COMPUTATION OF DERIVATIVES OF EIGENVALUES AND CORRESPONDING EIGENVECTORS OF PARAMETER-DEPENDENT MATRICES

Abstract

This thesis aims to study some numerical methods for the computation of derivatives of matrix eigensystems whose matrix elements depend on some parameters. In many problems in Engineering and Physical Sciences, we are usually interested in knowing how the sensitivity of the physical quantity behaves for small changes in the parameters. To find this sensitivity, eigenvalue and eigenvector derivatives with respect to these parameters would often need to be computed.

We explore and study the simple eigenvalue case, which is well-known, and its existence is also well-established and understood in the literature. We examine the various types of methods for their computations, classified as Modal methods, Direct methods and Iterative methods in the literature, looking at their formulations, computations, and comparing their efficiencies and limitations. We then extend this to the more difficult but important problem of solving the semi-simple case, where some of the eigenvalues of interest are repeated.