Improved Nelson's Method for Computing Eigenvector Derivatives with Distinct and Repeated Eigenvalues

Abstract

T HE computation of derivatives of eigenvalues and eigenvectors with respect to model parameters is crucial for many applications [1]. Nelson [2] proposed an approach for general real eigensystemswith distinct eigenvalueswhere only the eigenvector of interest was required. Ojalvo [3], Mills-Curran [4], Dailey [5], and Shaw and Jayasuriya [6] developed Nelson’s method for real symmetric eigensystems with repeated eigenvalues. For computation of a particular solution, themethods described in [3,5]may fail in some circumstances. Other methods have been suggested by MillsCurran [4] and Song et al. [7], but they are difficult to implement. When a system of linear equations is solved, one hopes that the calculated solution is a close representation of the true solution. The computed solution for a system with a large condition number may be inexact due to extreme sensitivity of the solution to small changes in its coefficient matrix and right column vectors [8]. Because some diagonal elements in the coefficient matrix for particular solutions are set to units, its coefficients are not all of the same order of magnitude. So the coefficient matrix has a large condition number. In this Note, Nelson’s method is first extended to the case of repeated eigenvalues for the symmetric real eigensystems and its improvements are then presented. The condition number of the coefficient matrix of the improved method is considerably reduced.