Volume 43, pp. 188-212, 2014-2015.

On the location of the Ritz values in the Arnoldi process

Gérard Meurant

Abstract

In this paper we give a necessary and sufficient condition for a set of complex values ${\theta }_1,\dots ,{\theta }_k$ to be the Arnoldi Ritz values at iteration $k$ for a general diagonalizable matrix $A$. Then we consider normal matrices and, in particular, real normal matrices with a real starting vector. We study in detail the case $k=2,$ for which we characterize the boundary of the region in the complex plane where pairs of complex conjugate Ritz values are located. Several examples with computations of the boundary of the feasible region are given. Finally we formulate some conjectures and open problems for the location of the Arnoldi Ritz values in the case $k>2$ for real normal matrices.

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Key words

Arnoldi algorithm, eigenvalues, Ritz values, normal matrices

AMS subject classifications

65F15, 65F18, 15A18