SLASD4  -  subroutine computes the square root of the I-th
       updated eigenvalue of a positive symmetric rank-one  modi­
       fication  to  a positive diagonal matrix whose entries are
       given as the squares of the corresponding entries  in  the
       array  d,  and  that  0 <= D(i) < D(j) for i < j  and that
       RHO > 0


SYNOPSIS

       SUBROUTINE SLASD4( N, I, D, Z, DELTA,  RHO,  SIGMA,  WORK,
                          INFO )

           INTEGER        I, INFO, N

           REAL           RHO, SIGMA

           REAL           D( * ), DELTA( * ), WORK( * ), Z( * )


PURPOSE

       This  subroutine  computes  the  square  root  of the I-th
       updated eigenvalue of a positive symmetric rank-one  modi­
       fication  to  a positive diagonal matrix whose entries are
       given as the squares of the corresponding entries  in  the
       array  d, and that 0 <= D(i) < D(j) for i < j and that RHO
       > 0. This is arranged by the calling routine,  and  is  no
       loss in generality.  The rank-one modified system is thus

              diag( D ) * diag( D ) +  RHO *  Z * Z_transpose.

       where we assume the Euclidean norm of Z is 1.

       The  method  consists  of approximating the rational func­
       tions in the secular  equation  by  simpler  interpolating
       rational functions.


ARGUMENTS

       N      (input) INTEGER
              The length of all arrays.

       I      (input) INTEGER
              The index of the eigenvalue to be computed.  1 <= I
              <= N.

       D      (input) REAL array, dimension ( N )
              The original eigenvalues.  It is assumed that  they
              are in order, 0 <= D(I) < D(J)  for I < J.

       Z      (input) REAL array, dimension ( N )
              The components of the updating vector.

       DELTA  (output) REAL array, dimension ( N )
              If N .ne. 1, DELTA contains (D(j) - sigma_I) in its
              j-th component.  If N = 1, then DELTA(1) = 1.   The

              construct the (singular) eigenvectors.

       RHO    (input) REAL
              The scalar in the symmetric updating formula.

       SIGMA  (output) REAL
              The computed lambda_I, the I-th updated eigenvalue.

       WORK   (workspace) REAL array, dimension ( N )
              If  N .ne. 1, WORK contains (D(j) + sigma_I) in its
              j-th component.  If N = 1, then WORK( 1 ) = 1.

       INFO   (output) INTEGER
              = 0:  successful exit
              > 0:  if INFO = 1, the updating process failed.


PARAMETERS

       Logical variable ORGATI (origin-at-i?) is used for distin­
       guishing  whether D(i) or D(i+1) is treated as the origin.

       ORGATI = .true.    origin at i ORGATI =  .false.    origin
       at i+1

       Logical  variable SWTCH3 (switch-for-3-poles?) is for not­
       ing if we are working with THREE poles!

       MAXIT is the maximum number of iterations allowed for each
       eigenvalue.

       Further Details ===============

       Based  on  contributions  by Ren-Cang Li, Computer Science
       Division, University of California at Berkeley, USA


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