SLASD4(3)      LAPACK routine of NEC Numeric Library Collection      SLASD4(3)



NAME
       SLASD4

SYNOPSIS
       SUBROUTINE SLASD4 (N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO)



PURPOSE
            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.




ARGUMENTS
           N         (input)
                     N is INTEGER
                    The length of all arrays.

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

           D         (input)
                     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         (input)
                     Z is REAL array, dimension ( N )
                    The components of the updating vector.

           DELTA     (output)
                     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       (input)
                     RHO is REAL
                    The scalar in the symmetric updating formula.

           SIGMA     (output)
                     SIGMA is REAL
                    The computed sigma_I, the I-th updated eigenvalue.

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



       Internal Parameters:


             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.



LAPACK routine                  31 October 2017                      SLASD4(3)