Local convergence of Newton-like methods for degenerate eigenvalues of nonlinear eigenproblems. I. Classical algorithms

Abstract

We study the local convergence rates of several most widely used single-vector Newton-like methods for the solution of a degenerate eigenvalue of nonlinear algebraic eigenvalue problems of the form $$T(\lambda )v=0$$ T ( ? ) v = 0 . This problem has not been completely understood, since the Jacobian associated with Newton's method is singular at the desired eigenpair, and the standard convergence theory is not applicable. In fact, Newton's method generally converges only linearly towards singular roots. In this paper, we show that the local convergence of inverse iteration, Rayleigh functional iteration and the Jacobi---Davidson method are at least quadratic for semi-simple eigenvalues. For defective eigenvalues, Newton-like methods converge only linearly in general. The results are illustrated by numerical experiments.