SRC/sgesvj.f(3)

NAME
SRC/sgesvj.f

SYNOPSIS

  Functions/Subroutines
subroutine sgesvj (joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, work, lwork, info)
SGESVJ

Function/Subroutine Documentation

  subroutine sgesvj (character*1 joba, character*1 jobu, character*1 jobv, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( n ) sva, integer mv, real, dimension( ldv, * ) v, integer ldv, real, dimension( lwork ) work, integer lwork, integer info)

SGESVJ

Purpose:

     SGESVJ computes the singular value decomposition (SVD) of a real
     M-by-N matrix A, where M >= N. The SVD of A is written as
                                        [++]   [xx]   [x0]   [xx]
                  A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]
                                        [++]   [xx]
     where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
     matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
     of SIGMA are the singular values of A. The columns of U and V are the
     left and the right singular vectors of A, respectively.
     SGESVJ 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 structure of A.
              = 'L': The input matrix A is lower triangular;
              = 'U': The input matrix A is upper triangular;
              = 'G': The input matrix A is general M-by-N matrix, M >= N.

    JOBU

              JOBU is CHARACTER*1
              Specifies whether to compute the left singular vectors
              (columns of U):
              = 'U': The left singular vectors corresponding to the nonzero
                     singular values are computed and returned in the leading
                     columns of A. See more details in the description of A.
                     The default numerical orthogonality threshold is set to
                     approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
              = 'C': Analogous to JOBU='U', except that user can control the
                     level of numerical orthogonality of the computed left
                     singular vectors. TOL can be set to TOL = CTOL*EPS, where
                     CTOL is given on input in the array WORK.
                     No CTOL smaller than ONE is allowed. CTOL greater
                     than 1 / EPS is meaningless. The option 'C'
                     can be used if M*EPS is satisfactory orthogonality
                     of the computed left singular vectors, so CTOL=M could
                     save few sweeps of Jacobi rotations.
                     See the descriptions of A and WORK(1).
              = 'N': The matrix U is not computed. However, see the
                     description of A.

    JOBV

              JOBV is CHARACTER*1
              Specifies whether to compute the right singular vectors, that
              is, the matrix V:
              = 'V':  the matrix V is computed and returned in the array V
              = 'A':  the Jacobi rotations are applied to the MV-by-N
                      array V. In other words, the right singular vector
                      matrix V is not computed explicitly; instead it is
                      applied to an MV-by-N matrix initially stored in the
                      first MV rows of V.
              = 'N':  the matrix V is not computed and the array V is not
                      referenced

    M

              M is INTEGER
              The number of rows of the input matrix A. 1/SLAMCH('E') > 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.
              On exit,
              If JOBU = 'U' .OR. JOBU = 'C':
                     If INFO = 0:
                     RANKA orthonormal columns of U are returned in the
                     leading RANKA columns of the array A. Here RANKA <= N
                     is the number of computed singular values of A that are
                     above the underflow threshold SLAMCH('S'). The singular
                     vectors corresponding to underflowed or zero singular
                     values are not computed. The value of RANKA is returned
                     in the array WORK as RANKA=NINT(WORK(2)). Also see the
                     descriptions of SVA and WORK. The computed columns of U
                     are mutually numerically orthogonal up to approximately
                     TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
                     see the description of JOBU.
                     If INFO > 0,
                     the procedure SGESVJ did not converge in the given number
                     of iterations (sweeps). In that case, the computed
                     columns of U may not be orthogonal up to TOL. The output
                     U (stored in A), SIGMA (given by the computed singular
                     values in SVA(1:N)) and V is still a decomposition of the
                     input matrix A in the sense that the residual
                     ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
              If JOBU = 'N':
                     If INFO = 0:
                     Note that the left singular vectors are 'for free' in the
                     one-sided Jacobi SVD algorithm. However, if only the
                     singular values are needed, the level of numerical
                     orthogonality of U is not an issue and iterations are
                     stopped when the columns of the iterated matrix are
                     numerically orthogonal up to approximately M*EPS. Thus,
                     on exit, A contains the columns of U scaled with the
                     corresponding singular values.
                     If INFO > 0:
                     the procedure SGESVJ did not converge in the given number
                     of iterations (sweeps).

