A numerical method for polynomial eigenvalue problems using contour integral
ABSTRACT We propose a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs). The method finds eigenvalues contained in a certain domain which is defined by a surrounding integral path. By evaluating the contour integral numerically along the path, the method reduces the original PEP into a small generalized eigenvalue problem, which has the identical eigenvalues in the domain. Error analysis indicates that the error of the eigenvalues is not uniform: inner eigenvalues are less erroneous. Four numerical examples are presented, which confirm the theoretical predictions.
Article: Local convergence of Newton-like methods for degenerate eigenvalues of nonlinear algebraic eigenproblems.[show abstract] [hide abstract]
ABSTRACT: We study the local convergence rates of several single-vector Newton-like methods for the solution of a semi-simple or defective eigenvalue of nonlinear algebraic eigenvalue problems of the form $T(\lambda)v=0$. This problem has not been fully understood, since the Jacobian associated with the single-vector Newton's method is singular at the desired eigenpair, and the standard convergence theory is not applicable. In addition, Newton's method generally converges only linearly towards singular roots. In this paper, we show that faster convergence can be achieved for degenerate eigenvalues. For semi-simple eigenvalues, we show that the convergence of Newton's method, Rayleigh functional iteration and the Jacobi-Davidson method are quadratic, and the latter two converge cubically for locally symmetric problems. For defective eigenvalues, all these methods converge only linearly in general. We then study two accelerated methods for defective eigenvalues, which exhibit quadratic convergence and require the solution of two linear systems per iteration. The results are illustrated by numerical experiments.02/2013;