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 calling 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 functions in the
 secular equation by simpler interpolating rational functions.

          N is INTEGER
         The length of all arrays.

          I is INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

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

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

          DELTA is DOUBLE PRECISION array, dimension (N)
         If N .GT. 2, DELTA contains (D(j) - lambda_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
         for detail. The vector DELTA contains the information necessary
         to construct the eigenvectors by DLAED3 and DLAED9.

          RHO is DOUBLE PRECISION
         The scalar in the symmetric updating formula.

          DLAM is DOUBLE PRECISION
         The computed lambda_I, the I-th updated eigenvalue.

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

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  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.