General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Stoil I Ivanov

doi:10.3390/math8091599

Abstract

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

Highlights

Introduction and PreliminariesLet (K, | · |) be a normed field and f : K → K be an arbitrary function
A general convergence theory of the Picard iteration (1) in cone metric spaces and in n-dimensional vector spaces has been developed by Proinov [8,9,10,11]
A detailed local convergence analysis of these illustrious methods applied to multiple polynomial zeros can be found in the papers [8,14,15,16]

Summary

Introduction and Preliminaries

Let (K, | · |) be a normed field and f : K → K be an arbitrary function. It is well known that one of the most commonly used tools for finding the zeros of f is the Picard iteration xk+1 = Txk , k = 0, 1, 2, . . . ,. In this manner, we reduce the convergence analysis of (1) up to studying of some simple properties of the iteration function T.

General Local Convergence Theorems

Local Convergence of the Super-Halley Method for Multiple Polynomial Zeros

Local Convergence of the First Type

Local Convergence of the Second Type

Findings

Conclusions