lasd6(3)

NAME
lasd6 - lasd6: D&C step: merge subproblems

SYNOPSIS

  Functions

subroutine dlasd6 (icompq, nl, nr, sqre, d, vf, vl, alpha, beta, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, iwork, info)
DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
subroutine slasd6 (icompq, nl, nr, sqre, d, vf, vl, alpha, beta, idxq, perm, givptr, givcol, ldgcol, givnum, ldgnum, poles, difl, difr, z, k, c, s, work, iwork, info)
SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.

Detailed Description

Function Documentation

  subroutine dlasd6 (integer icompq, integer nl, integer nr, integer sqre, double precision, dimension( * ) d, double precision, dimension( * ) vf, double precision, dimension( * ) vl, double precision alpha, double precision beta, integer, dimension( * ) idxq, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, double precision, dimension( ldgnum, * ) givnum, integer ldgnum, double precision, dimension( ldgnum, * ) poles, double precision, dimension( * ) difl, double precision, dimension( * ) difr, double precision, dimension( * ) z, integer k, double precision c, double precision s, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.

Purpose:

    !>
    !> DLASD6 computes the SVD of an updated upper bidiagonal matrix B
    !> obtained by merging two smaller ones by appending a row. This
    !> routine is used only for the problem which requires all singular
    !> values and optionally singular vector matrices in factored form.
    !> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
    !> A related subroutine, DLASD1, handles the case in which all singular
    !> values and singular vectors of the bidiagonal matrix are desired.
    !>
    !> DLASD6 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 singular values of B can be computed using D1, D2, the first
    !> components of all the right singular vectors of the lower block, and
    !> the last components of all the right singular vectors of the upper
    !> block. These components are stored and updated in VF and VL,
    !> respectively, in DLASD6. Hence U and VT are not explicitly
    !> referenced.
    !>
    !> The singular values are stored in D. The algorithm consists of two
    !> stages:
    !>
    !>       The first stage consists of deflating the size of the problem
    !>       when there are multiple singular values or if there is a zero
    !>       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 DLASD7.
    !>
    !>       The second stage consists of calculating the updated
    !>       singular values. This is done by finding the roots of the
    !>       secular equation via the routine DLASD4 (as called by DLASD8).
    !>       This routine also updates VF and VL and computes the distances
    !>       between the updated singular values and the old singular
    !>       values.
    !>
    !> DLASD6 is called from DLASDA.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed in
    !>         factored form:
    !>         = 0: Compute singular values only.
    !>         = 1: Compute singular vectors in factored form as well.
    !>

    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 ( 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.
    !>

    VF

    !>          VF is DOUBLE PRECISION array, dimension ( M )
    !>         On entry, VF(1:NL+1) contains the first components of all
    !>         right singular vectors of the upper block; and VF(NL+2:M)
    !>         contains the first components of all right singular vectors
    !>         of the lower block. On exit, VF contains the first components
    !>         of all right singular vectors of the bidiagonal matrix.
    !>

    VL

    !>          VL is DOUBLE PRECISION array, dimension ( M )
    !>         On entry, VL(1:NL+1) contains the  last components of all
    !>         right singular vectors of the upper block; and VL(NL+2:M)
    !>         contains the last components of all right singular vectors of
    !>         the lower block. On exit, VL contains the last components of
    !>         all right singular vectors of the bidiagonal 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.
    !>

    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.
    !>

    PERM

    !>          PERM is INTEGER array, dimension ( N )
    !>         The permutations (from deflation and sorting) to be applied
    !>         to each block. Not referenced if ICOMPQ = 0.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER
    !>         The number of Givens rotations which took place in this
    !>         subproblem. Not referenced if ICOMPQ = 0.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
    !>         Each pair of numbers indicates a pair of columns to take place
    !>         in a Givens rotation. Not referenced if ICOMPQ = 0.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER
    !>         leading dimension of GIVCOL, must be at least N.
    !>

    GIVNUM

    !>          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
    !>         Each number indicates the C or S value to be used in the
    !>         corresponding Givens rotation. Not referenced if ICOMPQ = 0.
    !>

    LDGNUM

    !>          LDGNUM is INTEGER
    !>         The leading dimension of GIVNUM and POLES, must be at least N.
    !>

    POLES

    !>          POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
    !>         On exit, POLES(1,*) is an array containing the new singular
    !>         values obtained from solving the secular equation, and
    !>         POLES(2,*) is an array containing the poles in the secular
    !>         equation. Not referenced if ICOMPQ = 0.
    !>

    DIFL

    !>          DIFL is DOUBLE PRECISION array, dimension ( N )
    !>         On exit, DIFL(I) is the distance between I-th updated
    !>         (undeflated) singular value and the I-th (undeflated) old
    !>         singular value.
    !>

    DIFR

    !>          DIFR is DOUBLE PRECISION array,
    !>                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
    !>                   dimension ( K ) if ICOMPQ = 0.
    !>          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
    !>          defined and will not be referenced.
    !>
    !>          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
    !>          normalizing factors for the right singular vector matrix.
    !>
    !>         See DLASD8 for details on DIFL and DIFR.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension ( M )
    !>         The first elements of this array contain the components
    !>         of the deflation-adjusted updating row vector.
    !>

    K

    !>          K is INTEGER
    !>         Contains the dimension of the non-deflated matrix,
    !>         This is the order of the related secular equation. 1 <= K <=N.
    !>

    C

    !>          C is DOUBLE PRECISION
    !>         C contains garbage if SQRE =0 and the C-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    S

    !>          S is DOUBLE PRECISION
    !>         S contains garbage if SQRE =0 and the S-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension ( 4 * M )
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension ( 3 * 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 slasd6 (integer icompq, integer nl, integer nr, integer sqre, real, dimension( * ) d, real, dimension( * ) vf, real, dimension( * ) vl, real alpha, real beta, integer, dimension( * ) idxq, integer, dimension( * ) perm, integer givptr, integer, dimension( ldgcol, * ) givcol, integer ldgcol, real, dimension( ldgnum, * ) givnum, integer ldgnum, real, dimension( ldgnum, * ) poles, real, dimension( * ) difl, real, dimension( * ) difr, real, dimension( * ) z, integer k, real c, real s, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.

