lasd1(3)

NAME
lasd1 - lasd1: D&C step: merge subproblems

SYNOPSIS

  Functions

subroutine dlasd1 (nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info)
DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
subroutine slasd1 (nl, nr, sqre, d, alpha, beta, u, ldu, vt, ldvt, idxq, iwork, work, info)
SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

Detailed Description

Function Documentation

  subroutine dlasd1 (integer nl, integer nr, integer sqre, double precision, dimension( * ) d, double precision alpha, double precision beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * ) idxq, integer, dimension( * ) iwork, double precision, dimension( * ) work, integer info)

DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

Purpose:

    !>
    !> DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
    !> where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.
    !>
    !> A related subroutine DLASD7 handles the case in which the singular
    !> values (and the singular vectors in factored form) are desired.
    !>
    !> DLASD1 computes the SVD as follows:
    !>
    !>               ( D1(in)    0    0       0 )
    !>   B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
    !>               (   0       0   D2(in)   0 )
    !>
    !>     = U(out) * ( D(out) 0) * VT(out)
    !>
    !> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
    !> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
    !> elsewhere; and the entry b is empty if SQRE = 0.
    !>
    !> The left singular vectors of the original matrix are stored in U, and
    !> the transpose of the right singular vectors are stored in VT, and the
    !> singular values are in D.  The algorithm consists of three stages:
    !>
    !>    The first stage consists of deflating the size of the problem
    !>    when there are multiple singular values or when there are zeros in
    !>    the Z vector.  For each such occurrence the dimension of the
    !>    secular equation problem is reduced by one.  This stage is
    !>    performed by the routine DLASD2.
    !>
    !>    The second stage consists of calculating the updated
    !>    singular values. This is done by finding the square roots of the
    !>    roots of the secular equation via the routine DLASD4 (as called
    !>    by DLASD3). This routine also calculates the singular vectors of
    !>    the current problem.
    !>
    !>    The final stage consists of computing the updated singular vectors
    !>    directly using the updated singular values.  The singular vectors
    !>    for the current problem are multiplied with the singular vectors
    !>    from the overall problem.
    !>

Parameters
    NL

    !>          NL is INTEGER
    !>         The row dimension of the upper block.  NL >= 1.
    !>

    NR

    !>          NR is INTEGER
    !>         The row dimension of the lower block.  NR >= 1.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         = 0: the lower block is an NR-by-NR square matrix.
    !>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
    !>
    !>         The bidiagonal matrix has row dimension N = NL + NR + 1,
    !>         and column dimension M = N + SQRE.
    !>

    D

    !>          D is DOUBLE PRECISION array,
    !>                        dimension (N = NL+NR+1).
    !>         On entry D(1:NL,1:NL) contains the singular values of the
    !>         upper block; and D(NL+2:N) contains the singular values of
    !>         the lower block. On exit D(1:N) contains the singular values
    !>         of the modified matrix.
    !>

    ALPHA

    !>          ALPHA is DOUBLE PRECISION
    !>         Contains the diagonal element associated with the added row.
    !>

    BETA

    !>          BETA is DOUBLE PRECISION
    !>         Contains the off-diagonal element associated with the added
    !>         row.
    !>

    U

    !>          U is DOUBLE PRECISION array, dimension(LDU,N)
    !>         On entry U(1:NL, 1:NL) contains the left singular vectors of
    !>         the upper block; U(NL+2:N, NL+2:N) contains the left singular
    !>         vectors of the lower block. On exit U contains the left
    !>         singular vectors of the bidiagonal matrix.
    !>

    LDU

    !>          LDU is INTEGER
    !>         The leading dimension of the array U.  LDU >= max( 1, N ).
    !>

    VT

    !>          VT is DOUBLE PRECISION array, dimension(LDVT,M)
    !>         where M = N + SQRE.
    !>         On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
    !>         vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
    !>         the right singular vectors of the lower block. On exit
    !>         VT**T contains the right singular vectors of the
    !>         bidiagonal matrix.
    !>

    LDVT

    !>          LDVT is INTEGER
    !>         The leading dimension of the array VT.  LDVT >= max( 1, M ).
    !>

    IDXQ

    !>          IDXQ is INTEGER array, dimension(N)
    !>         This contains the permutation which will reintegrate the
    !>         subproblem just solved back into sorted order, i.e.
    !>         D( IDXQ( I = 1, N ) ) will be in ascending order.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension( 4 * N )
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
    !>

