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

          N is INTEGER


         The length of all arrays.

I

          I is INTEGER


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

D

          D is REAL array, dimension (N)


         The original eigenvalues.  It is assumed that they are in


         order, D(I) < D(J)  for I < J.

Z

          Z is REAL array, dimension (N)


         The components of the updating vector.

DELTA

          DELTA is REAL 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 SLAED5


         for detail. The vector DELTA contains the information necessary


         to construct the eigenvectors by SLAED3 and SLAED9.

RHO

          RHO is REAL


         The scalar in the symmetric updating formula.

DLAM

          DLAM is REAL


         The computed lambda_I, the I-th updated eigenvalue.

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA