A new approach for solving symmetric eigenvalue problems

Abstract

In this paper, we present a new approach for the solution to a series of slightly perturbed symmetric eigenvalue problems ( A + BS i B T ) x = λx, 0 ≤ i ≤ m, where A = A T ϵ R n × n , B ϵ R n × p , and S i = S T i ϵ R p × p , p ⪡ n. The matrix B is assumed to have full column rank. The main idea of our approach lies in a specific choice of starting vectors used in the block Lanczos algorithm so that the effect of the perturbations is confined to lie in the first diagonal block of the block tridiagonal matrix that is produced by the block Lanczos algorithm. Subsequently, for the perturbed eigenvalue problems under our consideration, the block Lanczos scheme needs to be applied to the original (unperturbed) matrix only once and then the first diagonal block updated for each perturbation so that for low-rank perturbations, the algorithm presented in this paper results in significant savings. Numerical examples based on finite element vibration analysis illustrate the advantages of this approach.