Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices

This manuscript is concerned with the determination of the rightmost eigenvalues of large sparse real nonsymmetric matrices. Specifically, the use of subspace iteration preconditioned by the Cayley transform and/or shift-invert is discussed. The convergence properties of subspace iteration are used to construct a strategy to validate the rightmost eigenvalue, which is computed by an iterative method. The motivation behind this paper is that rational preconditioners are very reliable in general but they can miss rightmost eigenvalues with large imaginary part. Numerical examples are given to illustrate the theory.

[1]  Richard A. Silverman,et al.  Introductory Complex Analysis , 1968 .

[2]  T. Manteuffel.  An Iterative Method for Solving Nonsymmetric Linear Systems With Dynamic Estimation of Parameters , 1975 .

[3]  G. Stewart.  Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices , 1976 .

[4]  T. Manteuffel.  The Tchebychev iteration for nonsymmetric linear systems , 1977 .

[5]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[6]  R. F. Heinemann,et al.  Multiplicity, stability, and oscillatory dynamics of the tubular reactor , 1981 .

[7]  Y. Saad,et al.  Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems , 1984 .

[8]  K. H Winters,et al.  Convergence properties of the finite-element method for Bénard Convection in an infinite layer , 1985 .

[9]  Y. Saad,et al.  Complex shift and invert strategies for real matrices , 1987 .

[10]  L. E. Scriven,et al.  Finding leading modes of a viscous free surface flow: An asymmetric generalized eigenproblem , 1988, J. Sci. Comput..

[11]  Y. Saad,et al.  Numerical solution of large nonsymmetric eigenvalue problems , 1989 .

[12]  A. Spence,et al.  Two Methods for the Numerical Detection of Hopf Bifurcations , 1991 .

[13]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[14]  Computing selected eigenvalues of sparse unsymmetric matrices using subspace iteration , 1993, TOMS.

[15]  Alastair Spence,et al.  A GENERALISED CAYLEY TRANSFORM FOR THE NUMERICAL DETECTION OF HOPF BIFURCATIONS IN LARGE SYSTEMS , 1993 .

[16]  Axel Ruhe.  Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems , 1994 .