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A new iterative three-point method for solving systems of nonlinear equations

Carlos E. Cadenas R., Yorman J. Mendoza N

TL;DR

The paper develops a vector generalization of a scalar sixth‑order three‑point method to solve systems of nonlinear equations $\mathbf{F}(\mathbf{x})=\mathbf{0}$, providing a concrete iterative scheme that achieves six‑order convergence. Convergence is established through a Taylor‑based analysis of $\mathbf{F}$ and its Fréchet derivatives, yielding an explicit six‑th order error term. Numerical experiments on two problems confirm both the theoretical and computational convergence orders. An efficiency framework based on the Ostrowski index shows the proposed method outperforms a fifth‑order competitor for large systems, but is outperformed by another sixth‑order scheme due to higher Jacobian inversions, guiding future work toward reducing inversions and evaluations. Overall, the work advances high‑order, three‑point methods for vector nonlinear systems with a clear path toward practical efficiency improvements.

Abstract

A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that the new method also has a sixth order of convergence. It is confirmed that the theoretical order of convergence coincides with the computational order of convergence by the numerical solution of two problems. Finally, its computational efficiency is calculated and subsequently compared with that of other three-point methods of fifth and sixth order convergence that also solve systems of non-linear equations.

A new iterative three-point method for solving systems of nonlinear equations

TL;DR

The paper develops a vector generalization of a scalar sixth‑order three‑point method to solve systems of nonlinear equations , providing a concrete iterative scheme that achieves six‑order convergence. Convergence is established through a Taylor‑based analysis of and its Fréchet derivatives, yielding an explicit six‑th order error term. Numerical experiments on two problems confirm both the theoretical and computational convergence orders. An efficiency framework based on the Ostrowski index shows the proposed method outperforms a fifth‑order competitor for large systems, but is outperformed by another sixth‑order scheme due to higher Jacobian inversions, guiding future work toward reducing inversions and evaluations. Overall, the work advances high‑order, three‑point methods for vector nonlinear systems with a clear path toward practical efficiency improvements.

Abstract

A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of in , it is shown that the new method also has a sixth order of convergence. It is confirmed that the theoretical order of convergence coincides with the computational order of convergence by the numerical solution of two problems. Finally, its computational efficiency is calculated and subsequently compared with that of other three-point methods of fifth and sixth order convergence that also solve systems of non-linear equations.
Paper Structure (7 sections, 2 theorems, 67 equations, 2 figures, 3 tables)

This paper contains 7 sections, 2 theorems, 67 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Definitions relating to convergence
  3. Development of the method
  4. Numerical tests
  5. Computational efficiency
  6. Comparison of efficiencies
  7. Conclusions

Key Result

Lemma 1

If $\textbf{F}:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$ is a function $p$-times Fréchet differentiable in the convex set $D\subseteq\mathbb{R}^{n}$ then for $\textbf{x},\,\textbf{y}\in D$ we have to: where: , and

Figures (2)

  • Figure 1: graph of $g$.
  • Figure 2: Graph of $f(n,l)=2n^{2}+9n+3l(n+1)-17$.

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Lemma 1
  • Theorem 1