LAPACK  3.4.2
LAPACK: Linear Algebra PACKage
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double
Collaboration diagram for double:

Functions/Subroutines

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

Detailed Description

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


Function/Subroutine 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

Download DGEJSV + dependencies [TGZ] [ZIP] [TXT]
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.
Parameters:
[in]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 defficient. 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'.
[in]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.
[in]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.
[in]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.
[in]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.
[in]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
[in]M
          M is INTEGER
         The number of rows of the input matrix A.  M >= 0.
[in]N
          N is INTEGER
         The number of columns of the input matrix A. M >= N >= 0.
[in,out]A
          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).
[out]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.
[out]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.
[in]LDU
          LDU is INTEGER
          The leading dimension of the array U,  LDU >= 1.
          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.
[out]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.
[in]LDV
          LDV is INTEGER
          The leading dimension of the array V,  LDV >= 1.
          If JOBV = 'V' or 'J' or 'W', then LDV >= N.
[out]WORK
          WORK is DOUBLE PRECISION array, dimension at least 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.
[in]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, DGELQ,
               DORMLQ. In general, the optimal length LWORK is computed as
               LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), 
                       N+LWORK(DGELQ), 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.
[out]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.
[out]INFO
          INFO is INTEGER
           < 0  : if INFO = -i, then the i-th argument had an illegal value.
           = 0 :  successfull 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:
September 2012
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 defficient cases. It will be available in the SIGMA library [4].
  If M is much larger than N, it is obvious that the inital 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.nosp@m.@mat.nosp@m.h.hr. Thank you.

Definition at line 473 of file dgejsv.f.

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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

Download DGESDD + dependencies [TGZ] [ZIP] [TXT]
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:
[in]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.
[in]M
          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns of the input matrix A.  N >= 0.
[in,out]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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).
[out]S
          S is DOUBLE PRECISION array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).
[out]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.
[in]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.
[out]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.
[in]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).
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= 1.
          If JOBZ = 'N',
            LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)).
          If JOBZ = 'O',
            LWORK >= 3*min(M,N) + 
                     max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
          If JOBZ = 'S' or 'A'
            LWORK >= 3*min(M,N) +
                     max(max(M,N),4*min(M,N)*min(M,N)+3*min(M,N)+max(M,N)).
          For good performance, LWORK should generally be larger.
          If LWORK = -1 but other input arguments are legal, WORK(1)
          returns the optimal LWORK.
[out]IWORK
          IWORK is INTEGER array, dimension (8*min(M,N))
[out]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:
September 2012
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 217 of file dgesdd.f.

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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

Download DGESVD + dependencies [TGZ] [ZIP] [TXT]
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:
[in]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.
[in]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'.
[in]M
          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns of the input matrix A.  N >= 0.
[in,out]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.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).
[out]S
          S is DOUBLE PRECISION array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).
[out]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.
[in]LDU
          LDU is INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBU = 'S' or 'A', LDU >= M.
[out]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.
[in]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).
[out]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.
[in]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.
[out]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 211 of file dgesvd.f.

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