Newton's Method in Floating Point Arithmetic and Iterative Refinement of Generalized Eigenvalue Problems

Abstract

We examine the behavior of Newton's method in floating point arithmetic, allowing for extended precision in computation of the residual, inaccurate evaluation of the Jacobian and unstable solution of the linear systems. We bound the limiting accuracy and the smallest norm of the residual. The application that motivates this work is iterative refinement for the generalized eigenvalue problem. We show that iterative refinement by Newton's method can be used to improve the forward and backward errors of computed eigenpairs.