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Numerical Linear Algebra with Applications by William Ford

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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 Hm 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 Hm, the Ritz values. If uk is an eigenvector associated with a Ritz value λk, then ...

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