complex16GEsing(3)

NAME
complex16GEsing - complex16

SYNOPSIS

  Functions
subroutine zgejsv (joba, jobu, jobv, jobr, jobt, jobp, m, n, a, lda, sva, u, ldu, v, ldv, cwork, lwork, rwork, lrwork, iwork, info)
ZGEJSV
subroutine zgesdd (jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)
ZGESDD
subroutine zgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, info)
ZGESVD computes the singular value decomposition (SVD) 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
subroutine zgesvdx (jobu, jobvt, range, m, n, a, lda, vl, vu, il, iu, ns, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)
ZGESVDX computes the singular value decomposition (SVD) for GE matrices
subroutine zggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork, info)
ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Detailed Description

This is the group of complex16 singular value driver functions for GE matrices

Function Documentation

  subroutine zgejsv (character*1 joba, character*1 jobu, character*1 jobv, character*1 jobr, character*1 jobt, character*1 jobp, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( n ) sva, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( lwork ) cwork, integer lwork, double precision, dimension( lrwork ) rwork, integer lrwork, integer, dimension( * ) iwork, integer info)

ZGEJSV

Purpose:

     ZGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
     matrix [A], where M >= N. The SVD of [A] is written as
                  [A] = [U] * [SIGMA] * [V]^*,
     where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
     diagonal elements, [U] is an M-by-N (or M-by-M) unitary matrix, and
     [V] is an N-by-N unitary matrix. The diagonal elements of [SIGMA] are
     the singular values of [A]. The columns of [U] and [V] are the left and
     the right singular vectors of [A], respectively. The matrices [U] and [V]
     are computed and stored in the arrays U and V, respectively. The diagonal
     of [SIGMA] is computed and stored in the array SVA.

Arguments:

Parameters
    JOBA

              JOBA is CHARACTER*1
             Specifies the level of accuracy:
           = 'C': This option works well (high relative accuracy) if A = B * D,
                  with well-conditioned B and arbitrary diagonal matrix D.
                  The accuracy cannot be spoiled by COLUMN scaling. The
                  accuracy of the computed output depends on the condition of
                  B, and the procedure aims at the best theoretical accuracy.
                  The relative error max_{i=1:N}|d sigma_i| / sigma_i is
                  bounded by f(M,N)*epsilon* cond(B), independent of D.
                  The input matrix is preprocessed with the QRF with column
                  pivoting. This initial preprocessing and preconditioning by
                  a rank revealing QR factorization is common for all values of
                  JOBA. Additional actions are specified as follows:
           = 'E': Computation as with 'C' with an additional estimate of the
                  condition number of B. It provides a realistic error bound.
           = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
                  D1, D2, and well-conditioned matrix C, this option gives
                  higher accuracy than the 'C' option. If the structure of the
                  input matrix is not known, and relative accuracy is
                  desirable, then this option is advisable. The input matrix A
                  is preprocessed with QR factorization with FULL (row and
                  column) pivoting.
           = 'G': Computation as with 'F' with an additional estimate of the
                  condition number of B, where A=B*D. If A has heavily weighted
                  rows, then using this condition number gives too pessimistic
                  error bound.
           = 'A': Small singular values are not well determined by the data
                  and are considered as noisy; the matrix is treated as
                  numerically rank deficient. The error in the computed
                  singular values is bounded by f(m,n)*epsilon*||A||.
                  The computed SVD A = U * S * V^* restores A up to
                  f(m,n)*epsilon*||A||.
                  This gives the procedure the licence to discard (set to zero)
                  all singular values below N*epsilon*||A||.
           = 'R': Similar as in 'A'. Rank revealing property of the initial
                  QR factorization is used do reveal (using triangular factor)
                  a gap sigma_{r+1} < epsilon * sigma_r in which case the
                  numerical RANK is declared to be r. The SVD is computed with
                  absolute error bounds, but more accurately than with 'A'.

    JOBU

              JOBU is CHARACTER*1
             Specifies whether to compute the columns of U:
           = 'U': N columns of U are returned in the array U.
           = 'F': full set of M left sing. vectors is returned in the array U.
           = 'W': U may be used as workspace of length M*N. See the description
                  of U.
           = 'N': U is not computed.

    JOBV

              JOBV is CHARACTER*1
             Specifies whether to compute the matrix V:
           = 'V': N columns of V are returned in the array V; Jacobi rotations
                  are not explicitly accumulated.
           = 'J': N columns of V are returned in the array V, but they are
                  computed as the product of Jacobi rotations, if JOBT = 'N'.
           = 'W': V may be used as workspace of length N*N. See the description
                  of V.
           = 'N': V is not computed.

