Numerical Linear Algebra with Applications

Chapter 22

Large Sparse Eigenvalue Problems

Abstract

This chapter presents the difficult problem of computing a few eigenvalues and associated eigenvectors of a large, sparse, matrix A. The power method is a Krylov subspace method and can be used to compute the largest eigenvalue in magnitude and its corresponding eigenvector, assuming there is a dominant eigenvalue. It turns out that some of the largest and smallest eigenvalues of A can be approximated in some cases by the eigenvalues of H_m in the Arnoldi decomposition of A. For a nonsymmetric matrix, the approach is to find the Arnoldi decomposition and compute some of the largest or smallest eigenvalues of H_m, the Ritz values. If uk is an eigenvector associated with a Ritz value λk, then ...