gesvdq(3)

NAME
gesvdq - gesvdq: SVD, QR with pivoting

SYNOPSIS

  Functions

subroutine cgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u, ldu, v, ldv, numrank, iwork, liwork, cwork, lcwork, rwork, lrwork, info)
CGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices
subroutine dgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u, ldu, v, ldv, numrank, iwork, liwork, work, lwork, rwork, lrwork, info)
DGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices
subroutine sgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u, ldu, v, ldv, numrank, iwork, liwork, work, lwork, rwork, lrwork, info)
SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices
subroutine zgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u, ldu, v, ldv, numrank, iwork, liwork, cwork, lcwork, rwork, lrwork, info)
ZGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

Detailed Description

Function Documentation

  subroutine cgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) s, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, complex, dimension( * ) cwork, integer lcwork, real, dimension( * ) rwork, integer lrwork, integer info)

CGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

Purpose:

    !>
    !> CGESVDQ 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.
    !>

Parameters
    JOBA

    !>  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 = SLAMCH('Epsilon'). This authorises CGESVDQ 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)
    !>

    JOBP

    !>  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.
    !>

    JOBR

    !>          JOBR is CHARACTER*1
    !>          = 'T' After the initial pivoted QR factorization, CGESVD 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.
    !>

    JOBU

    !>          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.
    !>

    JOBV

    !>          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.
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the input matrix A.  M >= N >= 0.
    !>

    A

    !>          A is COMPLEX 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 CGEQP3. 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.
    !>

    LDA

    !>          LDA is INTEGER.
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    S

    !>          S is REAL array of dimension N.
    !>          The singular values of A, ordered so that S(i) >= S(i+1).
    !>

    U

    !>          U is COMPLEX 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.
    !>

    LDU

    !>          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.
    !>

    V

    !>          V is COMPLEX 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.
    !>

    LDV

    !>          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.
    !>

    NUMRANK

    !>          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.
    !>

    IWORK

    !>          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.
    !>

    LIWORK

    !>          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.
    !>

    CWORK

    !>          CWORK is COMPLEX 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
    !>          CGEQP3.
    !>
    !>          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.
    !>

    LCWORK

    !>          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.
    !>

    RWORK

    !>          RWORK is REAL 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 CGESVD applied to the upper triangular or trapezoidal
    !>          R (from the initial QR factorization). In case of early exit (no call to
    !>          CGESVD, 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.
    !>

    LRWORK

    !>          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.
    !>

    INFO

    !>          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 (computed in CGESVD) did not
    !>          converge to zero.
    !>

Further Details:

    !>
    !>   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, enclosed in [[double brackets]].
    !>

Bugs, examples and comments

    !>  Please report all bugs and send interesting examples and/or comments to
    !>  drmac@math.hr. Thank you.
    !>

References

    !>  [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
    !>

Contributors:

    !> Developed and coded by Zlatko Drmac, Department of Mathematics
    !>  University of Zagreb, Croatia, drmac@math.hr
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, double precision, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer info)

DGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

Purpose:

    !>
    !> DGESVDQ computes the singular value decomposition (SVD) of a real
    !> 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 orthogonal 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.
    !>

Parameters
    JOBA

    !>  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 DGESVDQ 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)
    !>

    JOBP

    !>  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.
    !>

    JOBR

    !>          JOBR is CHARACTER*1
    !>          = 'T' After the initial pivoted QR factorization, DGESVD is applied to
    !>          the transposed R**T 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 DGESVD. This may be
    !>          preferred as it involves less data movement.
    !>

    JOBU

    !>          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**T , 0)**T. 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.
    !>

    JOBV

    !>          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.
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the input matrix A.  M >= N >= 0.
    !>

    A

    !>          A is DOUBLE PRECISION 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 DGEQP3. If JOBU = 'F', these Householder
    !>          vectors together with WORK(1:N) can be used to restore the Q factors from
    !>          the initial pivoted QR factorization of A. See the description of U.
    !>

    LDA

    !>          LDA is INTEGER.
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    S

    !>          S is DOUBLE PRECISION array of dimension N.
    !>          The singular values of A, ordered so that S(i) >= S(i+1).
    !>

    U

    !>          U is DOUBLE PRECISION 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 orthogonal matrix that can be used to form the left
    !>          singular vectors.
    !>          If JOBU = 'N', U is not referenced.
    !>

    LDU

    !>          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.
    !>

    V

    !>          V is DOUBLE PRECISION 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 orthogonal matrix  V**T;
    !>          If JOBV = 'R', V contains the first NUMRANK rows of V**T (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.
    !>

    LDV

    !>          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.
    !>

    NUMRANK

    !>          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 DGESVD. The final value of NUMRANK might be further reduced if
    !>          some singular values are computed as zeros.
    !>

