slasd4.f - Man Page

subroutine slasd4 (N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO)
SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.

SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc.  

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.

Parameters:

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.

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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Contributors:

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

Definition at line 155 of file slasd4.f.