This subroutine computes the square root of the I-th updated


 eigenvalue of a positive symmetric rank-one modification 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 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, 0 <= 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 .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th


         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA


         contains the information necessary to construct the


         (singular) eigenvectors.

RHO

          RHO is REAL


         The scalar in the symmetric updating formula.

SIGMA

          SIGMA is REAL


         The computed sigma_I, the I-th updated eigenvalue.

WORK

          WORK is 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

          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.

December 2016

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