DLASD4(1) LAPACK auxiliary routine (version 3.2) DLASD4(1)[top]NAMEDLASD4 - subroutine compute the square root of the I-th updated eigen‐ value of a positive symmetric rank-one modification to a positive diag‐ onal 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 > 0SYNOPSISSUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO ) INTEGER I, INFO, N DOUBLE PRECISION RHO, SIGMA DOUBLE PRECISION D( * ), DELTA( * ), WORK( * ), Z( * )PURPOSEThis 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 sec‐ ular equation by simpler interpolating rational functions.ARGUMENTSN (input) INTEGER The length of all arrays. I (input) INTEGER The index of the eigenvalue to be computed. 1 <= I <= N. D (input) 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 (input) DOUBLE PRECISION array, dimension ( N ) The components of the updating vector. DELTA (output) DOUBLE PRECISION array, dimension ( N ) If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th compo‐ nent. If N = 1, then DELTA(1) = 1. The vector DELTA contains the information necessary to construct the (singular) eigenvec‐ tors. RHO (input) DOUBLE PRECISION The scalar in the symmetric updating formula. SIGMA (output) DOUBLE PRECISION The computed sigma_I, the I-th updated eigenvalue. WORK (workspace) DOUBLE PRECISION array, dimension ( N ) If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th compo‐ nent. If N = 1, then WORK( 1 ) = 1. INFO (output) INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed.PARAMETERSLogical 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 LAPACK auxiliary routine (versioNovember 2008 DLASD4(1)

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