    JOBR

              JOBR is CHARACTER*1
             Specifies the RANGE for the singular values. Issues the licence to
             set to zero small positive singular values if they are outside
             specified range. If A .NE. 0 is scaled so that the largest singular
             value of c*A is around SQRT(BIG), BIG=DLAMCH('O'), then JOBR issues
             the licence to kill columns of A whose norm in c*A is less than
             SQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN,
             where SFMIN=DLAMCH('S'), EPSLN=DLAMCH('E').
           = 'N': Do not kill small columns of c*A. This option assumes that
                  BLAS and QR factorizations and triangular solvers are
                  implemented to work in that range. If the condition of A
                  is greater than BIG, use ZGESVJ.
           = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
                  (roughly, as described above). This option is recommended.
                                                 ===========================
             For computing the singular values in the FULL range [SFMIN,BIG]
             use ZGESVJ.

    JOBT

              JOBT is CHARACTER*1
             If the matrix is square then the procedure may determine to use
             transposed A if A^* seems to be better with respect to convergence.
             If the matrix is not square, JOBT is ignored.
             The decision is based on two values of entropy over the adjoint
             orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
           = 'T': transpose if entropy test indicates possibly faster
             convergence of Jacobi process if A^* is taken as input. If A is
             replaced with A^*, then the row pivoting is included automatically.
           = 'N': do not speculate.
             The option 'T' can be used to compute only the singular values, or
             the full SVD (U, SIGMA and V). For only one set of singular vectors
             (U or V), the caller should provide both U and V, as one of the
             matrices is used as workspace if the matrix A is transposed.
             The implementer can easily remove this constraint and make the
             code more complicated. See the descriptions of U and V.
             In general, this option is considered experimental, and 'N'; should
             be preferred. This is subject to changes in the future.

    JOBP

              JOBP is CHARACTER*1
             Issues the licence to introduce structured perturbations to drown
             denormalized numbers. This licence should be active if the
             denormals are poorly implemented, causing slow computation,
             especially in cases of fast convergence (!). For details see [1,2].
             For the sake of simplicity, this perturbations are included only
             when the full SVD or only the singular values are requested. The
             implementer/user can easily add the perturbation for the cases of
             computing one set of singular vectors.
           = 'P': introduce perturbation
           = 'N': do not perturb

    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, dimension (LDA,N)
              On entry, the M-by-N matrix A.

    LDA

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

    SVA

              SVA is DOUBLE PRECISION array, dimension (N)
              On exit,
              - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
                computation SVA contains Euclidean column norms of the
                iterated matrices in the array A.
              - For WORK(1) .NE. WORK(2): The singular values of A are
                (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
                sigma_max(A) overflows or if small singular values have been
                saved from underflow by scaling the input matrix A.
              - If JOBR='R' then some of the singular values may be returned
                as exact zeros obtained by 'set to zero' because they are
                below the numerical rank threshold or are denormalized numbers.

    U

              U is COMPLEX*16 array, dimension ( LDU, N )
              If JOBU = 'U', then U contains on exit the M-by-N matrix of
                             the left singular vectors.
              If JOBU = 'F', then U contains on exit the M-by-M matrix of
                             the left singular vectors, including an ONB
                             of the orthogonal complement of the Range(A).
              If JOBU = 'W'  .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N),
                             then U is used as workspace if the procedure
                             replaces A with A^*. In that case, [V] is computed
                             in U as left singular vectors of A^* and then
                             copied back to the V array. This 'W' option is just
                             a reminder to the caller that in this case U is
                             reserved as workspace of length N*N.
              If JOBU = 'N'  U is not referenced, unless JOBT='T'.

    LDU

              LDU is INTEGER
              The leading dimension of the array U,  LDU >= 1.
              IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.

    V

              V is COMPLEX*16 array, dimension ( LDV, N )
              If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
                             the right singular vectors;
              If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N),
                             then V is used as workspace if the pprocedure
                             replaces A with A^*. In that case, [U] is computed
                             in V as right singular vectors of A^* and then
                             copied back to the U array. This 'W' option is just
                             a reminder to the caller that in this case V is
                             reserved as workspace of length N*N.
              If JOBV = 'N'  V is not referenced, unless JOBT='T'.

    LDV

              LDV is INTEGER
              The leading dimension of the array V,  LDV >= 1.
              If JOBV = 'V' or 'J' or 'W', then LDV >= N.

    CWORK

              CWORK is COMPLEX*16 array, dimension (MAX(2,LWORK))
              If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
              LRWORK=-1), then on exit CWORK(1) contains the required length of
              CWORK for the job parameters used in the call.

