Numerical Solution of Linear Eigenvalue Problems

We review numerical methods for computing eigenvalues of matrices. We start by considering the computation of the dominant eigenpair of a general dense matrix using the power method, and then generalize to orthogonal iterations and the QR iteration with shifts. We also consider divide-and-conquer algorithms for tridiagonal matrices. The second part of this survey involves the computation of eigenvalues of large and sparse matrices. The Lanczos and Arnoldi methods are developed and described within the context of Krylov subspace eigensolvers. We also present the idea of the Jacobi–Davidson method.