Composite Arnoldi-Newton methods for large nonsymmetric eigenvalue problems

Research Report MRR97-052

Composite Arnoldi-Newton methods for large nonsymmetric eigenvalue problems

David L. Harrar II and Michael R. Osborne

Abstract: In [Osborne \& Michaelson, Computer J. 7, pp. 66-71, 1964] it was shown that it is possible to construct eigenvalue algorithms based on Newton's method and inverse iteration which can result in second- and even third- order rates of convergence, with applications to nonlinear problems also possible. Recently, the use of these methods for large-scale problems was considered in [Cleary \& Osborne, Proc. of Computational Techniques and Appls. Conf. CTAC '93, pp. 140-147, World Scientific, 1994]. The difficulty is that these methods require matrix factorization at each iteration and become prohibitively expensive if good initial estimates are not available. Arnoldi's method is effective for computing a few extreme eigenvalues. In order to compute interior eigenvalues, it is necessary to use, for example, a shift-invert transformation, which necessitates matrix factorization. However, it is often satisfactory to perform several Arnoldi iterations prior to refactorization. Our aim is to see if composite algorithms can be constructed in which, for example, Arnoldi's method is used to obtain good initial eigenpair estimates for use in conjunction with Newton's method. The hope is that a composite method can achieve the same accuracy but at a lower computational cost. We apply our methods to a problem arising in the study of chemical reactions in a tubular reactor and to a problem from the optical sciences, that of determining guided mode solutions for optical fibers. Numerical experiments are carried out on a Fujitsu VPP300 supercomputer.

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