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dlasd6 examples
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Man Page Details
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NAME
       DLASD6  -  the  SVD of an updated upper bidiagonal matrix B obtained by
       merging two smaller ones by appending a row

SYNOPSIS
       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 )

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

           DOUBLE         PRECISION ALPHA, BETA, C, S

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

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

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 vec-
       tors of the bidiagonal matrix are desired.

       DLASD6 computes the SVD as follows:

                     ( D1(in)  0    0     0 )
         B = U(in) * (   Z1'   a   Z2'    b ) * VT(in)
                     (   0     0   D2(in) 0 )

           = U(out) * ( D(out) 0) * VT(out)

       where Z' = (Z1' a Z2' b) = u' VT', and u is a  vector  of  dimension  M
       with  ALPHA  and  BETA  in the NL+1 and NL+2 th entries and zeros else-
       where; and the entry b is empty if SQRE = 0.

       The singular values of B can be computed using D1, D2, the first compo-
       nents  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 occurence 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.


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) 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     (input/output) 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)  con-
              tains  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) 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) con-
              tains 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) DOUBLE PRECISION
              Contains the diagonal element associated with the added row.

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

       IDXQ   (output) INTEGER array, dimension ( N )
              This  contains  the  permutation which will reintegrate the sub-
              problem 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) DOUBLE PRECISION array, dimension ( LDGNUM, 2  )
              Each  number indicates the C or S value to be used in the corre-
              sponding Givens rotation. Not referenced if ICOMPQ = 0.

              LDGNUM (input) INTEGER  The  leading  dimension  of  GIVNUM  and
              POLES, must be at least N.

       POLES  (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
              On exit, POLES(1,*) is an array containing the new singular val-
              ues 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) DOUBLE PRECISION array, dimension ( N )
              On exit, DIFL(I) is the distance  between  I-th  updated  (unde-
              flated)  singular  value  and the I-th (undeflated) old singular
              value.

       DIFR   (output) DOUBLE PRECISION 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 normaliz-
              ing factors for the right singular vector matrix.

              See DLASD8 for details on DIFL and DIFR.

       Z      (output) DOUBLE PRECISION 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) DOUBLE PRECISION
              C contains garbage if SQRE =0 and the C-value of a Givens  rota-
              tion related to the right null space if SQRE = 1.

       S      (output) DOUBLE PRECISION
              S  contains garbage if SQRE =0 and the S-value of a Givens rota-
              tion related to the right null space if SQRE = 1.

       WORK   (workspace) DOUBLE PRECISION 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, an singular value did not converge