    LWORK

              LWORK is INTEGER
              Length of CWORK to confirm proper allocation of workspace.
              LWORK depends on the job:
              1. If only SIGMA is needed ( JOBU = 'N', JOBV = 'N' ) and
                1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
                   LWORK >= 2*N+1. This is the minimal requirement.
                   ->> For optimal performance (blocked code) the optimal value
                   is LWORK >= N + (N+1)*NB. Here NB is the optimal
                   block size for ZGEQP3 and ZGEQRF.
                   In general, optimal LWORK is computed as
                   LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ)).
                1.2. .. an estimate of the scaled condition number of A is
                   required (JOBA='E', or 'G'). In this case, LWORK the minimal
                   requirement is LWORK >= N*N + 2*N.
                   ->> For optimal performance (blocked code) the optimal value
                   is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
                   In general, the optimal length LWORK is computed as
                   LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ),
                                N*N+LWORK(ZPOCON)).
              2. If SIGMA and the right singular vectors are needed (JOBV = 'V'),
                 (JOBU = 'N')
                2.1   .. no scaled condition estimate requested (JOBE = 'N'):
                -> the minimal requirement is LWORK >= 3*N.
                -> For optimal performance,
                   LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
                   where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,
                   ZUNMLQ. In general, the optimal length LWORK is computed as
                   LWORK >= max(N+LWORK(ZGEQP3), N+LWORK(ZGESVJ),
                           N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).
                2.2 .. an estimate of the scaled condition number of A is
                   required (JOBA='E', or 'G').
                -> the minimal requirement is LWORK >= 3*N.
                -> For optimal performance,
                   LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
                   where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,
                   ZUNMLQ. In general, the optimal length LWORK is computed as
                   LWORK >= max(N+LWORK(ZGEQP3), LWORK(ZPOCON), N+LWORK(ZGESVJ),
                           N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).
              3. If SIGMA and the left singular vectors are needed
                3.1  .. no scaled condition estimate requested (JOBE = 'N'):
                -> the minimal requirement is LWORK >= 3*N.
                -> For optimal performance:
                   if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
                   where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.
                   In general, the optimal length LWORK is computed as
                   LWORK >= max(N+LWORK(ZGEQP3), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).
                3.2  .. an estimate of the scaled condition number of A is
                   required (JOBA='E', or 'G').
                -> the minimal requirement is LWORK >= 3*N.
                -> For optimal performance:
                   if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
                   where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.
                   In general, the optimal length LWORK is computed as
                   LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZPOCON),
                            2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).
              4. If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
                4.1. if JOBV = 'V'
                   the minimal requirement is LWORK >= 5*N+2*N*N.
                4.2. if JOBV = 'J' the minimal requirement is
                   LWORK >= 4*N+N*N.
                In both cases, the allocated CWORK can accommodate blocked runs
                of ZGEQP3, ZGEQRF, ZGELQF, SUNMQR, ZUNMLQ.
              If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
              LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
              minimal length of CWORK for the job parameters used in the call.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (MAX(7,LWORK))
              On exit,
              RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
                        such that SCALE*SVA(1:N) are the computed singular values
                        of A. (See the description of SVA().)
              RWORK(2) = See the description of RWORK(1).
              RWORK(3) = SCONDA is an estimate for the condition number of
                        column equilibrated A. (If JOBA = 'E' or 'G')
                        SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
                        It is computed using ZPOCON. It holds
                        N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
                        where R is the triangular factor from the QRF of A.
                        However, if R is truncated and the numerical rank is
                        determined to be strictly smaller than N, SCONDA is
                        returned as -1, thus indicating that the smallest
                        singular values might be lost.
              If full SVD is needed, the following two condition numbers are
              useful for the analysis of the algorithm. They are provided for
              a developer/implementer who is familiar with the details of
              the method.
              RWORK(4) = an estimate of the scaled condition number of the
                        triangular factor in the first QR factorization.
              RWORK(5) = an estimate of the scaled condition number of the
                        triangular factor in the second QR factorization.
              The following two parameters are computed if JOBT = 'T'.
              They are provided for a developer/implementer who is familiar
              with the details of the method.
              RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
                        of diag(A^* * A) / Trace(A^* * A) taken as point in the
                        probability simplex.
              RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
              If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or
              LRWORK=-1), then on exit RWORK(1) contains the required length of
              RWORK for the job parameters used in the call.

