A Family of Newton Type Iterative Methods for Solving Nonlinear Equations

Xiaofeng Wang,Sheng Zhang,Xiaodong Fan,Yuping Qin,Weiyi Qian

doi:10.3390/a8030786

Abstract

In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory is 2n requiring the evaluations of n functions and one first-order derivative in per full iteration, which implies that this family is optimal according to Kung and Traub’s conjecture (1974). Its error equations and asymptotic convergence constants are obtained. The n-point iterative methods with memory are obtained by using a self-accelerating parameter, which achieve much faster convergence than the corresponding n-point methods without memory. The increase of convergence order is attained without any additional calculations so that the n-point Newton type iterative methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

Highlights

IntroductionSolving nonlinear equations by iterative methods have been of great interest to numerical analysts
Solving nonlinear equations by iterative methods have been of great interest to numerical analysts.The most famous one-point iterative method is probably Newton’s Equation [1]: xk+1 = xk − f/f 0, which converges quadratically
Equation [6], we derive a family of n-point iterative methods without memory for solving nonlinear equations

Summary

Introduction

Solving nonlinear equations by iterative methods have been of great interest to numerical analysts. Equation [6], we derive a family of n-point iterative methods without memory for solving nonlinear equations. We prove that the order of convergence of the n-point methods without memory is 2n requiring the evaluations of n functions and one first-order derivative in per full iteration. Based on Wu’s Equation [2] and Petković’s n-point methods [6], we derive a general optimal 2n th order family and write it in the following form:. Using the Taylor series and symbolic computation in the programming package Mathematica, we can find the order of convergence and the asymptotic error constant (AEC) of the n-point methods. The n-point family Equation (4) converges with at least 2n th order and satisfies the error relation ek+1 = ek,n = yk,n − a = (c2 + λ) n−2 n n +1 dn e2k + O(e2k.

New Families of Iterative Methods with Memory

Numerical Results

Conclusions