doubleGEsing(3)

NAME
doubleGEsing

SYNOPSIS

  Functions
subroutine dgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
DGEJSV
subroutine dgesdd (JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)
DGESDD
subroutine dgesvd (JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO)
DGESVD computes the singular value decomposition (SVD) for GE matrices
subroutine dgesvdx (JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU, IL, IU, NS, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)
DGESVDX computes the singular value decomposition (SVD) for GE matrices
subroutine dggsvd3 (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, IWORK, INFO)
DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Detailed Description

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

Function Documentation

  subroutine dgejsv (character*1 JOBA, character*1 JOBU, character*1 JOBV, character*1 JOBR, character*1 JOBT, character*1 JOBP, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( n ) SVA, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( lwork ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

DGEJSV

Purpose:

     DGEJSV computes the singular value decomposition (SVD) of a real M-by-N
     matrix [A], where M >= N. The SVD of [A] is written as
                  [A] = [U] * [SIGMA] * [V]^t,
     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) 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. 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.
     DGEJSV can sometimes compute tiny singular values and their singular vectors much
     more accurately than other SVD routines, see below under Further Details.

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=D*B. If A has heavily weighted
                 rows, then using this condition number gives too pessimistic
                 error bound.
           = 'A': Small singular values are the noise and 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^t 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. This option is
                 allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
           = '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 DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
            the licence to kill columns of A whose norm in c*A is less than
            DSQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
            where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('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 DGESVJ.
           = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]
                 (roughly, as described above). This option is recommended.
                                                ~~~~~~~~~~~~~~~~~~~~~~~~~~~
            For computing the singular values in the FULL range [SFMIN,BIG]
            use DGESVJ.

    JOBT

              JOBT is CHARACTER*1
            If the matrix is square then the procedure may determine to use
            transposed A if A^t seems to be better with respect to convergence.
            If the matrix is not square, JOBT is ignored. This is subject to
            changes in the future.
            The decision is based on two values of entropy over the adjoint
            orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
           = 'T': transpose if entropy test indicates possibly faster
            convergence of Jacobi process if A^t is taken as input. If A is
            replaced with A^t, then the row pivoting is included automatically.
           = 'N': do not speculate.
            This option 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.

    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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
                             then U is used as workspace if the procedure
                             replaces A with A^t. In that case, [V] is computed
                             in U as left singular vectors of A^t 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 DOUBLE PRECISION 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.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
                             then V is used as workspace if the pprocedure
                             replaces A with A^t. In that case, [U] is computed
                             in V as right singular vectors of A^t 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.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)
              On exit, if N.GT.0 .AND. M.GT.0 (else not referenced),
              WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
                        that SCALE*SVA(1:N) are the computed singular values
                        of A. (See the description of SVA().)
              WORK(2) = See the description of WORK(1).
              WORK(3) = SCONDA is an estimate for the condition number of
                        column equilibrated A. (If JOBA .EQ. 'E' or 'G')
                        SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
                        It is computed using DPOCON. 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 provied for
              a developer/implementer who is familiar with the details of
              the method.
              WORK(4) = an estimate of the scaled condition number of the
                        triangular factor in the first QR factorization.
              WORK(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 .EQ. 'T'.
              They are provided for a developer/implementer who is familiar
              with the details of the method.
              WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
                        of diag(A^t*A) / Trace(A^t*A) taken as point in the
                        probability simplex.
              WORK(7) = the entropy of A*A^t.

    LWORK

              LWORK is INTEGER
              Length of WORK to confirm proper allocation of work space.
              LWORK depends on the job:
              If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
                -> .. no scaled condition estimate required (JOBE.EQ.'N'):
                   LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
                   ->> For optimal performance (blocked code) the optimal value
                   is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
                   block size for DGEQP3 and DGEQRF.
                   In general, optimal LWORK is computed as
                   LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
                -> .. an estimate of the scaled condition number of A is
                   required (JOBA='E', 'G'). In this case, LWORK is the maximum
                   of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
                   ->> For optimal performance (blocked code) the optimal value
                   is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
                   In general, the optimal length LWORK is computed as
                   LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
                                                         N+N*N+LWORK(DPOCON),7).
              If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
                -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
                -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
                   where NB is the optimal block size for DGEQP3, DGEQRF, DGELQF,
                   DORMLQ. In general, the optimal length LWORK is computed as
                   LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
                           N+LWORK(DGELQF), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
              If SIGMA and the left singular vectors are needed
                -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
                -> For optimal performance:
                   if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
                   if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
                   where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
                   In general, the optimal length LWORK is computed as
                   LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
                            2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
                   Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or
                   M*NB (for JOBU.EQ.'F').
              If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
                -> if JOBV.EQ.'V'
                   the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
                -> if JOBV.EQ.'J' the minimal requirement is
                   LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
                -> For optimal performance, LWORK should be additionally
                   larger than N+M*NB, where NB is the optimal block size
                   for DORMQR.

    IWORK

              IWORK is INTEGER array, dimension (M+3*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).EQ.1 then some of the column norms of A
                         were denormalized floats. The requested high accuracy
                         is not warranted by the data.

    INFO

              INFO is INTEGER
               < 0  : if INFO = -i, then the i-th argument had an illegal value.
               = 0 :  successful exit;
               > 0 :  DGEJSV  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.