    LRWORK

              LRWORK is INTEGER
              Length of RWORK to confirm proper allocation of workspace.
              LRWORK depends on the job:
           1. If only the singular values are requested i.e. if
              LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
              then:
              1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
                   then: LRWORK = max( 7, 2 * M ).
              1.2. Otherwise, LRWORK  = max( 7,  N ).
           2. If singular values with the right singular vectors are requested
              i.e. if
              (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
              .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
              then:
              2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
              then LRWORK = max( 7, 2 * M ).
              2.2. Otherwise, LRWORK  = max( 7,  N ).
           3. If singular values with the left singular vectors are requested, i.e. if
              (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
              .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
              then:
              3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
              then LRWORK = max( 7, 2 * M ).
              3.2. Otherwise, LRWORK  = max( 7,  N ).
           4. If singular values with both the left and the right singular vectors
              are requested, i.e. if
              (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
              (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
              then:
              4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
              then LRWORK = max( 7, 2 * M ).
              4.2. Otherwise, LRWORK  = max( 7, N ).
              If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and
              the length of RWORK is returned in RWORK(1).

    IWORK

              IWORK is INTEGER array, of dimension at least 4, that further depends
              on the job:
              1. If only the singular values are requested then:
                 If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
                 then the length of IWORK is N+M; otherwise the length of IWORK is N.
              2. If the singular values and the right singular vectors are requested then:
                 If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
                 then the length of IWORK is N+M; otherwise the length of IWORK is N.
              3. If the singular values and the left singular vectors are requested then:
                 If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
                 then the length of IWORK is N+M; otherwise the length of IWORK is N.
              4. If the singular values with both the left and the right singular vectors
                 are requested, then:
                 4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
                      If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
                      then the length of IWORK is N+M; otherwise the length of IWORK is N.
                 4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
                      If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
                      then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.

              On exit,
              IWORK(1) = the numerical rank determined after the initial
                         QR factorization with pivoting. See the descriptions
                         of JOBA and JOBR.
              IWORK(2) = the number of the computed nonzero singular values
              IWORK(3) = if nonzero, a warning message:
                         If IWORK(3) = 1 then some of the column norms of A
                         were denormalized floats. The requested high accuracy
                         is not warranted by the data.
              IWORK(4) = 1 or -1. If IWORK(4) = 1, then the procedure used A^* to
                         do the job as specified by the JOB parameters.
              If the call to ZGEJSV is a workspace query (indicated by LWORK = -1 or
              LRWORK = -1), then on exit IWORK(1) contains the required length of
              IWORK for the job parameters used in the call.

    INFO

              INFO is INTEGER
               < 0:  if INFO = -i, then the i-th argument had an illegal value.
               = 0:  successful exit;
               > 0:  ZGEJSV  did not converge in the maximal allowed number
                     of sweeps. The computed values may be inaccurate.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      ZGEJSV implements a preconditioned Jacobi SVD algorithm. It uses ZGEQP3,
      ZGEQRF, and ZGELQF as preprocessors and preconditioners. Optionally, an
      additional row pivoting can be used as a preprocessor, which in some
      cases results in much higher accuracy. An example is matrix A with the
      structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
      diagonal matrices and C is well-conditioned matrix. In that case, complete
      pivoting in the first QR factorizations provides accuracy dependent on the
      condition number of C, and independent of D1, D2. Such higher accuracy is
      not completely understood theoretically, but it works well in practice.
      Further, if A can be written as A = B*D, with well-conditioned B and some
      diagonal D, then the high accuracy is guaranteed, both theoretically and
      in software, independent of D. For more details see [1], [2].
         The computational range for the singular values can be the full range
      ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
      & LAPACK routines called by ZGEJSV are implemented to work in that range.
      If that is not the case, then the restriction for safe computation with
      the singular values in the range of normalized IEEE numbers is that the
      spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
      overflow. This code (ZGEJSV) is best used in this restricted range,
      meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
      returned as zeros. See JOBR for details on this.
         Further, this implementation is somewhat slower than the one described
      in [1,2] due to replacement of some non-LAPACK components, and because
      the choice of some tuning parameters in the iterative part (ZGESVJ) is
      left to the implementer on a particular machine.
         The rank revealing QR factorization (in this code: ZGEQP3) should be
      implemented as in [3]. We have a new version of ZGEQP3 under development
      that is more robust than the current one in LAPACK, with a cleaner cut in
      rank deficient cases. It will be available in the SIGMA library [4].
      If M is much larger than N, it is obvious that the initial QRF with
      column pivoting can be preprocessed by the QRF without pivoting. That
      well known trick is not used in ZGEJSV because in some cases heavy row
      weighting can be treated with complete pivoting. The overhead in cases
      M much larger than N is then only due to pivoting, but the benefits in
      terms of accuracy have prevailed. The implementer/user can incorporate
      this extra QRF step easily. The implementer can also improve data movement
      (matrix transpose, matrix copy, matrix transposed copy) - this
      implementation of ZGEJSV uses only the simplest, naive data movement.

