SRC/sgejsv.f(3)

NAME
SRC/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)

SGEJSV

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  because they are
    !>            below the numerical rank threshold or are denormalized numbers.
    !>

    U

    !>          U is REAL array, dimension ( LDU, N ) or ( LDU, M )
    !>          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 procedure
    !>                         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 (MAX(7,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 SGEQP3 and SGEQRF.
    !>               In general, optimal LWORK is computed as
    !>               LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SGEQRF), 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(SGEQP3),N+LWORK(SGEQRF),
    !>                                                     N+N*N+LWORK(SPOCON),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 SGEQP3, SGEQRF, SGELQF,
    !>               SORMLQ. In general, the optimal length LWORK is computed as
    !>               LWORK >= max(2*M+N,N+LWORK(SGEQP3), N+LWORK(SPOCON),
    !>                       N+LWORK(SGELQF), 2*N+LWORK(SGEQRF), N+LWORK(SORMLQ)).
    !>
    !>          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 SGEQP3, SGEQRF, SORMQR.
    !>               In general, the optimal length LWORK is computed as
    !>               LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SPOCON),
    !>                        2*N+LWORK(SGEQRF), N+LWORK(SORMQR)).
    !>               Here LWORK(SORMQR) 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 SORMQR.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (MAX(3,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.