COMPUTATION OF SELECTED EIGENVALUES OF GENERALIZED EIGENVALUE PROBLEMS

We examine and develop techniques for obtaining a few selected eigenvalues of the generalized eigenvalue problem Ax = λBx, where A and B are n × n, nonysmmetric, banded complex matrices. One way of obtaining the desired eigenvalues is to use a direct method to compute all the eigenvalues. Direct methods are computationally intensive and destroy the sparsity of the matrices A and B. Iterative methods, on the other hand, maintain the sparsity of the matrices and compute only a few eigenvalues. The iterative algorithms that we consider are the Arnoldi and the Lanczos methods. We use a shift and invert strategy to increase the rate of convergence towards the desired eigenvalues. We compare these two approaches for a model problem, which arises from considering the linear stability of compressible boundary layers and some other test problems. We present a general scheme to compute the eigenvalues lying inside a “box” in the complex plane. We also outline a procedure to seperate the converged eigenvalues from spurious approximations. In addition, this procedure can also improve the approximations to the eigenvalues of interest. Numerical results obtained on a CRAY Y-MP are presented. © 1993 by Academic Press, Inc.