Contributor:
    Zlatko Drmac, Department of Mathematics, Faculty of Science, University of Zagreb (Zagreb, Croatia); drmac@math.hr

References:

     [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
         SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
         LAPACK Working note 169.
     [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
         SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
         LAPACK Working note 170.
     [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
         factorization software - a case study.
         ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
         LAPACK Working note 176.
     [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
         QSVD, (H,K)-SVD computations.
         Department of Mathematics, University of Zagreb, 2008, 2016.

Bugs, examples and comments:
    Please report all bugs and send interesting examples and/or comments to drmac@math.hr. Thank you.

Definition at line 566 of file zgejsv.f.

  subroutine zgesdd (character jobz, 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( ldvt, * ) vt, integer ldvt, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGESDD

Purpose:

     ZGESDD computes the singular value decomposition (SVD) of a complex
     M-by-N matrix A, optionally computing the left and/or right singular
     vectors, by using divide-and-conquer method. The SVD is written
          A = U * SIGMA * conjugate-transpose(V)
     where SIGMA is an M-by-N matrix which is zero except for its
     min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
     V is an N-by-N unitary matrix.  The diagonal elements of SIGMA
     are the singular values of A; they are real and non-negative, and
     are returned in descending order.  The first min(m,n) columns of
     U and V are the left and right singular vectors of A.
     Note that the routine returns VT = V**H, not V.
     The divide and conquer algorithm makes very mild assumptions about
     floating point arithmetic. It will work on machines with a guard
     digit in add/subtract, or on those binary machines without guard
     digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
     Cray-2. It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              Specifies options for computing all or part of the matrix U:
              = 'A':  all M columns of U and all N rows of V**H are
                      returned in the arrays U and VT;
              = 'S':  the first min(M,N) columns of U and the first
                      min(M,N) rows of V**H are returned in the arrays U
                      and VT;
              = 'O':  If M >= N, the first N columns of U are overwritten
                      in the array A and all rows of V**H are returned in
                      the array VT;
                      otherwise, all columns of U are returned in the
                      array U and the first M rows of V**H are overwritten
                      in the array A;
              = 'N':  no columns of U or rows of V**H are computed.

    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.  N >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if JOBZ = 'O',  A is overwritten with the first N columns
                              of U (the left singular vectors, stored
                              columnwise) if M >= N;
                              A is overwritten with the first M rows
                              of V**H (the right singular vectors, stored
                              rowwise) otherwise.
              if JOBZ .ne. 'O', the contents of A are destroyed.

    LDA

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

    S

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

    U

              U is COMPLEX*16 array, dimension (LDU,UCOL)
              UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
              UCOL = min(M,N) if JOBZ = 'S'.
              If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
              unitary matrix U;
              if JOBZ = 'S', U contains the first min(M,N) columns of U
              (the left singular vectors, stored columnwise);
              if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1;
              if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

    VT

              VT is COMPLEX*16 array, dimension (LDVT,N)
              If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
              N-by-N unitary matrix V**H;
              if JOBZ = 'S', VT contains the first min(M,N) rows of
              V**H (the right singular vectors, stored rowwise);
              if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1;
              if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
              if JOBZ = 'S', LDVT >= min(M,N).

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= 1.
              If LWORK = -1, a workspace query is assumed.  The optimal
              size for the WORK array is calculated and stored in WORK(1),
              and no other work except argument checking is performed.
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N', LWORK >= 2*mn + mx.
              If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
              If JOBZ = 'S', LWORK >=   mn*mn + 3*mn.
              If JOBZ = 'A', LWORK >=   mn*mn + 2*mn + mx.
              These are not tight minimums in all cases; see comments inside code.
              For good performance, LWORK should generally be larger;
              a query is recommended.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
              Let mx = max(M,N) and mn = min(M,N).
              If JOBZ = 'N',    LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
              else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
              else              LRWORK >= max( 5*mn*mn + 5*mn,
                                               2*mx*mn + 2*mn*mn + mn ).

