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NAME

       SLASD9 - find the square roots of the roots of the secular equation,

SYNOPSIS

       SUBROUTINE SLASD9( ICOMPQ, LDU, K, D, Z, VF, VL, DIFL, DIFR, DSIGMA, WORK, INFO )

           INTEGER        ICOMPQ, INFO, K, LDU

           REAL           D( * ), DIFL( * ), DIFR( LDU, * ), DSIGMA( * ), VF( * ), VL( * ), WORK(
                          * ), Z( * )

PURPOSE

       SLASD9 finds the square roots of the roots of the secular  equation,  as  defined  by  the
       values in DSIGMA and Z.  It makes the
       appropriate  calls  to SLASD4, and stores, for each  element in D, the distance to its two
       nearest poles (elements in DSIGMA). It also updates the arrays VF and VL,  the  first  and
       last components of all the right singular vectors of the original bidiagonal matrix.

       SLASD9 is called from SLASD7.

ARGUMENTS

       ICOMPQ  (input) INTEGER
               Specifies  whether  singular  vectors  are  to be computed in factored form in the
               calling routine:

               ICOMPQ = 0             Compute singular values only.

               ICOMPQ = 1             Compute singular vector matrices in factored form also.   K
               (input)  INTEGER  The  number  of  terms  in the rational function to be solved by
               SLASD4.  K >= 1.

       D       (output) REAL array, dimension(K)
               D(I) contains the updated singular values.

       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.

       Z       (input) REAL array, dimension (K)
               The  first  K  elements  of  this  array  contain the components of the deflation-
               adjusted updating row vector.

       VF      (input/output) REAL array, dimension(K)
               On entry, VF contains  information passed through SBEDE8.f On  exit,  VF  contains
               the  first  K  components of the first components of all right singular vectors of
               the bidiagonal matrix.

       VL      (input/output) REAL array, dimension(K)
               On entry, VL contains  information passed through SBEDE8.f On  exit,  VL  contains
               the first K components of the last components of all right singular vectors of the
               bidiagonal matrix.

       DIFL    (output) REAL array, dimension (K).
               On exit, DIFL(I) = D(I) - DSIGMA(I).

       DIFR    (output) REAL array,
               dimension (LDU, 2) if ICOMPQ =1 and dimension (K) if ICOMPQ = 0.  On exit, DIFR(I,
               1) = D(I) - DSIGMA(I+1), DIFR(K, 1) is not defined and will not be referenced.

               If ICOMPQ = 1, DIFR(1:K, 2) is an array containing the normalizing factors for the
               right singular vector matrix.

       WORK    (workspace) REAL array,
               dimension at least (3 * K) Workspace.

       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

FURTHER DETAILS

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