Computing several eigenvalues of nonlinear eigenvalue problems by selection

TL;DR

This paper introduces simple, efficient selection methods based on divided differences for computing multiple eigenvalues of nonlinear and multiparameter eigenproblems, avoiding the need for locking techniques and enhancing applicability.

Contribution

It presents new selection techniques using divided differences for nonlinear eigenproblems, generalizes methods for multiparameter problems, and adapts approaches for problems with infinite eigenvalues.

Findings

Effective for large sparse problems

Applicable to multiparameter eigenproblems

No need for locking techniques

Abstract

Computing more than one eigenvalue for (large sparse) one-parameter polynomial and general nonlinear eigenproblems, as well as for multiparameter linear and nonlinear eigenproblems, is a much harder task than for standard eigenvalue problems. We present simple but efficient selection methods based on divided differences to do this. In contrast to locking techniques, it is not necessary to keep converged eigenvectors in the search space, so that the entire search space may be devoted to new information. The techniques are applicable to many types of matrix eigenvalue problems; standard deflation is possible only for linear one-parameter problems. The methods are easy to understand and implement. Although divided differences are well-known in the context of nonlinear eigenproblems, the proposed selection techniques are new for one-parameter problems. For multiparameter problems, we improve on and generalize our previous work. We also show how to use divided differences in the framework of homogeneous coordinates, which may be appropriate for generalized eigenvalue problems with infinite eigenvalues. While the approaches are valuable alternatives for one-parameter nonlinear eigenproblems, they seem the only option for multiparameter problems.