Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  computes  the SVD of an updated upper bidiagonal matrix B obtained by merging
       two smaller ones by appending a row

SYNOPSIS

       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 )

           INTEGER        GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, NR, SQRE

           REAL           ALPHA, BETA, C, S

           INTEGER        GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), PERM( * )

           REAL           D( * ), DIFL( * ), DIFR( * ), GIVNUM( LDGNUM, * ), POLES( LDGNUM, *  ),
                          VF( * ), VL( * ), WORK( * ), Z( * )

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.

ARGUMENTS

        ICOMPQ (input) 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     (input) INTEGER
               The row dimension of the upper block.  NL >= 1.

        NR     (input) INTEGER
               The row dimension of the lower block.  NR >= 1.

        SQRE   (input) 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      (input/output) 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     (input/output) 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     (input/output) 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  (input/output) REAL
               Contains the diagonal element associated with the added row.

        BETA   (input/output) REAL
               Contains the off-diagonal element associated with the added
               row.

        IDXQ   (output) 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   (output) INTEGER array, dimension ( N )
               The permutations (from deflation and sorting) to be applied
               to each block. Not referenced if ICOMPQ = 0.
               GIVPTR (output) INTEGER
               The number of Givens rotations which took place in this
               subproblem. Not referenced if ICOMPQ = 0.
               GIVCOL (output) 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 (input) INTEGER
               leading dimension of GIVCOL, must be at least N.
               GIVNUM (output) 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 (input) INTEGER
               The leading dimension of GIVNUM and POLES, must be at least N.

        POLES  (output) 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   (output) 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   (output) 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      (output) REAL array, dimension ( M )
               The first elements of this array contain the components
               of the deflation-adjusted updating row vector.

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

        C      (output) 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      (output) 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   (workspace) REAL array, dimension ( 4 * M )

        IWORK  (workspace) INTEGER array, dimension ( 3 * N )

        INFO   (output) 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

FURTHER DETAILS

        Based on contributions by
           Ming Gu and Huan Ren, Computer Science Division, University of
           California at Berkeley, USA

 LAPACK auxiliary routine (version 3.3.0)   April 2011                            SLASD6(3lapack)