Purpose:

    !>
    !> SLASD6 computes the SVD of an updated upper bidiagonal matrix B
    !> obtained by merging two smaller ones by appending a row. This
    !> routine is used only for the problem which requires all singular
    !> values and optionally singular vector matrices in factored form.
    !> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
    !> A related subroutine, SLASD1, handles the case in which all singular
    !> values and singular vectors of the bidiagonal matrix are desired.
    !>
    !> SLASD6 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 singular values of B can be computed using D1, D2, the first
    !> components of all the right singular vectors of the lower block, and
    !> the last components of all the right singular vectors of the upper
    !> block. These components are stored and updated in VF and VL,
    !> respectively, in SLASD6. Hence U and VT are not explicitly
    !> referenced.
    !>
    !> The singular values are stored in D. The algorithm consists of two
    !> stages:
    !>
    !>       The first stage consists of deflating the size of the problem
    !>       when there are multiple singular values or if there is a zero
    !>       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 SLASD7.
    !>
    !>       The second stage consists of calculating the updated
    !>       singular values. This is done by finding the roots of the
    !>       secular equation via the routine SLASD4 (as called by SLASD8).
    !>       This routine also updates VF and VL and computes the distances
    !>       between the updated singular values and the old singular
    !>       values.
    !>
    !> SLASD6 is called from SLASDA.
    !>

Parameters
    ICOMPQ

    !>          ICOMPQ is INTEGER
    !>         Specifies whether singular vectors are to be computed in
    !>         factored form:
    !>         = 0: Compute singular values only.
    !>         = 1: Compute singular vectors in factored form as well.
    !>

    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).
    !>         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.
    !>

    VF

    !>          VF is REAL array, dimension (M)
    !>         On entry, VF(1:NL+1) contains the first components of all
    !>         right singular vectors of the upper block; and VF(NL+2:M)
    !>         contains the first components of all right singular vectors
    !>         of the lower block. On exit, VF contains the first components
    !>         of all right singular vectors of the bidiagonal matrix.
    !>

    VL

    !>          VL is REAL array, dimension (M)
    !>         On entry, VL(1:NL+1) contains the  last components of all
    !>         right singular vectors of the upper block; and VL(NL+2:M)
    !>         contains the last components of all right singular vectors of
    !>         the lower block. On exit, VL contains the last components of
    !>         all right singular vectors of the bidiagonal 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.
    !>

    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.
    !>

    PERM

    !>          PERM is INTEGER array, dimension ( N )
    !>         The permutations (from deflation and sorting) to be applied
    !>         to each block. Not referenced if ICOMPQ = 0.
    !>

    GIVPTR

    !>          GIVPTR is INTEGER
    !>         The number of Givens rotations which took place in this
    !>         subproblem. Not referenced if ICOMPQ = 0.
    !>

    GIVCOL

    !>          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
    !>         Each pair of numbers indicates a pair of columns to take place
    !>         in a Givens rotation. Not referenced if ICOMPQ = 0.
    !>

    LDGCOL

    !>          LDGCOL is INTEGER
    !>         leading dimension of GIVCOL, must be at least N.
    !>

    GIVNUM

    !>          GIVNUM is REAL array, dimension ( LDGNUM, 2 )
    !>         Each number indicates the C or S value to be used in the
    !>         corresponding Givens rotation. Not referenced if ICOMPQ = 0.
    !>

    LDGNUM

    !>          LDGNUM is INTEGER
    !>         The leading dimension of GIVNUM and POLES, must be at least N.
    !>

    POLES

    !>          POLES is REAL array, dimension ( LDGNUM, 2 )
    !>         On exit, POLES(1,*) is an array containing the new singular
    !>         values obtained from solving the secular equation, and
    !>         POLES(2,*) is an array containing the poles in the secular
    !>         equation. Not referenced if ICOMPQ = 0.
    !>

    DIFL

    !>          DIFL is REAL array, dimension ( N )
    !>         On exit, DIFL(I) is the distance between I-th updated
    !>         (undeflated) singular value and the I-th (undeflated) old
    !>         singular value.
    !>

    DIFR

    !>          DIFR is REAL array,
    !>                   dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
    !>                   dimension ( K ) if ICOMPQ = 0.
    !>          On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
    !>          defined and will not be referenced.
    !>
    !>          If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
    !>          normalizing factors for the right singular vector matrix.
    !>
    !>         See SLASD8 for details on DIFL and DIFR.
    !>

    Z

    !>          Z is REAL array, dimension ( M )
    !>         The first elements of this array contain the components
    !>         of the deflation-adjusted updating row vector.
    !>

    K

    !>          K is INTEGER
    !>         Contains the dimension of the non-deflated matrix,
    !>         This is the order of the related secular equation. 1 <= K <=N.
    !>

    C

    !>          C is REAL
    !>         C contains garbage if SQRE =0 and the C-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    S

    !>          S is REAL
    !>         S contains garbage if SQRE =0 and the S-value of a Givens
    !>         rotation related to the right null space if SQRE = 1.
    !>

    WORK

    !>          WORK is REAL array, dimension ( 4 * M )
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension ( 3 * 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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