A power method for nonlinear operators

Abstract

In this article we consider the problem of computing the dominant eigenvalue of the linearization of a nonlinear operator. We define a power method that converges under natural conditions on the nonlinear operator. This nonlinear power method does not require the linearization itself, but only the action of the nonlinear operator on arbitrary functions. We apply this method to investigate the stability of equilibrium solutions of differential equations.

Keywords:

• Power method
• Eigenvalues
• Eigenvectors

Keywords:

• AMS 2000 Subject Classifications
• 15A18
• 65F15
• 46N40

Acknowledgments

The authors gratefully acknowledge G. Stewart's help in understanding the convergence of the perturbed power method. D. Estep's research is supported in part by the National Science Foundation through grants DMS–0107832, DGE–0221595003, and MSPA–CSE–0434354, the Department of Energy through grants DE–FG02–04ER25620 and DE–FG02–05ER25699, and the National Aeronautics and Space Administration through grant NNGH04GH63G. S. Eastman's research was supported in part by the National Science Foundation through grant DMS–0107832 and AASU Research and Scholarship grant #727180.