Arnoldi versus Nonsymmetric Lanczos Algorithms for Solving Matrix Eigenvalue Problems

We obtain several results which may be useful in determining the convergence behavior of eigenvalue algorithms based upon Arnoldi and nonsymmetric Lanczos recursions. We derive a relationship between nonsymmetric Lanczos eigenvalue procedures and Arnoldi eigenvalue procedures. We demonstrate that the Arnoldi recursions preserve a property which characterizes normal matrices, and that if we could determine the appropriate starting vectors, we could mimic the nonsymmetric Lanczos eigenvalue convergence on a general diagonalizable matrix by its convergence on related normal matrices. Using a unitary equivalence for each of these Krylov subspace methods, we define sets of test problems where we can easily vary certain spectral properties of the matrices. We use these and other test problems to examine the behavior of an Arnoldi and of a nonsymmetric Lanczos procedure. The results of these tests suggest that to completely characterize the behavior of these methods on nonnormal problems it is not sufficient to know the singular values of the eigenvector matrix. They also suggest a potential source of numerical difficulties for both types of methods.