NAME¶

SLASD4 - subroutine compute 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

SYNOPSIS¶

SUBROUTINE SLASD4(

N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )

INTEGER I, INFO, N REAL RHO, SIGMA REAL D( * ), DELTA( * ), WORK( * ), Z( * )

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 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) INTEGER

The length of all arrays.

I (input) INTEGER

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

D (input) 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) REAL array, dimension (N)

The components of the updating vector.

DELTA (output) 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) REAL

The scalar in the symmetric updating formula.

SIGMA (output) REAL

The computed sigma_I, the I-th updated eigenvalue.

WORK (workspace) 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) INTEGER

= 0: successful exit
> 0: if INFO = 1, the updating process failed.

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. Further Details =============== Based on contributions by Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA