General Algorithms for Computing Derivatives of Repeated Eigenvalues and Eigenvectors of Symmetric Quadratic Eigenvalue Problems

The numerical computation of derivatives of eigenvalues and eigenvectors has been an active research topic due to its wide applications in engineering and the physical sciences. There are many numerical methods available in the literatures for computing derivatives of eigenvalues and eigenvectors for standard eigenvalue problems and quadratic eigenvalue problems. However, almost all existing methods for quadratic eigenvalue problems have a common limitation, that is, they are based on the assumption that repeated eigenvalues have distinct first order derivatives. In this paper, we lift this assumption and develop general algorithms for computing derivatives, of any arbitrary order, of repeated eigenvalues and corresponding eigenvectors of quadratic eigenvalue problems, under much more general conditions than existing methods. The effectiveness of our algorithms are illustrated by some numerical examples.