZUNMLQ, ZUNMQR, XERBLA
* ..
* .. External Functions (BLAS, LAPACK)
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION ZLANGE, DZNRM2, DLAMCH
EXTERNAL LSAME, ZLANGE, IDAMAX, DZNRM2, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, MIN, DBLE, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
WNTUS = LSAME( JOBU, 'S' ) .OR. LSAME( JOBU, 'U' )
WNTUR = LSAME( JOBU, 'R' )
WNTUA = LSAME( JOBU, 'A' )
WNTUF = LSAME( JOBU, 'F' )
LSVC0 = WNTUS .OR. WNTUR .OR. WNTUA
LSVEC = LSVC0 .OR. WNTUF
DNTWU = LSAME( JOBU, 'N' )
*
WNTVR = LSAME( JOBV, 'R' )
WNTVA = LSAME( JOBV, 'A' ) .OR. LSAME( JOBV, 'V' )
RSVEC = WNTVR .OR. WNTVA
DNTWV = LSAME( JOBV, 'N' )
*
ACCLA = LSAME( JOBA, 'A' )
ACCLM = LSAME( JOBA, 'M' )
CONDA = LSAME( JOBA, 'E' )
ACCLH = LSAME( JOBA, 'H' ) .OR. CONDA
*
ROWPRM = LSAME( JOBP, 'P' )
RTRANS = LSAME( JOBR, 'T' )
*
IF ( ROWPRM ) THEN
IMINWRK = MAX( 1, N + M - 1 )
RMINWRK = MAX( 2, M, 5*N )
ELSE
IMINWRK = MAX( 1, N )
RMINWRK = MAX( 2, 5*N )
END IF
LQUERY = (LIWORK .EQ. -1 .OR. LCWORK .EQ. -1 .OR. LRWORK .EQ. -1)
INFO = 0
IF ( .NOT. ( ACCLA .OR. ACCLM .OR. ACCLH ) ) THEN
INFO = -1
ELSE IF ( .NOT.( ROWPRM .OR. LSAME( JOBP, 'N' ) ) ) THEN
INFO = -2
ELSE IF ( .NOT.( RTRANS .OR. LSAME( JOBR, 'N' ) ) ) THEN
INFO = -3
ELSE IF ( .NOT.( LSVEC .OR. DNTWU ) ) THEN
INFO = -4
ELSE IF ( WNTUR .AND. WNTVA ) THEN
INFO = -5
ELSE IF ( .NOT.( RSVEC .OR. DNTWV )) THEN
INFO = -5
ELSE IF ( M.LT.0 ) THEN
INFO = -6
ELSE IF ( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
INFO = -7
ELSE IF ( LDA.LT.MAX( 1, M ) ) THEN
INFO = -9
ELSE IF ( LDU.LT.1 .OR. ( LSVC0 .AND. LDU.LT.M ) .OR.
$ ( WNTUF .AND. LDU.LT.N ) ) THEN
INFO = -12
ELSE IF ( LDV.LT.1 .OR. ( RSVEC .AND. LDV.LT.N ) .OR.
$ ( CONDA .AND. LDV.LT.N ) ) THEN
INFO = -14
ELSE IF ( LIWORK .LT. IMINWRK .AND. .NOT. LQUERY ) THEN
INFO = -17
END IF
*
*
IF ( INFO .EQ. 0 ) THEN
* .. compute the minimal and the optimal workspace lengths
* [[The expressions for computing the minimal and the optimal
* values of LCWORK are written with a lot of redundancy and
* can be simplified. However, this detailed form is easier for
* maintenance and modifications of the code.]]
