Third order iterative methods free from second derivative for nonlinear systems

Abstract

In this work we present a family of predictor–corrector methods free from second derivative for solving nonlinear systems. We prove that the methods of this family are of third order convergence. We also perform numerical tests that allow us to compare these methods with Newton’s method. In addition, the numerical examples improve theoretical results, showing super cubic convergence for some methods of this family.

Introduction

Let us consider the problem of finding a real zero of a function $F:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$, that is, a real solution $\alpha$ of the nonlinear system

$$F(x)=0\text{,}$$

of n equations with n unknowns. The most known iterative method is the classical Newton’s method that converges quadratically under certain conditions. Recently, for $n=1$, many robust and efficient methods have been proposed with higher convergence order. In this paper, the third order method [1] for single equations described by:

Predictor-step:

$$\begin{array}{cc}\hfill & y_n=x_n-\frac{f(x_n)}{f^{\prime }(x_{n)}}\text{,}\hfill \\ \hfill & z_n=-\frac{(y_n-x_n{)}^2}{2f^{\prime }(x_n)}{f}^{″}(x_n)\text{,}\hfill \end{array}$$

Corrector-step:

$$x_{n+1}=y_n-\frac{(y_n+z_n-x_n{)}^2}{2f^{\prime }(x_n)}{f}^{″}(x_n)\text{,}$$

is generalized to nonlinear systems using the predictor–corrector technique instead of the decomposition method and the nonlinear operator $N(x)$ proposed in [1].

The method obtained uses the second derivative, what is a serious drawback, specially for functions of several variables. To remove the second derivative, we extend the technique due to Kou et al. [2], to the multidimensional case. As in [2], [3], [4] for single equations, we obtain a family of predictor–corrector methods for nonlinear systems.

The paper is structured as follows. In Section 2 we describe the numerical methods that we propose. In Section 3 we analysis the convergence of these methods and we prove that, under certain conditions, they have cubic convergence. The last section is devoted to the numerical results obtained by applying the described methods to several nonlinear systems. The experimental results suggest that the iterates converge at a rate higher than three.

In the next sections we use the same notation for the n-dimensional case as for the one-dimensional case by simply interpreting the symbols appropriately.

In order to explore the convergence properties of the family methods that we describe in the next section, we recall the following result of Taylor’s expression on vector functions (see [5]).

Lemma 1

Let $F:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a $C^p$ function at each point of a convex set $D_0\subset D$, then for any $x\text{,}y\in D_0$

$$F(y)=F(x)+\sum _{k=1}^{p-1}\frac{1}{k!}{F}^{(k)}(x)(y-x{)}^k+R_p\text{,}$$

where

$$\Vert R_p\Vert ⩽\frac{1}{p!}\underset{0⩽t⩽1}{\sup }\Vert F^{(p)}(x+t(y-x))\Vert \Vert y-x{\Vert }^p\text{.}$$