    LDA

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

    SVA

              SVA is REAL array, dimension (N)
              On exit,
              If INFO = 0 :
              depending on the value SCALE = WORK(1), we have:
                     If SCALE = ONE:
                     SVA(1:N) contains the computed singular values of A.
                     During the computation SVA contains the Euclidean column
                     norms of the iterated matrices in the array A.
                     If SCALE .NE. ONE:
                     The singular values of A are SCALE*SVA(1:N), and this
                     factored representation is due to the fact that some of the
                     singular values of A might underflow or overflow.
              If INFO > 0 :
              the procedure SGESVJ did not converge in the given number of
              iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.

    MV

              MV is INTEGER
              If JOBV = 'A', then the product of Jacobi rotations in SGESVJ
              is applied to the first MV rows of V. See the description of JOBV.

    V

              V is REAL array, dimension (LDV,N)
              If JOBV = 'V', then V contains on exit the N-by-N matrix of
                             the right singular vectors;
              If JOBV = 'A', then V contains the product of the computed right
                             singular vector matrix and the initial matrix in
                             the array V.
              If JOBV = 'N', then V is not referenced.

    LDV

              LDV is INTEGER
              The leading dimension of the array V, LDV >= 1.
              If JOBV = 'V', then LDV >= max(1,N).
              If JOBV = 'A', then LDV >= max(1,MV) .

    WORK

              WORK is REAL array, dimension (LWORK)
              On entry,
              If JOBU = 'C' :
              WORK(1) = CTOL, where CTOL defines the threshold for convergence.
                        The process stops if all columns of A are mutually
                        orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
                        It is required that CTOL >= ONE, i.e. it is not
                        allowed to force the routine to obtain orthogonality
                        below EPSILON.
              On exit,
              WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
                        are the computed singular vcalues of A.
                        (See description of SVA().)
              WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
                        singular values.
              WORK(3) = NINT(WORK(3)) is the number of the computed singular
                        values that are larger than the underflow threshold.
              WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
                        rotations needed for numerical convergence.
              WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
                        This is useful information in cases when SGESVJ did
                        not converge, as it can be used to estimate whether
                        the output is still useful and for post festum analysis.
              WORK(6) = the largest absolute value over all sines of the
                        Jacobi rotation angles in the last sweep. It can be
                        useful for a post festum analysis.

    LWORK

              LWORK is INTEGER
             length of WORK, WORK >= MAX(6,M+N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, then the i-th argument had an illegal value
              > 0:  SGESVJ did not converge in the maximal allowed number (30)
                    of sweeps. The output may still be useful. See the
                    description of WORK.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:
    The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane rotations. The rotations are implemented as fast scaled rotations of Anda and Park [1]. In the case of underflow of the Jacobi angle, a modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses column interchanges of de Rijk [2]. The relative accuracy of the computed singular values and the accuracy of the computed singular vectors (in angle metric) is as guaranteed by the theory of Demmel and Veselic [3]. The condition number that determines the accuracy in the full rank case is essentially min_{D=diag} kappa(A*D), where kappa(.) is the spectral condition number. The best performance of this Jacobi SVD procedure is achieved if used in an accelerated version of Drmac and Veselic [5,6], and it is the kernel routine in the SIGMA library [7]. Some tuning parameters (marked with [TP]) are available for the implementer.
     The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits.

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

References:
    [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
     SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
     [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer.
     SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
     [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
     [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic.
     SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
     [5] 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.
     [6] 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.
     [7] 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 test examples and comments to drmac@math.hr. Thank you.

Definition at line 321 of file sgesvj.f.

Author

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