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NAME
     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( * )



     #include <sunperf.h>

     void dlaed4(int n, int i,  double  *d,  double  *dz,  double
               *delta, double drho, double *dlam, int *info) ;

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 no loss in  gen-
     erality.  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  functions
     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  neces-
               sary to construct the eigenvectors.

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

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

     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  noting
     if we are working with THREE poles!

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

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