    IWORK

    !>          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, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
    !>          IWORK(1) returns the minimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          LIWORK >= N + M - 1,     if JOBP = 'P' and JOBA .NE. 'E';
    !>          LIWORK >= N              if JOBP = 'N' and JOBA .NE. 'E';
    !>          LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E';
    !>          LIWORK >= N + N          if JOBP = 'N' and JOBA = 'E'.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates and returns the optimal and minimal sizes
    !>          for the WORK, IWORK, and RWORK arrays, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (max(2, LWORK)), used as a workspace.
    !>          On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters
    !>          needed to recover the Q factor from the QR factorization computed by
    !>          DGEQP3.
    !>
    !>          If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
    !>          WORK(1) returns the optimal LWORK, and
    !>          WORK(2) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. It is determined as follows:
    !>          Let  LWQP3 = 3*N+1,  LWCON = 3*N, and let
    !>          LWORQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'
    !>                  { MAX( M, 1 ),  if JOBU = 'A'
    !>          LWSVD = MAX( 5*N, 1 )
    !>          LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ),
    !>          LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 )
    !>          Then the minimal value of LWORK 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, LWORQ ) if the singular values and the left
    !>                                   singular vectors are requested;
    !>          = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) 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, LWORQ ) if the full SVD is requested with JOBV = 'R';
    !>                                   independent of JOBR;
    !>          = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested,
    !>                                   JOBV = 'R' and, also a scaled condition
    !>                                   estimate requested; independent of JOBR;
    !>          = MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
    !>         N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the
    !>                         full SVD is requested with JOBV = 'A' or 'V', and
    !>                         JOBR ='N'
    !>          = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
    !>         N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) )
    !>                         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, LWORQ ),
    !>         N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the
    !>                         full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
    !>          = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
    !>         N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) )
    !>                         if the full SVD is requested with JOBV = 'A' or 'V', and
    !>                         JOBR ='T', and also a scaled condition number estimate
    !>                         requested.
    !>          Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ).
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates and returns the optimal and minimal sizes
    !>          for the WORK, IWORK, and RWORK arrays, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    RWORK

    !>          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 DGESVD applied to the upper triangular or trapezoidal
    !>          R (from the initial QR factorization). In case of early exit (no call to
    !>          DGESVD, such as in the case of zero matrix) RWORK(2) = -1.
    !>
    !>          If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
    !>          RWORK(1) returns the minimal LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER.
    !>          The dimension of the array RWORK.
    !>          If JOBP ='P', then LRWORK >= MAX(2, M).
    !>          Otherwise, LRWORK >= 2
    !>
    !>          If LRWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates and returns the optimal and minimal sizes
    !>          for the WORK, IWORK, and RWORK arrays, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if DBDSQR did not converge, INFO specifies how many superdiagonals
    !>          of an intermediate bidiagonal form B (computed in DGESVD) did not
    !>          converge to zero.
    !>

Further Details:

    !>
    !>   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, enclosed in [[double brackets]].
    !>

Bugs, examples and comments

    !>  Please report all bugs and send interesting examples and/or comments to
    !>  drmac@math.hr. Thank you.
    !>

References

    !>  [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
    !>

Contributors:

    !> Developed and coded by Zlatko Drmac, Department of Mathematics
    !>  University of Zagreb, Croatia, drmac@math.hr
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine sgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) s, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, real, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer info)

SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

Purpose:

    !>
    !> SGESVDQ computes the singular value decomposition (SVD) of a real
    !> 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 orthogonal 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.
    !>

Parameters
    JOBA

    !>  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 = SLAMCH('Epsilon'). This authorises CGESVDQ 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)
    !>

    JOBP

    !>  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.
    !>

    JOBR

    !>          JOBR is CHARACTER*1
    !>          = 'T' After the initial pivoted QR factorization, SGESVD is applied to
    !>          the transposed R**T 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 SGESVD. This may be
    !>          preferred as it involves less data movement.
    !>

    JOBU

    !>          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**T , 0)**T. 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.
    !>

    JOBV

    !>          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.
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the input matrix A.  M >= N >= 0.
    !>

    A

    !>          A is REAL 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 SGEQP3. If JOBU = 'F', these Householder
    !>          vectors together with WORK(1:N) can be used to restore the Q factors from
    !>          the initial pivoted QR factorization of A. See the description of U.
    !>

    LDA

    !>          LDA is INTEGER.
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    S

    !>          S is REAL array of dimension N.
    !>          The singular values of A, ordered so that S(i) >= S(i+1).
    !>

    U

    !>          U is REAL 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 orthogonal matrix that can be used to form the left
    !>          singular vectors.
    !>          If JOBU = 'N', U is not referenced.
    !>

    LDU

    !>          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.
    !>

    V

    !>          V is REAL 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 orthogonal matrix  V**T;
    !>          If JOBV = 'R', V contains the first NUMRANK rows of V**T (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.
    !>

    LDV

    !>          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.
    !>

    NUMRANK

    !>          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 SGESVD. The final value of NUMRANK might be further reduced if
    !>          some singular values are computed as zeros.
    !>

    IWORK

    !>          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, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
    !>          IWORK(1) returns the minimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK.
    !>          LIWORK >= N + M - 1,     if JOBP = 'P' and JOBA .NE. 'E';
    !>          LIWORK >= N              if JOBP = 'N' and JOBA .NE. 'E';
    !>          LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E';
    !>          LIWORK >= N + N          if JOBP = 'N' and JOBA = 'E'.
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates and returns the optimal and minimal sizes
    !>          for the WORK, IWORK, and RWORK arrays, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    WORK