*
* .. minimal workspace length for ZGEQP3 of an M x N matrix
LWQP3 = N+1
* .. minimal workspace length for ZUNMQR to build left singular vectors
IF ( WNTUS .OR. WNTUR ) THEN
LWUNQ = MAX( N , 1 )
ELSE IF ( WNTUA ) THEN
LWUNQ = MAX( M , 1 )
END IF
* .. minimal workspace length for ZPOCON of an N x N matrix
LWCON = 2 * N
* .. ZGESVD of an N x N matrix
LWSVD = MAX( 3 * N, 1 )
IF ( LQUERY ) THEN
CALL ZGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1,
$ RDUMMY, IERR )
LWRK_ZGEQP3 = INT( CDUMMY(1) )
IF ( WNTUS .OR. WNTUR ) THEN
CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
$ LDU, CDUMMY, -1, IERR )
LWRK_ZUNMQR = INT( CDUMMY(1) )
ELSE IF ( WNTUA ) THEN
CALL ZUNMQR( 'L', 'N', M, M, N, A, LDA, CDUMMY, U,
$ LDU, CDUMMY, -1, IERR )
LWRK_ZUNMQR = INT( CDUMMY(1) )
ELSE
LWRK_ZUNMQR = 0
END IF
END IF
MINWRK = 2
OPTWRK = 2
IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN
* .. minimal and optimal sizes of the complex workspace if
* only the singular values are requested
IF ( CONDA ) THEN
MINWRK = MAX( N+LWQP3, LWCON, LWSVD )
ELSE
MINWRK = MAX( N+LWQP3, LWSVD )
END IF
IF ( LQUERY ) THEN
CALL ZGESVD( 'N', 'N', N, N, A, LDA, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
LWRK_ZGESVD = INT( CDUMMY(1) )
IF ( CONDA ) THEN
OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON, LWRK_ZGESVD )
ELSE
OPTWRK = MAX( N+LWRK_ZGEQP3, LWRK_ZGESVD )
END IF
END IF
ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN
* .. minimal and optimal sizes of the complex workspace if the
* singular values and the left singular vectors are requested
IF ( CONDA ) THEN
MINWRK = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ )
ELSE
MINWRK = N + MAX( LWQP3, LWSVD, LWUNQ )
END IF
IF ( LQUERY ) THEN
IF ( RTRANS ) THEN
CALL ZGESVD( 'N', 'O', N, N, A, LDA, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
ELSE
CALL ZGESVD( 'O', 'N', N, N, A, LDA, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
END IF
LWRK_ZGESVD = INT( CDUMMY(1) )
IF ( CONDA ) THEN
OPTWRK = N + MAX( LWRK_ZGEQP3, LWCON, LWRK_ZGESVD,
$ LWRK_ZUNMQR )
ELSE
OPTWRK = N + MAX( LWRK_ZGEQP3, LWRK_ZGESVD,
$ LWRK_ZUNMQR )
END IF
END IF
ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN
* .. minimal and optimal sizes of the complex workspace if the
* singular values and the right singular vectors are requested
IF ( CONDA ) THEN
MINWRK = N + MAX( LWQP3, LWCON, LWSVD )
ELSE
MINWRK = N + MAX( LWQP3, LWSVD )
END IF
IF ( LQUERY ) THEN
IF ( RTRANS ) THEN
CALL ZGESVD( 'O', 'N', N, N, A, LDA, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
ELSE
CALL ZGESVD( 'N', 'O', N, N, A, LDA, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
END IF
LWRK_ZGESVD = INT( CDUMMY(1) )
IF ( CONDA ) THEN
OPTWRK = N + MAX( LWRK_ZGEQP3, LWCON, LWRK_ZGESVD )
ELSE
OPTWRK = N + MAX( LWRK_ZGEQP3, LWRK_ZGESVD )
END IF
END IF
ELSE
* .. minimal and optimal sizes of the complex workspace if the
* full SVD is requested
IF ( RTRANS ) THEN
MINWRK = MAX( LWQP3, LWSVD, LWUNQ )
IF ( CONDA ) MINWRK = MAX( MINWRK, LWCON )
MINWRK = MINWRK + N
IF ( WNTVA ) THEN
* .. minimal workspace length for N x N/2 ZGEQRF
LWQRF = MAX( N/2, 1 )
* .. minimal workspace lengt for N/2 x N/2 ZGESVD
LWSVD2 = MAX( 3 * (N/2), 1 )
LWUNQ2 = MAX( N, 1 )
MINWRK2 = MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2,
$ N/2+LWUNQ2, LWUNQ )
IF ( CONDA ) MINWRK2 = MAX( MINWRK2, LWCON )
MINWRK2 = N + MINWRK2
MINWRK = MAX( MINWRK, MINWRK2 )
END IF
