!>
!> 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension ( N )
!>         The components of the updating vector.
!> 

DELTA

!>          DELTA is DOUBLE PRECISION 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 DOUBLE PRECISION
!>         The scalar in the symmetric updating formula.
!> 

SIGMA

!>          SIGMA is DOUBLE PRECISION
!>         The computed sigma_I, the I-th updated eigenvalue.
!> 

WORK

!>          WORK is DOUBLE PRECISION 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

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