    IWORK

              IWORK is INTEGER array, dimension (8*min(M,N))

    INFO

              INFO is INTEGER
              <  0:  if INFO = -i, the i-th argument had an illegal value.
              = -4:  if A had a NAN entry.
              >  0:  The updating process of DBDSDC did not converge.
              =  0:  successful exit.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 225 of file zgesdd.f.

  subroutine zgesvd (character jobu, character jobvt, 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( ldvt, * ) vt, integer ldvt, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZGESVD computes the singular value decomposition (SVD) for GE matrices

Purpose:

     ZGESVD computes the singular value decomposition (SVD) of a complex
     M-by-N matrix A, optionally computing the left and/or right singular
     vectors. The SVD is written
          A = U * SIGMA * conjugate-transpose(V)
     where SIGMA is an M-by-N matrix which is zero except for its
     min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
     V is an N-by-N unitary matrix.  The diagonal elements of SIGMA
     are the singular values of A; they are real and non-negative, and
     are returned in descending order.  The first min(m,n) columns of
     U and V are the left and right singular vectors of A.
     Note that the routine returns V**H, not V.

Parameters
    JOBU

              JOBU is CHARACTER*1
              Specifies options for computing all or part of the matrix U:
              = 'A':  all M columns of U are returned in array U:
              = 'S':  the first min(m,n) columns of U (the left singular
                      vectors) are returned in the array U;
              = 'O':  the first min(m,n) columns of U (the left singular
                      vectors) are overwritten on the array A;
              = 'N':  no columns of U (no left singular vectors) are
                      computed.

    JOBVT

              JOBVT is CHARACTER*1
              Specifies options for computing all or part of the matrix
              V**H:
              = 'A':  all N rows of V**H are returned in the array VT;
              = 'S':  the first min(m,n) rows of V**H (the right singular
                      vectors) are returned in the array VT;
              = 'O':  the first min(m,n) rows of V**H (the right singular
                      vectors) are overwritten on the array A;
              = 'N':  no rows of V**H (no right singular vectors) are
                      computed.
              JOBVT and JOBU cannot both be 'O'.

    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.  N >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit,
              if JOBU = 'O',  A is overwritten with the first min(m,n)
                              columns of U (the left singular vectors,
                              stored columnwise);
              if JOBVT = 'O', A is overwritten with the first min(m,n)
                              rows of V**H (the right singular vectors,
                              stored rowwise);
              if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
                              are destroyed.

    LDA

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

    S

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

    U

              U is COMPLEX*16 array, dimension (LDU,UCOL)
              (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
              If JOBU = 'A', U contains the M-by-M unitary matrix U;
              if JOBU = 'S', U contains the first min(m,n) columns of U
              (the left singular vectors, stored columnwise);
              if JOBU = 'N' or 'O', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1; if
              JOBU = 'S' or 'A', LDU >= M.

    VT

              VT is COMPLEX*16 array, dimension (LDVT,N)
              If JOBVT = 'A', VT contains the N-by-N unitary matrix
              V**H;
              if JOBVT = 'S', VT contains the first min(m,n) rows of
              V**H (the right singular vectors, stored rowwise);
              if JOBVT = 'N' or 'O', VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1; if
              JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              LWORK >=  MAX(1,2*MIN(M,N)+MAX(M,N)).
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (5*min(M,N))
              On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the
              unconverged superdiagonal elements of an upper bidiagonal
              matrix B whose diagonal is in S (not necessarily sorted).
              B satisfies A = U * B * VT, so it has the same singular
              values as A, and singular vectors related by U and VT.

    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
                    did not converge to zero. See the description of RWORK
                    above for details.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 212 of file zgesvd.f.

  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, enlosed 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.

Definition at line 410 of file zgesvdq.f.

  subroutine zgesvdx (character jobu, character jobvt, character range, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision vl, double precision vu, integer il, integer iu, integer ns, double precision, dimension( * ) s, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldvt, * ) vt, integer ldvt, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGESVDX computes the singular value decomposition (SVD) for GE matrices

Purpose:

      ZGESVDX computes the singular value decomposition (SVD) of a complex
      M-by-N matrix A, optionally computing the left and/or right singular
      vectors. The SVD is written
          A = U * SIGMA * transpose(V)
      where SIGMA is an M-by-N matrix which is zero except for its
      min(m,n) diagonal elements, U is an M-by-M unitary matrix, and
      V is an N-by-N unitary matrix.  The diagonal elements of SIGMA
      are the singular values of A; they are real and non-negative, and
      are returned in descending order.  The first min(m,n) columns of
      U and V are the left and right singular vectors of A.
      ZGESVDX uses an eigenvalue problem for obtaining the SVD, which
      allows for the computation of a subset of singular values and
      vectors. See DBDSVDX for details.
      Note that the routine returns V**T, not V.

