lasda(3)

NAME
lasda - lasda: D&C step: top level solver

SYNOPSIS

  Functions

subroutine dlasda (icompq, smlsiz, n, sqre, d, e, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info)
DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.
subroutine slasda (icompq, smlsiz, n, sqre, d, e, u, ldu, vt, k, difl, difr, z, poles, givptr, givcol, ldgcol, perm, givnum, c, s, work, iwork, info)
SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.

Detailed Description

Function Documentation

  subroutine dlasda (integer icompq, integer smlsiz, integer n, integer sqre, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldu, * ) vt, integer, dimension( * ) k, double precision, dimension( ldu, * ) difl, double precision, dimension( ldu, * ) difr, double precision, dimension( ldu, * ) z, double precision, dimension( ldu, * ) poles, integer, dimension( * ) givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, integer, dimension( ldgcol, * ) perm, double precision, dimension( ldu, * ) givnum, double precision, dimension( * ) c, double precision, dimension( * ) s, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.

Purpose:

    !>
    !> Using a divide and conquer approach, DLASDA computes the singular
    !> value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
    !> B with diagonal D and offdiagonal E, where M = N + SQRE. The
    !> algorithm computes the singular values in the SVD B = U * S * VT.
    !> The orthogonal matrices U and VT are optionally computed in
    !> compact form.
    !>
    !> A related subroutine, DLASD0, computes the singular values and
    !> the singular vectors in explicit form.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed
    !>         in compact form, as follows
    !>         = 0: Compute singular values only.
    !>         = 1: Compute singular vectors of upper bidiagonal
    !>              matrix in compact form.
    !>

    SMLSIZ

    !>          SMLSIZ is INTEGER
    !>         The maximum size of the subproblems at the bottom of the
    !>         computation tree.
    !>

    N

    !>          N is INTEGER
    !>         The row dimension of the upper bidiagonal matrix. This is
    !>         also the dimension of the main diagonal array D.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         Specifies the column dimension of the bidiagonal matrix.
    !>         = 0: The bidiagonal matrix has column dimension M = N;
    !>         = 1: The bidiagonal matrix has column dimension M = N + 1.
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension ( N )
    !>         On entry D contains the main diagonal of the bidiagonal
    !>         matrix. On exit D, if INFO = 0, contains its singular values.
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension ( M-1 )
    !>         Contains the subdiagonal entries of the bidiagonal matrix.
    !>         On exit, E has been destroyed.
    !>

    U

    !>          U is DOUBLE PRECISION array,
    !>         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
    !>         singular vector matrices of all subproblems at the bottom
    !>         level.
    !>

    LDU

    !>          LDU is INTEGER, LDU = > N.
    !>         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
    !>         GIVNUM, and Z.
    !>

    VT

    !>          VT is DOUBLE PRECISION array,
    !>         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
    !>         singular vector matrices of all subproblems at the bottom
    !>         level.
    !>

    K

    !>          K is INTEGER array,
    !>         dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
    !>         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
    !>         secular equation on the computation tree.
    !>

    DIFL

    !>          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ),
    !>         where NLVL = floor(log_2 (N/SMLSIZ))).
    !>

    DIFR

    !>          DIFR is DOUBLE PRECISION array,
    !>                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
    !>                  dimension ( N ) if ICOMPQ = 0.
    !>         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
    !>         record distances between singular values on the I-th
    !>         level and singular values on the (I -1)-th level, and
    !>         DIFR(1:N, 2 * I ) contains the normalizing factors for
    !>         the right singular vector matrix. See DLASD8 for details.
    !>

    Z

    !>          Z is DOUBLE PRECISION array,
    !>                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
    !>                  dimension ( N ) if ICOMPQ = 0.
    !>         The first K elements of Z(1, I) contain the components of
    !>         the deflation-adjusted updating row vector for subproblems
    !>         on the I-th level.
    !>

    POLES

    !>          POLES is DOUBLE PRECISION array,
    !>         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
    !>         POLES(1, 2*I) contain  the new and old singular values
    !>         involved in the secular equations on the I-th level.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER array,
    !>         dimension ( N ) if ICOMPQ = 1, and not referenced if
    !>         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
    !>         the number of Givens rotations performed on the I-th
    !>         problem on the computation tree.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array,
    !>         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
    !>         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
    !>         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
    !>         of Givens rotations performed on the I-th level on the
    !>         computation tree.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER, LDGCOL = > N.
    !>         The leading dimension of arrays GIVCOL and PERM.
    !>

    PERM

    !>          PERM is INTEGER array,
    !>         dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
    !>         permutations done on the I-th level of the computation tree.
    !>

    GIVNUM

    !>          GIVNUM is DOUBLE PRECISION array,
    !>         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
    !>         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
    !>         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
    !>         values of Givens rotations performed on the I-th level on
    !>         the computation tree.
    !>

    C

    !>          C is DOUBLE PRECISION array,
    !>         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
    !>         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
    !>         C( I ) contains the C-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension ( N ) if
    !>         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
    !>         and the I-th subproblem is not square, on exit, S( I )
    !>         contains the S-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension
    !>         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (7*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = 1, a singular value did not converge
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

  subroutine slasda (integer icompq, integer smlsiz, integer n, integer sqre, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) vt, integer, dimension( * ) k, real, dimension( ldu, * ) difl, real, dimension( ldu, * ) difr, real, dimension( ldu, * ) z, real, dimension( ldu, * ) poles, integer, dimension( * ) givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, integer, dimension( ldgcol, * ) perm, real, dimension( ldu, * ) givnum, real, dimension( * ) c, real, dimension( * ) s, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc.