ELSE
MINWRK = MAX( LWQP3, LWSVD, LWUNQ )
IF ( CONDA ) MINWRK = MAX( MINWRK, LWCON )
MINWRK = MINWRK + N
IF ( WNTVA ) THEN
* .. minimal workspace length for N/2 x N ZGELQF
LWLQF = MAX( N/2, 1 )
LWSVD2 = MAX( 3 * (N/2), 1 )
LWUNLQ = MAX( N , 1 )
MINWRK2 = MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2,
$ N/2+LWUNLQ, LWUNQ )
IF ( CONDA ) MINWRK2 = MAX( MINWRK2, LWCON )
MINWRK2 = N + MINWRK2
MINWRK = MAX( MINWRK, MINWRK2 )
END IF
END IF
IF ( LQUERY ) THEN
IF ( RTRANS ) THEN
CALL ZGESVD( 'O', 'A', N, N, A, LDA, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
LWRK_ZGESVD = INT( CDUMMY(1) )
OPTWRK = MAX(LWRK_ZGEQP3,LWRK_ZGESVD,LWRK_ZUNMQR)
IF ( CONDA ) OPTWRK = MAX( OPTWRK, LWCON )
OPTWRK = N + OPTWRK
IF ( WNTVA ) THEN
CALL ZGEQRF(N,N/2,U,LDU,CDUMMY,CDUMMY,-1,IERR)
LWRK_ZGEQRF = INT( CDUMMY(1) )
CALL ZGESVD( 'S', 'O', N/2,N/2, V,LDV, S, U,LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
LWRK_ZGESVD2 = INT( CDUMMY(1) )
CALL ZUNMQR( 'R', 'C', N, N, N/2, U, LDU, CDUMMY,
$ V, LDV, CDUMMY, -1, IERR )
LWRK_ZUNMQR2 = INT( CDUMMY(1) )
OPTWRK2 = MAX( LWRK_ZGEQP3, N/2+LWRK_ZGEQRF,
$ N/2+LWRK_ZGESVD2, N/2+LWRK_ZUNMQR2 )
IF ( CONDA ) OPTWRK2 = MAX( OPTWRK2, LWCON )
OPTWRK2 = N + OPTWRK2
OPTWRK = MAX( OPTWRK, OPTWRK2 )
END IF
ELSE
CALL ZGESVD( 'S', 'O', N, N, A, LDA, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
LWRK_ZGESVD = INT( CDUMMY(1) )
OPTWRK = MAX(LWRK_ZGEQP3,LWRK_ZGESVD,LWRK_ZUNMQR)
IF ( CONDA ) OPTWRK = MAX( OPTWRK, LWCON )
OPTWRK = N + OPTWRK
IF ( WNTVA ) THEN
CALL ZGELQF(N/2,N,U,LDU,CDUMMY,CDUMMY,-1,IERR)
LWRK_ZGELQF = INT( CDUMMY(1) )
CALL ZGESVD( 'S','O', N/2,N/2, V, LDV, S, U, LDU,
$ V, LDV, CDUMMY, -1, RDUMMY, IERR )
LWRK_ZGESVD2 = INT( CDUMMY(1) )
CALL ZUNMLQ( 'R', 'N', N, N, N/2, U, LDU, CDUMMY,
$ V, LDV, CDUMMY,-1,IERR )
LWRK_ZUNMLQ = INT( CDUMMY(1) )
OPTWRK2 = MAX( LWRK_ZGEQP3, N/2+LWRK_ZGELQF,
$ N/2+LWRK_ZGESVD2, N/2+LWRK_ZUNMLQ )
IF ( CONDA ) OPTWRK2 = MAX( OPTWRK2, LWCON )
OPTWRK2 = N + OPTWRK2
OPTWRK = MAX( OPTWRK, OPTWRK2 )
END IF
END IF
END IF
END IF
*
MINWRK = MAX( 2, MINWRK )
OPTWRK = MAX( 2, OPTWRK )
IF ( LCWORK .LT. MINWRK .AND. (.NOT.LQUERY) ) INFO = -19
*
END IF
*
IF (INFO .EQ. 0 .AND. LRWORK .LT. RMINWRK .AND. .NOT. LQUERY) THEN
INFO = -21
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGESVDQ', -INFO )
RETURN
ELSE IF ( LQUERY ) THEN
*
* Return optimal workspace
*
IWORK(1) = IMINWRK
CWORK(1) = OPTWRK
CWORK(2) = MINWRK
RWORK(1) = RMINWRK
RETURN
END IF
*
* Quick return if the matrix is void.
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) ) THEN
* .. all output is void.
RETURN
END IF
*
BIG = DLAMCH('O')
ASCALED = .FALSE.
IF ( ROWPRM ) THEN
* .. reordering the rows in decreasing sequence in the
* ell-infinity norm - this enhances numerical robustness in
* the case of differently scaled rows.
DO 1904 p = 1, M
* RWORK(p) = ABS( A(p,IZAMAX(N,A(p,1),LDA)) )
* [[ZLANGE will return NaN if an entry of the p-th row is Nan]]
RWORK(p) = ZLANGE( 'M', 1, N, A(p,1), LDA, RDUMMY )
* .. check for NaN's and Inf's
IF ( ( RWORK(p) .NE. RWORK(p) ) .OR.
$ ( (RWORK(p)*ZERO) .NE. ZERO ) ) THEN
INFO = -8
CALL XERBLA( 'ZGESVDQ', -INFO )
RETURN
END IF
1904 CONTINUE
DO 1952 p = 1, M - 1
q = IDAMAX( M-p+1, RWORK(p), 1 ) + p - 1
IWORK(N+p) = q
IF ( p .NE. q ) THEN
RTMP = RWORK(p)
RWORK(p) = RWORK(q)
RWORK(q) = RTMP
END IF
1952 CONTINUE
*
IF ( RWORK(1) .EQ. ZERO ) THEN
* Quick return: A is the M x N zero matrix.