    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 slasd1 (integer nl, integer nr, integer sqre, real, dimension( * ) d, real alpha, real beta, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt, integer ldvt, integer, dimension( * ) idxq, integer, dimension( * ) iwork, real, dimension( * ) work, integer info)

SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

Purpose:

    !>
    !> SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
    !> where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.
    !>
    !> A related subroutine SLASD7 handles the case in which the singular
    !> values (and the singular vectors in factored form) are desired.
    !>
    !> SLASD1 computes the SVD as follows:
    !>
    !>               ( D1(in)    0    0       0 )
    !>   B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
    !>               (   0       0   D2(in)   0 )
    !>
    !>     = U(out) * ( D(out) 0) * VT(out)
    !>
    !> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
    !> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
    !> elsewhere; and the entry b is empty if SQRE = 0.
    !>
    !> The left singular vectors of the original matrix are stored in U, and
    !> the transpose of the right singular vectors are stored in VT, and the
    !> singular values are in D.  The algorithm consists of three stages:
    !>
    !>    The first stage consists of deflating the size of the problem
    !>    when there are multiple singular values or when there are zeros in
    !>    the Z vector.  For each such occurrence the dimension of the
    !>    secular equation problem is reduced by one.  This stage is
    !>    performed by the routine SLASD2.
    !>
    !>    The second stage consists of calculating the updated
    !>    singular values. This is done by finding the square roots of the
    !>    roots of the secular equation via the routine SLASD4 (as called
    !>    by SLASD3). This routine also calculates the singular vectors of
    !>    the current problem.
    !>
    !>    The final stage consists of computing the updated singular vectors
    !>    directly using the updated singular values.  The singular vectors
    !>    for the current problem are multiplied with the singular vectors
    !>    from the overall problem.
    !>

Parameters
    NL

    !>          NL is INTEGER
    !>         The row dimension of the upper block.  NL >= 1.
    !>

    NR

    !>          NR is INTEGER
    !>         The row dimension of the lower block.  NR >= 1.
    !>

    SQRE

    !>          SQRE is INTEGER
    !>         = 0: the lower block is an NR-by-NR square matrix.
    !>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
    !>
    !>         The bidiagonal matrix has row dimension N = NL + NR + 1,
    !>         and column dimension M = N + SQRE.
    !>

    D

    !>          D is REAL array, dimension (NL+NR+1).
    !>         N = NL+NR+1
    !>         On entry D(1:NL,1:NL) contains the singular values of the
    !>         upper block; and D(NL+2:N) contains the singular values of
    !>         the lower block. On exit D(1:N) contains the singular values
    !>         of the modified matrix.
    !>

    ALPHA

    !>          ALPHA is REAL
    !>         Contains the diagonal element associated with the added row.
    !>

    BETA

    !>          BETA is REAL
    !>         Contains the off-diagonal element associated with the added
    !>         row.
    !>

    U

    !>          U is REAL array, dimension (LDU,N)
    !>         On entry U(1:NL, 1:NL) contains the left singular vectors of
    !>         the upper block; U(NL+2:N, NL+2:N) contains the left singular
    !>         vectors of the lower block. On exit U contains the left
    !>         singular vectors of the bidiagonal matrix.
    !>

    LDU

    !>          LDU is INTEGER
    !>         The leading dimension of the array U.  LDU >= max( 1, N ).
    !>

    VT

    !>          VT is REAL array, dimension (LDVT,M)
    !>         where M = N + SQRE.
    !>         On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
    !>         vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
    !>         the right singular vectors of the lower block. On exit
    !>         VT**T contains the right singular vectors of the
    !>         bidiagonal matrix.
    !>

    LDVT

    !>          LDVT is INTEGER
    !>         The leading dimension of the array VT.  LDVT >= max( 1, M ).
    !>

    IDXQ

    !>          IDXQ is INTEGER array, dimension (N)
    !>         This contains the permutation which will reintegrate the
    !>         subproblem just solved back into sorted order, i.e.
    !>         D( IDXQ( I = 1, N ) ) will be in ascending order.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (4*N)
    !>

    WORK

    !>          WORK is REAL array, dimension (3*M**2+2*M)
    !>

    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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