Date:
    June 2016

Further Details:

      DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3,
      DGEQRF, and DGELQF 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 DGEJSV 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 (DGEJSV) 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 (DGESVJ) is
      left to the implementer on a particular machine.
         The rank revealing QR factorization (in this code: DGEQP3) should be
      implemented as in [3]. We have a new version of DGEQP3 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 DGEJSV 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 DGEJSV uses only the simplest, naive data movement.

Contributors:
    Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

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.

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

Definition at line 478 of file dgejsv.f.

  subroutine dgesdd (character JOBZ, 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( ldvt, * ) VT, integer LDVT, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

DGESDD

Purpose:

     DGESDD computes the singular value decomposition (SVD) of a real
     M-by-N matrix A, optionally computing the left and right singular
     vectors.  If singular vectors are desired, it uses a
     divide-and-conquer algorithm.
     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 orthogonal matrix, and
     V is an N-by-N orthogonal 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**T, 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**T 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**T are returned in the arrays U
                      and VT;
              = 'O':  If M >= N, the first N columns of U are overwritten
                      on the array A and all rows of V**T are returned in
                      the array VT;
                      otherwise, all columns of U are returned in the
                      array U and the first M rows of V**T are overwritten
                      in the array A;
              = 'N':  no columns of U or rows of V**T 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 DOUBLE PRECISION 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**T (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 DOUBLE PRECISION 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
              orthogonal 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 DOUBLE PRECISION array, dimension (LDVT,N)
              If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
              N-by-N orthogonal matrix V**T;
              if JOBZ = 'S', VT contains the first min(M,N) rows of
              V**T (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 DOUBLE PRECISION 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 >= 3*mn + max( mx, 7*mn ).
              If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
              If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
              If JOBZ = 'A', LWORK >= 4*mn*mn + 6*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.

    IWORK

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

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  DBDSDC did not converge, updating process failed.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    June 2016

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

Definition at line 220 of file dgesdd.f.

  subroutine dgesvd (character JOBU, character JOBVT, 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( ldvt, * ) VT, integer LDVT, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

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

Purpose:

     DGESVD computes the singular value decomposition (SVD) of a real
     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 orthogonal matrix, and
     V is an N-by-N orthogonal 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**T, 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**T:
              = 'A':  all N rows of V**T are returned in the array VT;
              = 'S':  the first min(m,n) rows of V**T (the right singular
                      vectors) are returned in the array VT;
              = 'O':  the first min(m,n) rows of V**T (the right singular
                      vectors) are overwritten on the array A;
              = 'N':  no rows of V**T (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 DOUBLE PRECISION 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**T (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 DOUBLE PRECISION 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 orthogonal 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 DOUBLE PRECISION array, dimension (LDVT,N)
              If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
              V**T;
              if JOBVT = 'S', VT contains the first min(m,n) rows of
              V**T (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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
              if INFO > 0, WORK(2:MIN(M,N)) 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.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
                 - PATH 1  (M much larger than N, JOBU='N')
                 - PATH 1t (N much larger than M, JOBVT='N')
              LWORK >= MAX(1,3*MIN(M,N) + MAX(M,N),5*MIN(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.

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

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    April 2012

Definition at line 213 of file dgesvd.f.

  subroutine dgesvdx (character JOBU, character JOBVT, character RANGE, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision VL, double precision VU, integer IL, integer IU, integer NS, double precision, dimension( * ) S, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldvt, * ) VT, integer LDVT, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

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

Purpose:

      DGESVDX computes the singular value decomposition (SVD) of a real
      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 orthogonal matrix, and
      V is an N-by-N orthogonal 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.
      DGESVDX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.

    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.

Date:
    June 2016

Definition at line 265 of file dgesvdx.f.

  subroutine dggsvd3 (character JOBU, character JOBV, character JOBQ, integer M, integer N, integer P, integer K, integer L, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) ALPHA, double precision, dimension( * ) BETA, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

     DGGSVD3 computes the generalized singular value decomposition (GSVD)
     of an M-by-N real matrix A and P-by-N real matrix B:
           U**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )
     where U, V and Q are orthogonal matrices.
     Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
     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 orthogonal
     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**T.
     If ( A**T,B**T)**T  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**T*A x = lambda* B**T*B x.
     In some literature, the GSVD of A and B is presented in the form
                      U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
     where U and V are orthogonal and X is nonsingular, 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':  Orthogonal matrix U is computed;
              = 'N':  U is not computed.

    JOBV

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

    JOBQ

              JOBQ is CHARACTER*1
              = 'Q':  Orthogonal 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**T,B**T)**T.

    A

              A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the P-by-N matrix B.
              On exit, B contains 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 DOUBLE PRECISION array, dimension (LDU,M)
              If JOBU = 'U', U contains the M-by-M orthogonal 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 DOUBLE PRECISION array, dimension (LDV,P)
              If JOBV = 'V', V contains the P-by-P orthogonal 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 DOUBLE PRECISION array, dimension (LDQ,N)
              If JOBQ = 'Q', Q contains the N-by-N orthogonal 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 DOUBLE PRECISION 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.

    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 DTGSJA.

Internal Parameters:

      TOLA    DOUBLE PRECISION
      TOLB    DOUBLE PRECISION
              TOLA and TOLB are the thresholds to determine the effective
              rank of (A**T,B**T)**T. 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.

Date:
    August 2015

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

Further Details:
    DGGSVD3 replaces the deprecated subroutine DGGSVD.

Definition at line 351 of file dggsvd3.f.

Author

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