NUMRANK = 0
CALL DLASET( 'G', N, 1, ZERO, ZERO, S, N )
IF ( WNTUS ) CALL ZLASET('G', M, N, CZERO, CONE, U, LDU)
IF ( WNTUA ) CALL ZLASET('G', M, M, CZERO, CONE, U, LDU)
IF ( WNTVA ) CALL ZLASET('G', N, N, CZERO, CONE, V, LDV)
IF ( WNTUF ) THEN
CALL ZLASET( 'G', N, 1, CZERO, CZERO, CWORK, N )
CALL ZLASET( 'G', M, N, CZERO, CONE, U, LDU )
END IF
DO 5001 p = 1, N
IWORK(p) = p
5001 CONTINUE
IF ( ROWPRM ) THEN
DO 5002 p = N + 1, N + M - 1
IWORK(p) = p - N
5002 CONTINUE
END IF
IF ( CONDA ) RWORK(1) = -1
RWORK(2) = -1
RETURN
END IF
*
IF ( RWORK(1) .GT. BIG / SQRT(DBLE(M)) ) THEN
* .. to prevent overflow in the QR factorization, scale the
* matrix by 1/sqrt(M) if too large entry detected
CALL ZLASCL('G',0,0,SQRT(DBLE(M)),ONE, M,N, A,LDA, IERR)
ASCALED = .TRUE.
END IF
CALL ZLASWP( N, A, LDA, 1, M-1, IWORK(N+1), 1 )
END IF
*
* .. At this stage, preemptive scaling is done only to avoid column
* norms overflows during the QR factorization. The SVD procedure should
* have its own scaling to save the singular values from overflows and
* underflows. That depends on the SVD procedure.
*
IF ( .NOT.ROWPRM ) THEN
RTMP = ZLANGE( 'M', M, N, A, LDA, RWORK )
IF ( ( RTMP .NE. RTMP ) .OR.
$ ( (RTMP*ZERO) .NE. ZERO ) ) THEN
INFO = -8
CALL XERBLA( 'ZGESVDQ', -INFO )
RETURN
END IF
IF ( RTMP .GT. BIG / SQRT(DBLE(M)) ) THEN
* .. to prevent overflow in the QR factorization, scale the
* matrix by 1/sqrt(M) if too large entry detected
CALL ZLASCL('G',0,0, SQRT(DBLE(M)),ONE, M,N, A,LDA, IERR)
ASCALED = .TRUE.
END IF
END IF
*
* .. QR factorization with column pivoting
*
* A * P = Q * [ R ]
* [ 0 ]
*
DO 1963 p = 1, N
* .. all columns are free columns
IWORK(p) = 0
1963 CONTINUE
CALL ZGEQP3( M, N, A, LDA, IWORK, CWORK, CWORK(N+1), LCWORK-N,
$ RWORK, IERR )
*
* If the user requested accuracy level allows truncation in the
* computed upper triangular factor, the matrix R is examined and,
* if possible, replaced with its leading upper trapezoidal part.
*
EPSLN = DLAMCH('E')
SFMIN = DLAMCH('S')
* SMALL = SFMIN / EPSLN
NR = N
*
IF ( ACCLA ) THEN
*
* Standard absolute error bound suffices. All sigma_i with
* sigma_i < N*EPS*||A||_F are flushed to zero. This is an
* aggressive enforcement of lower numerical rank by introducing a
* backward error of the order of N*EPS*||A||_F.
NR = 1
RTMP = SQRT(DBLE(N))*EPSLN
DO 3001 p = 2, N
IF ( ABS(A(p,p)) .LT. (RTMP*ABS(A(1,1))) ) GO TO 3002
NR = NR + 1
3001 CONTINUE
3002 CONTINUE
*
ELSEIF ( ACCLM ) THEN
* .. similarly as above, only slightly more gentle (less aggressive).
* Sudden drop on the diagonal of R is used as the criterion for being
* close-to-rank-deficient. The threshold is set to EPSLN=DLAMCH('E').
* [[This can be made more flexible by replacing this hard-coded value
* with a user specified threshold.]] Also, the values that underflow
* will be truncated.
NR = 1
DO 3401 p = 2, N
IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
$ ( ABS(A(p,p)) .LT. SFMIN ) ) GO TO 3402
NR = NR + 1
3401 CONTINUE
3402 CONTINUE
*
ELSE
* .. RRQR not authorized to determine numerical rank except in the
* obvious case of zero pivots.
* .. inspect R for exact zeros on the diagonal;
* R(i,i)=0 => R(i:N,i:N)=0.
NR = 1
DO 3501 p = 2, N
IF ( ABS(A(p,p)) .EQ. ZERO ) GO TO 3502
NR = NR + 1
3501 CONTINUE
3502 CONTINUE
*
IF ( CONDA ) THEN
* Estimate the scaled condition number of A. Use the fact that it is
* the same as the scaled condition number of R.
* .. V is used as workspace
CALL ZLACPY( 'U', N, N, A, LDA, V, LDV )
* Only the leading NR x NR submatrix of the triangular factor
* is considered. Only if NR=N will this give a reliable error
* bound. However, even for NR < N, this can be used on an
* expert level and obtain useful information in the sense of
* perturbation theory.
