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

Definition at line 309 of file dlasd6.f.

  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

Definition at line 309 of file slasd6.f.

Author

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