DLAED4  -  subroutine computes the I-th updated eigenvalue
       of a symmetric rank-one modification to a diagonal  matrix
       whose  elements are given in the array d, and that  D(i) <
       D(j) for i < j  and that RHO > 0


SYNOPSIS

       SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO )

           INTEGER        I, INFO, N

           DOUBLE         PRECISION DLAM, RHO

           DOUBLE         PRECISION D( * ), DELTA( * ), Z( * )


PURPOSE

       This subroutine computes the I-th updated eigenvalue of  a
       symmetric rank-one modification to a diagonal matrix whose
       elements are given in the array d, and that  D(i)  <  D(j)
       for  i < j and that RHO > 0. This is arranged by the call­
       ing routine, and is no loss in generality.   The  rank-one
       modified system is thus

                  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) DOUBLE PRECISION array, dimension (N)
              The original eigenvalues.  It is assumed that  they
              are in order, D(I) < D(J)  for I < J.

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

       DELTA  (output) DOUBLE PRECISION array, dimension (N)
              If  N  .ne.  1, DELTA contains (D(j) - lambda_I) in
              its  j-th component.  If N = 1, then DELTA(1) =  1.
              The vector DELTA contains the information necessary
              to construct the eigenvectors.

              The scalar in the symmetric updating formula.

       DLAM   (output) DOUBLE PRECISION
              The computed lambda_I, the I-th updated eigenvalue.

       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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