DO 3053 p = 1, NR
RTMP = DZNRM2( p, V(1,p), 1 )
CALL ZDSCAL( p, ONE/RTMP, V(1,p), 1 )
3053 CONTINUE
IF ( .NOT. ( LSVEC .OR. RSVEC ) ) THEN
CALL ZPOCON( 'U', NR, V, LDV, ONE, RTMP,
$ CWORK, RWORK, IERR )
ELSE
CALL ZPOCON( 'U', NR, V, LDV, ONE, RTMP,
$ CWORK(N+1), RWORK, IERR )
END IF
SCONDA = ONE / SQRT(RTMP)
* For NR=N, SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1),
* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
* See the reference [1] for more details.
END IF
*
ENDIF
*
IF ( WNTUR ) THEN
N1 = NR
ELSE IF ( WNTUS .OR. WNTUF) THEN
N1 = N
ELSE IF ( WNTUA ) THEN
N1 = M
END IF
*
IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
*.......................................................................
* .. only the singular values are requested
*.......................................................................
IF ( RTRANS ) THEN
*
* .. compute the singular values of R**H = [A](1:NR,1:N)**H
* .. set the lower triangle of [A] to [A](1:NR,1:N)**H and
* the upper triangle of [A] to zero.
DO 1146 p = 1, MIN( N, NR )
A(p,p) = CONJG(A(p,p))
DO 1147 q = p + 1, N
A(q,p) = CONJG(A(p,q))
IF ( q .LE. NR ) A(p,q) = CZERO
1147 CONTINUE
1146 CONTINUE
*
CALL ZGESVD( 'N', 'N', N, NR, A, LDA, S, U, LDU,
$ V, LDV, CWORK, LCWORK, RWORK, INFO )
*
ELSE
*
* .. compute the singular values of R = [A](1:NR,1:N)
*
IF ( NR .GT. 1 )
$ CALL ZLASET( 'L', NR-1,NR-1, CZERO,CZERO, A(2,1), LDA )
CALL ZGESVD( 'N', 'N', NR, N, A, LDA, S, U, LDU,
$ V, LDV, CWORK, LCWORK, RWORK, INFO )
*
END IF
*
ELSE IF ( LSVEC .AND. ( .NOT. RSVEC) ) THEN
*.......................................................................
* .. the singular values and the left singular vectors requested
*.......................................................................""""""""
IF ( RTRANS ) THEN
* .. apply ZGESVD to R**H
* .. copy R**H into [U] and overwrite [U] with the right singular
* vectors of R
DO 1192 p = 1, NR
DO 1193 q = p, N
U(q,p) = CONJG(A(p,q))
1193 CONTINUE
1192 CONTINUE
IF ( NR .GT. 1 )
$ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, U(1,2), LDU )
* .. the left singular vectors not computed, the NR right singular
* vectors overwrite [U](1:NR,1:NR) as conjugate transposed. These
* will be pre-multiplied by Q to build the left singular vectors of A.
CALL ZGESVD( 'N', 'O', N, NR, U, LDU, S, U, LDU,
$ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
*
DO 1119 p = 1, NR
U(p,p) = CONJG(U(p,p))
DO 1120 q = p + 1, NR
CTMP = CONJG(U(q,p))
U(q,p) = CONJG(U(p,q))
U(p,q) = CTMP
1120 CONTINUE
1119 CONTINUE
*
ELSE
* .. apply ZGESVD to R
* .. copy R into [U] and overwrite [U] with the left singular vectors
CALL ZLACPY( 'U', NR, N, A, LDA, U, LDU )
IF ( NR .GT. 1 )
$ CALL ZLASET( 'L', NR-1, NR-1, CZERO, CZERO, U(2,1), LDU )
* .. the right singular vectors not computed, the NR left singular
* vectors overwrite [U](1:NR,1:NR)
CALL ZGESVD( 'O', 'N', NR, N, U, LDU, S, U, LDU,
$ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
* .. now [U](1:NR,1:NR) contains the NR left singular vectors of
* R. These will be pre-multiplied by Q to build the left singular
* vectors of A.
END IF
*
* .. assemble the left singular vector matrix U of dimensions
* (M x NR) or (M x N) or (M x M).
IF ( ( NR .LT. M ) .AND. ( .NOT.WNTUF ) ) THEN
CALL ZLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
IF ( NR .LT. N1 ) THEN
CALL ZLASET( 'A',NR,N1-NR,CZERO,CZERO,U(1,NR+1), LDU )
CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
$ U(NR+1,NR+1), LDU )
END IF
END IF
*
* The Q matrix from the first QRF is built into the left singular
* vectors matrix U.
*
IF ( .NOT.WNTUF )
$ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
$ LDU, CWORK(N+1), LCWORK-N, IERR )
IF ( ROWPRM .AND. .NOT.WNTUF )
$ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(N+1), -1 )
*
ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN
*.......................................................................
* .. the singular values and the right singular vectors requested
*.......................................................................
