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NAME

       slasd6.f -

SYNOPSIS

   Functions/Subroutines
       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.

Function/Subroutine Documentation

   subroutine slasd6 (integerICOMPQ, integerNL, integerNR, integerSQRE, real, dimension( * )D,
       real, dimension( * )VF, real, dimension( * )VL, realALPHA, realBETA, integer, dimension( *
       )IDXQ, integer, dimension( * )PERM, integerGIVPTR, integer, dimension( ldgcol, * )GIVCOL,
       integerLDGCOL, real, dimension( ldgnum, * )GIVNUM, integerLDGNUM, real, dimension( ldgnum,
       * )POLES, real, dimension( * )DIFL, real, dimension( * )DIFR, real, dimension( * )Z,
       integerK, realC, realS, real, dimension( * )WORK, integer, dimension( * )IWORK,
       integerINFO)
       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 occurence 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 ( LDGNUM, 2 ) if ICOMPQ = 1 and
                             dimension ( N ) if ICOMPQ = 0.
                    On exit, DIFR(I, 1) is the distance between I-th updated
                    (undeflated) singular value and the I+1-th (undeflated) old
                    singular value.

                    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.

       Date:
           September 2012

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

       Definition at line 312 of file slasd6.f.

Author

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