Sensitivity Analysis and Its Numerical Methods for Derivatives of Quadratic Eigenvalue Problems

Abstract

Derivatives of eigenvalues and eigenvectors of parameter-dependent matrix eigenproblems play a key role in the optimum design of structures in engineering, and in the solution of inverse problems, such as the problem of model updating, which arises, for example, when information on the normal modes of vibration of a structure is used to detect structural damage. Both these applications often involve quadratic eigenvalue problems. Most existing methods for the computation of derivatives of quadratic eigenvalue problems are based on the assumption that repeated eigenvalues have well separated first order derivatives. In this paper we propose new algorithms for computing derivatives of eigenvalues and eigenvectors for quadratic eigenvalue problems under much more general conditions than existing methods, whose effectiveness for repeated or tightly clustered eigenvalues are confirmed by some numerical examples.