IF ( RTRANS ) THEN
* .. apply ZGESVD to R**H
* .. copy R**H into V and overwrite V with the left singular vectors
DO 1165 p = 1, NR
DO 1166 q = p, N
V(q,p) = CONJG(A(p,q))
1166 CONTINUE
1165 CONTINUE
IF ( NR .GT. 1 )
$ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
* .. the left singular vectors of R**H overwrite V, the right singular
* vectors not computed
IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
CALL ZGESVD( 'O', 'N', N, NR, V, LDV, S, U, LDU,
$ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
*
DO 1121 p = 1, NR
V(p,p) = CONJG(V(p,p))
DO 1122 q = p + 1, NR
CTMP = CONJG(V(q,p))
V(q,p) = CONJG(V(p,q))
V(p,q) = CTMP
1122 CONTINUE
1121 CONTINUE
*
IF ( NR .LT. N ) THEN
DO 1103 p = 1, NR
DO 1104 q = NR + 1, N
V(p,q) = CONJG(V(q,p))
1104 CONTINUE
1103 CONTINUE
END IF
CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
ELSE
* .. need all N right singular vectors and NR < N
* [!] This is simple implementation that augments [V](1:N,1:NR)
* by padding a zero block. In the case NR << N, a more efficient
* way is to first use the QR factorization. For more details
* how to implement this, see the " FULL SVD " branch.
CALL ZLASET('G', N, N-NR, CZERO, CZERO, V(1,NR+1), LDV)
CALL ZGESVD( 'O', 'N', N, N, V, LDV, S, U, LDU,
$ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
*
DO 1123 p = 1, N
V(p,p) = CONJG(V(p,p))
DO 1124 q = p + 1, N
CTMP = CONJG(V(q,p))
V(q,p) = CONJG(V(p,q))
V(p,q) = CTMP
1124 CONTINUE
1123 CONTINUE
CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
END IF
*
ELSE
* .. aply ZGESVD to R
* .. copy R into V and overwrite V with the right singular vectors
CALL ZLACPY( 'U', NR, N, A, LDA, V, LDV )
IF ( NR .GT. 1 )
$ CALL ZLASET( 'L', NR-1, NR-1, CZERO, CZERO, V(2,1), LDV )
* .. the right singular vectors overwrite V, the NR left singular
* vectors stored in U(1:NR,1:NR)
IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
CALL ZGESVD( 'N', 'O', NR, N, V, LDV, S, U, LDU,
$ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
* .. now [V](1:NR,1:N) contains V(1:N,1:NR)**H
ELSE
* .. need all N right singular vectors and NR < N
* [!] This is simple implementation that augments [V](1:NR,1:N)
* by padding a zero block. In the case NR << N, a more efficient
* way is to first use the LQ factorization. For more details
* how to implement this, see the " FULL SVD " branch.
CALL ZLASET('G', N-NR, N, CZERO,CZERO, V(NR+1,1), LDV)
CALL ZGESVD( 'N', 'O', N, N, V, LDV, S, U, LDU,
$ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
END IF
* .. now [V] contains the adjoint of the matrix of the right singular
* vectors of A.
END IF
*
ELSE
*.......................................................................
* .. FULL SVD requested
*.......................................................................
IF ( RTRANS ) THEN
*
* .. apply ZGESVD to R**H [[this option is left for R&D&T]]
*
IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
* .. copy R**H into [V] and overwrite [V] with the left singular
* vectors of R**H
DO 1168 p = 1, NR
DO 1169 q = p, N
V(q,p) = CONJG(A(p,q))
1169 CONTINUE
1168 CONTINUE
IF ( NR .GT. 1 )
$ CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
*
* .. the left singular vectors of R**H overwrite [V], the NR right
* singular vectors of R**H stored in [U](1:NR,1:NR) as conjugate
* transposed
CALL ZGESVD( 'O', 'A', N, NR, V, LDV, S, V, LDV,
$ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
* .. assemble V
DO 1115 p = 1, NR
V(p,p) = CONJG(V(p,p))
DO 1116 q = p + 1, NR
CTMP = CONJG(V(q,p))
V(q,p) = CONJG(V(p,q))
V(p,q) = CTMP
1116 CONTINUE
1115 CONTINUE
IF ( NR .LT. N ) THEN
DO 1101 p = 1, NR
DO 1102 q = NR+1, N
V(p,q) = CONJG(V(q,p))
1102 CONTINUE
1101 CONTINUE
END IF
CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
*
DO 1117 p = 1, NR
U(p,p) = CONJG(U(p,p))
DO 1118 q = p + 1, NR
CTMP = CONJG(U(q,p))
U(q,p) = CONJG(U(p,q))
U(p,q) = CTMP
1118 CONTINUE
1117 CONTINUE
*
IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
CALL ZLASET('A', M-NR,NR, CZERO,CZERO, U(NR+1,1), LDU)
IF ( NR .LT. N1 ) THEN
CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
$ U(NR+1,NR+1), LDU )
END IF
END IF
*
ELSE
* .. need all N right singular vectors and NR < N
* .. copy R**H into [V] and overwrite [V] with the left singular
* vectors of R**H
* [[The optimal ratio N/NR for using QRF instead of padding
* with zeros. Here hard coded to 2; it must be at least
* two due to work space constraints.]]
