Acceleration of One-step Stationary Root-finding Algorithms

Abstract

A general method for increasing the order of one-step stationary root-finding algorithms is presented. The acceleration process is based on the formula Ψ(x) =: Φ(x) - [A(x)]−1F(Φ(x)), where F is a function whose domain and range are subsets of a Banach space X, the functions Ψ and Φ are iteration functions associated with one-step stationary root-finding methods, and A ε L(X, X). Using the concepts of order as defined in this paper, it is shown that, if the order of Φ is p and the order of A is r, then the order of Ψis min {p + r, 2p}. A second formula, A(x)=A(x){2I - [A(x)]−1F'(Φ(x)}}−1, is proposed to increase the order of Ā. In this case, the order of A is shown to be min {p, 2r}. These two formulae can be used in a ‘bootstrapping’ manner to create iteration functions of arbitrarily high order. Of particular interest is the application of this acceleration process to Newton's method. This leads to a general class of multistep methods which includes Traub's method, a class of Newton-Richardson methods, and a class of Newton-Moser methods. The theory developed unifies and simplifies the rate of convergence analysis of these algorithms.