Ritz values and Arnoldi convergence for nonsymmetric matrices

Abstract

The restarted Arnoldi method, useful for determining a few desired eigenvalues of a matrix, employs shifts to refine eigenvalue estimates. In the symmetric case, using selected Ritz values as shifts produces convergence due to interlacing. For nonsymmetric matrices the behavior of Ritz values is insufficiently understood, and hence no satisfactory general convergence theory exists. Towards developing such a theory, this work demonstrates that Ritz values of nonsymmetric matrices can obey certain geometric constraints, as illustrated through careful analysis of Jordan blocks. By constructing conditions for localizing the Ritz values of a matrix with one simple normal wanted eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. As Ritz values are the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for nonsymmetric matrices has the potential to inform a broad range of analysis.