* OPTRATIO = ILAENV(6, 'ZGESVD', 'S' // 'O', NR,N,0,0)
* OPTRATIO = MAX( OPTRATIO, 2 )
OPTRATIO = 2
IF ( OPTRATIO*NR .GT. N ) THEN
DO 1198 p = 1, NR
DO 1199 q = p, N
V(q,p) = CONJG(A(p,q))
1199 CONTINUE
1198 CONTINUE
IF ( NR .GT. 1 )
$ CALL ZLASET('U',NR-1,NR-1, CZERO,CZERO, V(1,2),LDV)
*
CALL ZLASET('A',N,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
CALL ZGESVD( 'O', 'A', N, N, V, LDV, S, V, LDV,
$ U, LDU, CWORK(N+1), LCWORK-N, RWORK, INFO )
*
DO 1113 p = 1, N
V(p,p) = CONJG(V(p,p))
DO 1114 q = p + 1, N
CTMP = CONJG(V(q,p))
V(q,p) = CONJG(V(p,q))
V(p,q) = CTMP
1114 CONTINUE
1113 CONTINUE
CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
* .. assemble the left singular vector matrix U of dimensions
* (M x N1), i.e. (M x N) or (M x M).
*
DO 1111 p = 1, N
U(p,p) = CONJG(U(p,p))
DO 1112 q = p + 1, N
CTMP = CONJG(U(q,p))
U(q,p) = CONJG(U(p,q))
U(p,q) = CTMP
1112 CONTINUE
1111 CONTINUE
*
IF ( ( N .LT. M ) .AND. .NOT.(WNTUF)) THEN
CALL ZLASET('A',M-N,N,CZERO,CZERO,U(N+1,1),LDU)
IF ( N .LT. N1 ) THEN
CALL ZLASET('A',N,N1-N,CZERO,CZERO,U(1,N+1),LDU)
CALL ZLASET('A',M-N,N1-N,CZERO,CONE,
$ U(N+1,N+1), LDU )
END IF
END IF
ELSE
* .. copy R**H into [U] and overwrite [U] with the right
* singular vectors of R
DO 1196 p = 1, NR
DO 1197 q = p, N
U(q,NR+p) = CONJG(A(p,q))
1197 CONTINUE
1196 CONTINUE
IF ( NR .GT. 1 )
$ CALL ZLASET('U',NR-1,NR-1,CZERO,CZERO,U(1,NR+2),LDU)
CALL ZGEQRF( N, NR, U(1,NR+1), LDU, CWORK(N+1),
$ CWORK(N+NR+1), LCWORK-N-NR, IERR )
DO 1143 p = 1, NR
DO 1144 q = 1, N
V(q,p) = CONJG(U(p,NR+q))
1144 CONTINUE
1143 CONTINUE
CALL ZLASET('U',NR-1,NR-1,CZERO,CZERO,V(1,2),LDV)
CALL ZGESVD( 'S', 'O', NR, NR, V, LDV, S, U, LDU,
$ V,LDV, CWORK(N+NR+1),LCWORK-N-NR,RWORK, INFO )
CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
CALL ZUNMQR('R','C', N, N, NR, U(1,NR+1), LDU,
$ CWORK(N+1),V,LDV,CWORK(N+NR+1),LCWORK-N-NR,IERR)
CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
* .. assemble the left singular vector matrix U of dimensions
* (M x NR) or (M x N) or (M x M).
IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
CALL ZLASET('A',M-NR,NR,CZERO,CZERO,U(NR+1,1),LDU)
IF ( NR .LT. N1 ) THEN
CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
$ U(NR+1,NR+1),LDU)
END IF
END IF
END IF
END IF
*
ELSE
*
* .. apply ZGESVD to R [[this is the recommended option]]
*
IF ( WNTVR .OR. ( NR .EQ. N ) ) THEN
* .. copy R into [V] and overwrite V with the right singular vectors
CALL ZLACPY( 'U', NR, N, A, LDA, V, LDV )
IF ( NR .GT. 1 )
$ CALL ZLASET( 'L', NR-1,NR-1, CZERO,CZERO, V(2,1), LDV )
* .. the right singular vectors of R overwrite [V], the NR left
* singular vectors of R stored in [U](1:NR,1:NR)
CALL ZGESVD( 'S', 'O', NR, N, V, LDV, S, U, LDU,
$ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
CALL ZLAPMT( .FALSE., NR, N, V, LDV, IWORK )
* .. now [V](1:NR,1:N) contains V(1:N,1:NR)**H
* .. assemble the left singular vector matrix U of dimensions
* (M x NR) or (M x N) or (M x M).
IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
CALL ZLASET('A', M-NR,NR, CZERO,CZERO, U(NR+1,1), LDU)
IF ( NR .LT. N1 ) THEN
CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
$ U(NR+1,NR+1), LDU )
END IF
END IF
*
ELSE
* .. need all N right singular vectors and NR < N
* .. the requested number of the left singular vectors
* is then N1 (N or M)
* [[The optimal ratio N/NR for using LQ instead of padding
* with zeros. Here hard coded to 2; it must be at least
* two due to work space constraints.]]
