.TH SLASD4 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME SLASD4 - 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 .SH SYNOPSIS .TP 19 SUBROUTINE SLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO ) .TP 19 .ti +4 INTEGER I, INFO, N .TP 19 .ti +4 REAL RHO, SIGMA .TP 19 .ti +4 REAL D( * ), DELTA( * ), WORK( * ), Z( * ) .SH 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. .br where we assume the Euclidean norm of Z is 1. .br The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions. .SH ARGUMENTS .TP 7 N (input) INTEGER The length of all arrays. .TP 7 I (input) INTEGER The index of the eigenvalue to be computed. 1 <= I <= N. .TP 7 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. .TP 7 Z (input) REAL array, dimension (N) The components of the updating vector. .TP 7 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. .TP 7 RHO (input) REAL The scalar in the symmetric updating formula. .TP 7 SIGMA (output) REAL The computed sigma_I, the I-th updated eigenvalue. .TP 7 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. .TP 7 INFO (output) INTEGER = 0: successful exit .br > 0: if INFO = 1, the updating process failed. .SH 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