SLASD7  -  merge  the two sets of singular values together
       into a single sorted set


SYNOPSIS

       SUBROUTINE SLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW,  VF,
                          VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX,
                          IDXP, IDXQ, PERM, GIVPTR, GIVCOL,  LDG­
                          COL, GIVNUM, LDGNUM, C, S, INFO )

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

           REAL           ALPHA, BETA, C, S

           INTEGER        GIVCOL( LDGCOL, * ), IDX( * ), IDXP(  *
                          ), IDXQ( * ), PERM( * )

           REAL           D(  * ), DSIGMA( * ), GIVNUM( LDGNUM, *
                          ), VF( * ), VFW( * ), VL( * ),  VLW(  *
                          ), Z( * ), ZW( * )


PURPOSE

       SLASD7  merges  the  two  sets of singular values together
       into a single sorted set. Then it  tries  to  deflate  the
       size of the problem. There are two ways in which deflation
       can occur:  when two or more  singular  values  are  close
       together  or if there is a tiny entry in the Z vector. For
       each such occurrence the  order  of  the  related  secular
       equation problem is reduced by one.

       SLASD7 is called from SLASD6.


ARGUMENTS

       ICOMPQ  (input) INTEGER
               Specifies  whether singular vectors are to be com­
               puted in compact form, as follows:
               = 0: Compute singular values only.
               = 1: Compute singular vectors of upper  bidiagonal
               matrix in compact form.

       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 N = NL + NR + 1 rows and


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

       D      (input/output) REAL array, dimension ( N )
              On entry D contains the singular values of the  two
              submatrices  to be combined. On exit D contains the
              trailing (N-K) updated singular values (those which
              were deflated) sorted into increasing order.

       Z      (output) REAL array, dimension ( M )
              On  exit  Z contains the updating row vector in the
              secular equation.

       ZW     (workspace) REAL array, dimension ( M )
              Workspace for Z.

       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  sin­
              gular vectors of the bidiagonal matrix.

       VFW    (workspace) REAL array, dimension ( M )
              Workspace for VF.

       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 singu­
              lar vectors of the bidiagonal matrix.

       VLW    (workspace) REAL array, dimension ( M )
              Workspace for VL.

       ALPHA  (input) REAL
              Contains the diagonal element associated  with  the
              added row.

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

              DSIGMA (output) REAL array, dimension (  N  )  Con­
              tains a copy of the diagonal elements (K-1 singular


       IDX    (workspace) INTEGER array, dimension ( N )
              This will contain the permutation used to sort  the
              contents of D into ascending order.

       IDXP   (workspace) INTEGER array, dimension ( N )
              This  will  contain  the  permutation used to place
              deflated values of D at the end of  the  array.  On
              output IDXP(2:K)
              points  to the nondeflated D-values and IDXP(K+1:N)
              points to the deflated singular values.

       IDXQ   (input) INTEGER array, dimension ( N )
              This  contains  the  permutation  which  separately
              sorts  the  two  sub-problems  in  D into ascending
              order.  Note that entries in the first half of this
              permutation  must first be moved one position back­
              ward; and entries in the  second  half  must  first
              have NL+1 added to their values.

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

              GIVPTR  (output) INTEGER The number of Givens rota­
              tions which took place in this subproblem. Not ref­
              erenced 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  The  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 refer­
              enced if ICOMPQ = 0.

              LDGNUM (input) INTEGER  The  leading  dimension  of
              GIVNUM, must be at least 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


       INFO   (output) INTEGER
              = 0:  successful exit.
              <  0:  if INFO = -i, the i-th argument had an ille­
              gal value.


FURTHER DETAILS

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


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