sgejsv.f(3)

NAME
sgejsv.f

SYNOPSIS

  Functions/Subroutines
subroutine sgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
SGEJSV

Function/Subroutine Documentation

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

Purpose:

     SGEJSV 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.
     SGEJSV 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 SQRT(BIG), BIG=SLAMCH('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=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 SGESVJ.
           = '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 SGESVJ.

    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 REAL 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 REAL 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 REAL 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^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 REAL 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^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 REAL array, dimension (LWORK)
              On exit,
              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 = 'E' or 'G')
                        SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
                        It is computed using SPOCON. 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.
              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 = '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 = 'N', JOBV = 'N' ) and
                -> .. no scaled condition estimate required (JOBE = '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 = '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 = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
                   if JOBU = '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 = 'U') or
                   M*NB (for JOBU = 'F').
              If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
                -> if JOBV = 'V'
                   the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
                -> if JOBV = '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) = 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:  SGEJSV  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:

      SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,
      SGEQRF, and SGELQF 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 SGEJSV 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 (SGEJSV) is best used in this restricted range,
      meaning that singular values of magnitude below ||A||_2 / SLAMCH('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 (SGESVJ) is
      left to the implementer on a particular machine.
         The rank revealing QR factorization (in this code: SGEQP3) should be
      implemented as in [3]. We have a new version of SGEQP3 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 SGEJSV 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 SGEJSV 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 473 of file sgejsv.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.