SLASD3  -  find  all  the square roots of the roots of the
       secular equation, as defined by the values in D and Z


SYNOPSIS

       SUBROUTINE SLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA,  U,
                          LDU,  U2,  LDU2,  VT, LDVT, VT2, LDVT2,
                          IDXC, CTOT, Z, INFO )

           INTEGER        INFO, K, LDQ, LDU, LDU2,  LDVT,  LDVT2,
                          NL, NR, SQRE

           INTEGER        CTOT( * ), IDXC( * )

           REAL           D(  *  ),  DSIGMA( * ), Q( LDQ, * ), U(
                          LDU, * ), U2( LDU2, * ), VT( LDVT, * ),
                          VT2( LDVT2, * ), Z( * )


PURPOSE

       SLASD3 finds all the square roots of the roots of the sec­
       ular equation, as defined by the values in  D  and  Z.  It
       makes the appropriate calls to SLASD4 and then updates the
       singular vectors by matrix multiplication.

       This code makes very mild assumptions about floating point
       arithmetic. It will work on machines with a guard digit in
       add/subtract, or on those binary  machines  without  guard
       digits  which subtract like the Cray XMP, Cray YMP, Cray C
       90, or Cray 2.  It could conceivably fail  on  hexadecimal
       or  decimal  machines without guard digits, but we know of
       none.

       SLASD3 is called from SLASD1.


ARGUMENTS

       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
              M = N + SQRE >= N columns.

       K      (input) INTEGER
              The size of the secular equation, 1 =< K = < N.

              On  exit the square roots of the roots of the secu­
              lar equation, in ascending order.

       Q      (workspace) REAL array,
              dimension at least (LDQ,K).

       LDQ    (input) INTEGER
              The leading dimension of the array Q.  LDQ >= K.

              DSIGMA (input) REAL array, dimension(K) The first K
              elements of this array contain the old roots of the
              deflated updating problem.  These are the poles  of
              the secular equation.

       U      (input) REAL array, dimension (LDU, N)
              The  last  N - K columns of this matrix contain the
              deflated left singular vectors.

       LDU    (input) INTEGER
              The leading dimension of the array U.  LDU >= N.

       U2     (input) REAL array, dimension (LDU2, N)
              The first K columns of this matrix contain the non-
              deflated  left singular vectors for the split prob­
              lem.

       LDU2   (input) INTEGER
              The leading dimension of the array U2.  LDU2 >=  N.

       VT     (input) REAL array, dimension (LDVT, M)
              The  last M - K columns of VT' contain the deflated
              right singular vectors.

       LDVT   (input) INTEGER
              The leading dimension of the array VT.  LDVT >=  N.

       VT2    (input) REAL array, dimension (LDVT2, N)
              The  first  K  columns  of  VT2'  contain  the non-
              deflated right singular vectors for the split prob­
              lem.

       LDVT2  (input) INTEGER
              The  leading  dimension of the array VT2.  LDVT2 >=
              N.

       IDXC   (input) INTEGER array, dimension ( N )
              The permutation used to arrange the  columns  of  U
              (and  rows  of  VT)  into  three groups:  the first
              group contains non-zero entries only at  and  above
              (or  before)  NL  +1;  the second contains non-zero
              entries only at and below (or after) NL+2; and  the
              third  is  dense. The first column of U and the row
              of VT are treated separately, however.

              must  be likewise permuted before the matrix multi­
              plies can take place.

       CTOT   (input) INTEGER array, dimension ( 4 )
              A count of the total number of the various types of
              columns in U (or rows in VT), as described in IDXC.
              The fourth column type is any column which has been
              deflated.

       Z      (input) REAL array, dimension (K)
              The first K elements of this array contain the com­
              ponents of the deflation-adjusted updating row vec­
              tor.

       INFO   (output) INTEGER
              = 0:  successful exit.
              <  0:  if INFO = -i, the i-th argument had an ille­
              gal value.
              > 0:  if INFO = 1, an singular value did  not  con­
              verge


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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