Purpose:

    !>
    !> Using a divide and conquer approach, SLASDA computes the singular
    !> value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
    !> B with diagonal D and offdiagonal E, where M = N + SQRE. The
    !> algorithm computes the singular values in the SVD B = U * S * VT.
    !> The orthogonal matrices U and VT are optionally computed in
    !> compact form.
    !>
    !> A related subroutine, SLASD0, computes the singular values and
    !> the singular vectors in explicit form.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed
    !>         in compact form, as follows
    !>         = 0: Compute singular values only.
    !>         = 1: Compute singular vectors of upper bidiagonal
    !>              matrix in compact form.
    !>

    SMLSIZ

    !>          SMLSIZ is INTEGER
    !>         The maximum size of the subproblems at the bottom of the
    !>         computation tree.
    !>

    N

    !>          N is INTEGER
    !>         The row dimension of the upper bidiagonal matrix. This is
    !>         also the dimension of the main diagonal array D.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         Specifies the column dimension of the bidiagonal matrix.
    !>         = 0: The bidiagonal matrix has column dimension M = N;
    !>         = 1: The bidiagonal matrix has column dimension M = N + 1.
    !>

    D

    !>          D is REAL array, dimension ( N )
    !>         On entry D contains the main diagonal of the bidiagonal
    !>         matrix. On exit D, if INFO = 0, contains its singular values.
    !>

    E

    !>          E is REAL array, dimension ( M-1 )
    !>         Contains the subdiagonal entries of the bidiagonal matrix.
    !>         On exit, E has been destroyed.
    !>

    U

    !>          U is REAL array,
    !>         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
    !>         singular vector matrices of all subproblems at the bottom
    !>         level.
    !>

    LDU

    !>          LDU is INTEGER, LDU = > N.
    !>         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
    !>         GIVNUM, and Z.
    !>

    VT

    !>          VT is REAL array,
    !>         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
    !>         singular vector matrices of all subproblems at the bottom
    !>         level.
    !>

    K

    !>          K is INTEGER array, dimension ( N )
    !>         if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
    !>         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
    !>         secular equation on the computation tree.
    !>

    DIFL

    !>          DIFL is REAL array, dimension ( LDU, NLVL ),
    !>         where NLVL = floor(log_2 (N/SMLSIZ))).
    !>

    DIFR

    !>          DIFR is REAL array,
    !>                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
    !>                  dimension ( N ) if ICOMPQ = 0.
    !>         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
    !>         record distances between singular values on the I-th
    !>         level and singular values on the (I -1)-th level, and
    !>         DIFR(1:N, 2 * I ) contains the normalizing factors for
    !>         the right singular vector matrix. See SLASD8 for details.
    !>

    Z

    !>          Z is REAL array,
    !>                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and
    !>                  dimension ( N ) if ICOMPQ = 0.
    !>         The first K elements of Z(1, I) contain the components of
    !>         the deflation-adjusted updating row vector for subproblems
    !>         on the I-th level.
    !>

    POLES

    !>          POLES is REAL array,
    !>         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
    !>         POLES(1, 2*I) contain  the new and old singular values
    !>         involved in the secular equations on the I-th level.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER array,
    !>         dimension ( N ) if ICOMPQ = 1, and not referenced if
    !>         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
    !>         the number of Givens rotations performed on the I-th
    !>         problem on the computation tree.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array,
    !>         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
    !>         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
    !>         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
    !>         of Givens rotations performed on the I-th level on the
    !>         computation tree.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER, LDGCOL = > N.
    !>         The leading dimension of arrays GIVCOL and PERM.
    !>

    PERM

    !>          PERM is INTEGER array, dimension ( LDGCOL, NLVL )
    !>         if ICOMPQ = 1, and not referenced
    !>         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
    !>         permutations done on the I-th level of the computation tree.
    !>

    GIVNUM

    !>          GIVNUM is REAL array,
    !>         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not
    !>         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
    !>         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
    !>         values of Givens rotations performed on the I-th level on
    !>         the computation tree.
    !>

    C

    !>          C is REAL array,
    !>         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
    !>         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
    !>         C( I ) contains the C-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    S

    !>          S is REAL array, dimension ( N ) if
    !>         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
    !>         and the I-th subproblem is not square, on exit, S( I )
    !>         contains the S-value of a Givens rotation related to
    !>         the right null space of the I-th subproblem.
    !>

    WORK

    !>          WORK is REAL array, dimension
    !>         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (7*N).
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = 1, a singular value did not converge
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Author

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