    !>          WORK is REAL array, dimension (max(2, LWORK)), used as a workspace.
    !>          On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters
    !>          needed to recover the Q factor from the QR factorization computed by
    !>          SGEQP3.
    !>
    !>          If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
    !>          WORK(1) returns the optimal LWORK, and
    !>          WORK(2) returns the minimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. It is determined as follows:
    !>          Let  LWQP3 = 3*N+1,  LWCON = 3*N, and let
    !>          LWORQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'
    !>                  { MAX( M, 1 ),  if JOBU = 'A'
    !>          LWSVD = MAX( 5*N, 1 )
    !>          LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ),
    !>          LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 )
    !>          Then the minimal value of LWORK 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, LWORQ ) if the singular values and the left
    !>                                   singular vectors are requested;
    !>          = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) 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, LWORQ ) if the full SVD is requested with JOBV = 'R';
    !>                                   independent of JOBR;
    !>          = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested,
    !>                                   JOBV = 'R' and, also a scaled condition
    !>                                   estimate requested; independent of JOBR;
    !>          = MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
    !>         N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the
    !>                         full SVD is requested with JOBV = 'A' or 'V', and
    !>                         JOBR ='N'
    !>          = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
    !>         N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) )
    !>                         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, LWORQ ),
    !>         N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the
    !>                         full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
    !>          = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
    !>         N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) )
    !>                         if the full SVD is requested with JOBV = 'A' or 'V', and
    !>                         JOBR ='T', and also a scaled condition number estimate
    !>                         requested.
    !>          Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ).
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates and returns the optimal and minimal sizes
    !>          for the WORK, IWORK, and RWORK arrays, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    RWORK

    !>          RWORK is REAL 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 SGESVD applied to the upper triangular or trapezoidal
    !>          R (from the initial QR factorization). In case of early exit (no call to
    !>          SGESVD, such as in the case of zero matrix) RWORK(2) = -1.
    !>
    !>          If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
    !>          RWORK(1) returns the minimal LRWORK.
    !>

    LRWORK

    !>          LRWORK is INTEGER.
    !>          The dimension of the array RWORK.
    !>          If JOBP ='P', then LRWORK >= MAX(2, M).
    !>          Otherwise, LRWORK >= 2
    !>
    !>          If LRWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates and returns the optimal and minimal sizes
    !>          for the WORK, IWORK, and RWORK arrays, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if SBDSQR did not converge, INFO specifies how many superdiagonals
    !>          of an intermediate bidiagonal form B (computed in SGESVD) did not
    !>          converge to zero.
    !>

Further Details:

    !>
    !>   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, enclosed in [[double brackets]].
    !>

Bugs, examples and comments

    !>  Please report all bugs and send interesting examples and/or comments to
    !>  drmac@math.hr. Thank you.
    !>

References

    !>  [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
    !>

Contributors:

    !> Developed and coded by Zlatko Drmac, Department of Mathematics
    !>  University of Zagreb, Croatia, drmac@math.hr
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, complex*16, dimension( * ) cwork, integer lcwork, double precision, dimension( * ) rwork, integer lrwork, integer info)

ZGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

Purpose:

    !>
    !> 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.
    !>

Parameters
    JOBA

    !>  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)
    !>

    JOBP

    !>  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.
    !>

    JOBR

    !>          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.
    !>

    JOBU

    !>          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.
    !>

    JOBV

    !>          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.
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A.  M >= 0.
    !>

    N

    !>          N is INTEGER
    !>          The number of columns of the input matrix A.  M >= N >= 0.
    !>

    A

    !>          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.
    !>

    LDA

    !>          LDA is INTEGER.
    !>          The leading dimension of the array A.  LDA >= max(1,M).
    !>

    S

    !>          S is DOUBLE PRECISION array of dimension N.
    !>          The singular values of A, ordered so that S(i) >= S(i+1).
    !>

    U

    !>          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.
    !>

    LDU

    !>          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.
    !>

    V

    !>          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.
    !>

    LDV

    !>          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.
    !>

    NUMRANK

    !>          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.
    !>

    IWORK

    !>          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.
    !>

    LIWORK

    !>          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.
    !>

    CWORK

    !>          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.
    !>

    LCWORK

    !>          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.
    !>

    RWORK

    !>          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 trapezoidal
    !>          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.
    !>

    LRWORK

    !>          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.
    !>

    INFO

    !>          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.
    !>

Further Details:

    !>
    !>   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, enclosed in [[double brackets]].
    !>

Bugs, examples and comments

    !>  Please report all bugs and send interesting examples and/or comments to
    !>  drmac@math.hr. Thank you.
    !>

References

    !>  [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
    !>

Contributors:

    !> Developed and coded by Zlatko Drmac, Department of Mathematics
    !>  University of Zagreb, Croatia, drmac@math.hr
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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