* OPTRATIO = ILAENV(6, 'ZGESVD', 'S' // 'O', NR,N,0,0)
* OPTRATIO = MAX( OPTRATIO, 2 )
OPTRATIO = 2
IF ( OPTRATIO * NR .GT. N ) THEN
CALL ZLACPY( 'U', NR, N, A, LDA, V, LDV )
IF ( NR .GT. 1 )
$ CALL ZLASET('L', NR-1,NR-1, CZERO,CZERO, V(2,1),LDV)
* .. the right singular vectors of R overwrite [V], the NR left
* singular vectors of R stored in [U](1:NR,1:NR)
CALL ZLASET('A', N-NR,N, CZERO,CZERO, V(NR+1,1),LDV)
CALL ZGESVD( 'S', 'O', N, N, V, LDV, S, U, LDU,
$ V, LDV, CWORK(N+1), LCWORK-N, RWORK, INFO )
CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
* .. now [V] contains the adjoint of the matrix of the right
* singular vectors of A. The leading N left singular vectors
* are in [U](1:N,1:N)
* .. assemble the left singular vector matrix U of dimensions
* (M x N1), i.e. (M x N) or (M x M).
IF ( ( N .LT. M ) .AND. .NOT.(WNTUF)) THEN
CALL ZLASET('A',M-N,N,CZERO,CZERO,U(N+1,1),LDU)
IF ( N .LT. N1 ) THEN
CALL ZLASET('A',N,N1-N,CZERO,CZERO,U(1,N+1),LDU)
CALL ZLASET( 'A',M-N,N1-N,CZERO,CONE,
$ U(N+1,N+1), LDU )
END IF
END IF
ELSE
CALL ZLACPY( 'U', NR, N, A, LDA, U(NR+1,1), LDU )
IF ( NR .GT. 1 )
$ CALL ZLASET('L',NR-1,NR-1,CZERO,CZERO,U(NR+2,1),LDU)
CALL ZGELQF( NR, N, U(NR+1,1), LDU, CWORK(N+1),
$ CWORK(N+NR+1), LCWORK-N-NR, IERR )
CALL ZLACPY('L',NR,NR,U(NR+1,1),LDU,V,LDV)
IF ( NR .GT. 1 )
$ CALL ZLASET('U',NR-1,NR-1,CZERO,CZERO,V(1,2),LDV)
CALL ZGESVD( 'S', 'O', NR, NR, V, LDV, S, U, LDU,
$ V, LDV, CWORK(N+NR+1), LCWORK-N-NR, RWORK, INFO )
CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
CALL ZUNMLQ('R','N',N,N,NR,U(NR+1,1),LDU,CWORK(N+1),
$ V, LDV, CWORK(N+NR+1),LCWORK-N-NR,IERR)
CALL ZLAPMT( .FALSE., N, N, V, LDV, IWORK )
* .. assemble the left singular vector matrix U of dimensions
* (M x NR) or (M x N) or (M x M).
IF ( ( NR .LT. M ) .AND. .NOT.(WNTUF)) THEN
CALL ZLASET('A',M-NR,NR,CZERO,CZERO,U(NR+1,1),LDU)
IF ( NR .LT. N1 ) THEN
CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,
$ U(NR+1,NR+1), LDU )
END IF
END IF
END IF
END IF
* .. end of the "R**H or R" branch
END IF
*
* The Q matrix from the first QRF is built into the left singular
* vectors matrix U.
*
IF ( .NOT. WNTUF )
$ CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,
$ LDU, CWORK(N+1), LCWORK-N, IERR )
IF ( ROWPRM .AND. .NOT.WNTUF )
$ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(N+1), -1 )
*
* ... end of the "full SVD" branch
END IF
*
* Check whether some singular values are returned as zeros, e.g.
* due to underflow, and update the numerical rank.
p = NR
DO 4001 q = p, 1, -1
IF ( S(q) .GT. ZERO ) GO TO 4002
NR = NR - 1
4001 CONTINUE
4002 CONTINUE
*
* .. if numerical rank deficiency is detected, the truncated
* singular values are set to zero.
IF ( NR .LT. N ) CALL DLASET( 'G', N-NR,1, ZERO,ZERO, S(NR+1), N )
* .. undo scaling; this may cause overflow in the largest singular
* values.
IF ( ASCALED )
$ CALL DLASCL( 'G',0,0, ONE,SQRT(DBLE(M)), NR,1, S, N, IERR )
IF ( CONDA ) RWORK(1) = SCONDA
RWORK(2) = p - NR
* .. p-NR is the number of singular values that are computed as
* exact zeros in ZGESVD() applied to the (possibly truncated)
* full row rank triangular (trapezoidal) factor of A.
NUMRANK = NR
*
RETURN
*
* End of ZGESVDQ
*
END