Parameters
    JOBU

              JOBU is CHARACTER*1
              Specifies options for computing all or part of the matrix U:
              = 'V':  the first min(m,n) columns of U (the left singular
                      vectors) or as specified by RANGE are returned in
                      the array U;
              = 'N':  no columns of U (no left singular vectors) are
                      computed.

    JOBVT

              JOBVT is CHARACTER*1
               Specifies options for computing all or part of the matrix
               V**T:
               = 'V':  the first min(m,n) rows of V**T (the right singular
                       vectors) or as specified by RANGE are returned in
                       the array VT;
               = 'N':  no rows of V**T (no right singular vectors) are
                       computed.

    RANGE

              RANGE is CHARACTER*1
              = 'A': all singular values will be found.
              = 'V': all singular values in the half-open interval (VL,VU]
                     will be found.
              = 'I': the IL-th through IU-th singular values will be found.

    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.  N >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the contents of A are destroyed.

    LDA

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

    VL

              VL is DOUBLE PRECISION
              If RANGE='V', the lower bound of the interval to
              be searched for singular values. VU > VL.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is DOUBLE PRECISION
              If RANGE='V', the upper bound of the interval to
              be searched for singular values. VU > VL.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest singular value to be returned.
              1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
              Not referenced if RANGE = 'A' or 'V'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest singular value to be returned.
              1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
              Not referenced if RANGE = 'A' or 'V'.

    NS

              NS is INTEGER
              The total number of singular values found,
              0 <= NS <= min(M,N).
              If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.

    S

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

    U

              U is COMPLEX*16 array, dimension (LDU,UCOL)
              If JOBU = 'V', U contains columns of U (the left singular
              vectors, stored columnwise) as specified by RANGE; if
              JOBU = 'N', U is not referenced.
              Note: The user must ensure that UCOL >= NS; if RANGE = 'V',
              the exact value of NS is not known in advance and an upper
              bound must be used.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1; if
              JOBU = 'V', LDU >= M.

    VT

              VT is COMPLEX*16 array, dimension (LDVT,N)
              If JOBVT = 'V', VT contains the rows of V**T (the right singular
              vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N',
              VT is not referenced.
              Note: The user must ensure that LDVT >= NS; if RANGE = 'V',
              the exact value of NS is not known in advance and an upper
              bound must be used.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1; if
              JOBVT = 'V', LDVT >= NS (see above).

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see
              comments inside the code):
                 - PATH 1  (M much larger than N)
                 - PATH 1t (N much larger than M)
              LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths.
              For good performance, LWORK should generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
              LRWORK >= MIN(M,N)*(MIN(M,N)*2+15*MIN(M,N)).

    IWORK

              IWORK is INTEGER array, dimension (12*MIN(M,N))
              If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0,
              then IWORK contains the indices of the eigenvectors that failed
              to converge in DBDSVDX/DSTEVX.

    INFO

         INFO is INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an illegal value
               > 0:  if INFO = i, then i eigenvectors failed to converge
                     in DBDSVDX/DSTEVX.
                     if INFO = N*2 + 1, an internal error occurred in
                     DBDSVDX

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 267 of file zgesvdx.f.

  subroutine zggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

     ZGGSVD3 computes the generalized singular value decomposition (GSVD)
     of an M-by-N complex matrix A and P-by-N complex matrix B:
           U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )
     where U, V and Q are unitary matrices.
     Let K+L = the effective numerical rank of the
     matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
     triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) 'diagonal'
     matrices and of the following structures, respectively:
     If M-K-L >= 0,
                         K  L
            D1 =     K ( I  0 )
                     L ( 0  C )
                 M-K-L ( 0  0 )
                       K  L
            D2 =   L ( 0  S )
                 P-L ( 0  0 )
                     N-K-L  K    L
       ( 0 R ) = K (  0   R11  R12 )
                 L (  0    0   R22 )
     where
       C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
       S = diag( BETA(K+1),  ... , BETA(K+L) ),
       C**2 + S**2 = I.
       R is stored in A(1:K+L,N-K-L+1:N) on exit.
     If M-K-L < 0,
                       K M-K K+L-M
            D1 =   K ( I  0    0   )
                 M-K ( 0  C    0   )
                         K M-K K+L-M
            D2 =   M-K ( 0  S    0  )
                 K+L-M ( 0  0    I  )
                   P-L ( 0  0    0  )
                        N-K-L  K   M-K  K+L-M
       ( 0 R ) =     K ( 0    R11  R12  R13  )
                   M-K ( 0     0   R22  R23  )
                 K+L-M ( 0     0    0   R33  )
     where
       C = diag( ALPHA(K+1), ... , ALPHA(M) ),
       S = diag( BETA(K+1),  ... , BETA(M) ),
       C**2 + S**2 = I.
       (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
       ( 0  R22 R23 )
       in B(M-K+1:L,N+M-K-L+1:N) on exit.
     The routine computes C, S, R, and optionally the unitary
     transformation matrices U, V and Q.
     In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
     A and B implicitly gives the SVD of A*inv(B):
                          A*inv(B) = U*(D1*inv(D2))*V**H.
     If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
     equal to the CS decomposition of A and B. Furthermore, the GSVD can
     be used to derive the solution of the eigenvalue problem:
                          A**H*A x = lambda* B**H*B x.
     In some literature, the GSVD of A and B is presented in the form
                      U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
     where U and V are orthogonal and X is nonsingular, and D1 and D2 are
     ``diagonal''.  The former GSVD form can be converted to the latter
     form by taking the nonsingular matrix X as
                           X = Q*(  I   0    )
                                 (  0 inv(R) )

