Fixed Point and Newton’s Methods in the Complex Plane

François Dubeau,Calvin Gnang

doi:10.1155/2018/7289092

Abstract

We revisit the necessary and sufficient conditions for linear and high order of convergence of fixed point and Newton’s methods in the complex plane. Schröder’s processes of the first and second kind are revisited and extended. Examples and numerical experiments are included.

Highlights

In this paper, we revisit fixed point and Newton’s methods to find a simple solution of a nonlinear equation in the complex plane
We present only proofs of theorems we have to modify compared to the real case
A fixed point method use an iteration function (IF) which is an analytic function mapping its domain of definition into itself

Summary

Introduction

We revisit fixed point and Newton’s methods to find a simple solution of a nonlinear equation in the complex plane. We present sufficient and necessary conditions for the convergence of fixed point and Newton’s methods. Based on these conditions we show how to obtain direct processes to recursively increase the order of convergence. For the fixed point method, we present a generalization of Schroder’s method of the first kind. These results lead to a generalization of the Schroder’s process of the first kind. Based on the necessary and sufficient conditions, we propose two ways to increase the order of convergence of the Newton’s method.

Analytic Function

Fixed Point Method

Newton’s Iteration Function

Examples

Concluding Remarks