The Convergence of a Class of Parallel Newton-Type Iterative Methods

Qinglong Huang

doi:10.1155/2017/4281684

Abstract

A general iterative process is proposed, from which a class of parallel Newton-type iterative methods can be derived. A unified convergence theorem for the general iterative process is established. The convergence of these Newton-type iterative methods is obtained from the unified convergence theorem. The results of efficiency analyses and numerical example are satisfactory.

Highlights

The convergence of these Newton-type iterative methods is obtained from the unified convergence theorem
Attempts to improve Newton method are the subject of many papers [1,2,3,4,5,6,7,8,9,10].Consider the following polynomial of degree n:n f (x) = ∏ (x − ri), (1)i=1 with simple zeros r1, r2, . . . , rn
A unified convergence theorem for the general iterative process is established. The convergence of these Newton-type iterative methods is obtained from the unified convergence theorem

Summary

Introduction

The convergence of these Newton-type iterative methods is obtained from the unified convergence theorem. If Newton iterative function is chosen as φ, that is, uj(k) = φ(xj(k)) = xj(k) + αj(k) and αj(k) are defined by (3), (4) is the method discussed in paper [3]. These special methods are all modifications to process (2); their convergence and convergence order are obtained via the unified general convergence Theorem 2.

Results

Conclusion