Parameters
    JOBU

              JOBU is CHARACTER*1
              = 'U':  Unitary matrix U is computed;
              = 'N':  U is not computed.

    JOBV

              JOBV is CHARACTER*1
              = 'V':  Unitary matrix V is computed;
              = 'N':  V is not computed.

    JOBQ

              JOBQ is CHARACTER*1
              = 'Q':  Unitary matrix Q is computed;
              = 'N':  Q is not computed.

    M

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

    N

              N is INTEGER
              The number of columns of the matrices A and B.  N >= 0.

    P

              P is INTEGER
              The number of rows of the matrix B.  P >= 0.

    K

              K is INTEGER

    L

              L is INTEGER
              On exit, K and L specify the dimension of the subblocks
              described in Purpose.
              K + L = effective numerical rank of (A**H,B**H)**H.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, A contains the triangular matrix R, or part of R.
              See Purpose for details.

    LDA

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

    B

              B is COMPLEX*16 array, dimension (LDB,N)
              On entry, the P-by-N matrix B.
              On exit, B contains part of the triangular matrix R if
              M-K-L < 0.  See Purpose for details.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,P).

    ALPHA

              ALPHA is DOUBLE PRECISION array, dimension (N)

    BETA

              BETA is DOUBLE PRECISION array, dimension (N)
              On exit, ALPHA and BETA contain the generalized singular
              value pairs of A and B;
                ALPHA(1:K) = 1,
                BETA(1:K)  = 0,
              and if M-K-L >= 0,
                ALPHA(K+1:K+L) = C,
                BETA(K+1:K+L)  = S,
              or if M-K-L < 0,
                ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
                BETA(K+1:M) =S, BETA(M+1:K+L) =1
              and
                ALPHA(K+L+1:N) = 0
                BETA(K+L+1:N)  = 0

    U

              U is COMPLEX*16 array, dimension (LDU,M)
              If JOBU = 'U', U contains the M-by-M unitary matrix U.
              If JOBU = 'N', U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U. LDU >= max(1,M) if
              JOBU = 'U'; LDU >= 1 otherwise.

    V

              V is COMPLEX*16 array, dimension (LDV,P)
              If JOBV = 'V', V contains the P-by-P unitary matrix V.
              If JOBV = 'N', V is not referenced.

    LDV

              LDV is INTEGER
              The leading dimension of the array V. LDV >= max(1,P) if
              JOBV = 'V'; LDV >= 1 otherwise.

    Q

              Q is COMPLEX*16 array, dimension (LDQ,N)
              If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
              If JOBQ = 'N', Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q. LDQ >= max(1,N) if
              JOBQ = 'Q'; LDQ >= 1 otherwise.

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (2*N)

    IWORK

              IWORK is INTEGER array, dimension (N)
              On exit, IWORK stores the sorting information. More
              precisely, the following loop will sort ALPHA
                 for I = K+1, min(M,K+L)
                     swap ALPHA(I) and ALPHA(IWORK(I))
                 endfor
              such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, the Jacobi-type procedure failed to
                    converge.  For further details, see subroutine ZTGSJA.

Internal Parameters:

      TOLA    DOUBLE PRECISION
      TOLB    DOUBLE PRECISION
              TOLA and TOLB are the thresholds to determine the effective
              rank of (A**H,B**H)**H. Generally, they are set to
                       TOLA = MAX(M,N)*norm(A)*MACHEPS,
                       TOLB = MAX(P,N)*norm(B)*MACHEPS.
              The size of TOLA and TOLB may affect the size of backward
              errors of the decomposition.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:
    ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

Definition at line 350 of file zggsvd3.f.

Author

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