改进不动点法:一种加速不动点法求解非线性代数方程收敛的极简单方法

IF 2.3 3区 数学 Q1 MATHEMATICS
Mathematics Pub Date : 2022-10-14 DOI:10.3390/math10203797
U. Filobello-Niño, H. Vázquez-Leal, J. Huerta-Chua, J. Martínez-Castillo, A. Herrera-May, M. Sandoval-Hernandez, V. Jiménez-Fernández
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引用次数: 0

摘要

这项工作提出了增强不动点方法(EFPM),作为对线性或非线性代数方程精确或近似解问题的直接修改。该提案包括提供一种易于使用和系统化的通用方法。因此,预计这项工作将有助于打破对已知方法进行有效修改必须漫长而复杂的范式。事实上,该方法用同一个方程来表达代数方程,但乘以一个适当的因子,这个因子在大多数情况下只是一个简单的数字因子。其主要思想是修改原始方程,对其他方程稍作修改,使两者都有相同的解。接下来,将修改后的方程表示为不动点问题,并使用所提出的参数来加速原始方程不动点问题的收敛。由于牛顿方法是由代数方程的一个可能的不动点问题产生的,我们将看到,相对容易获得收敛阶数大于2的牛顿方法的修改版本。我们将在这项工作中看到这一程序的便利性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations
This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure.
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来源期刊
Mathematics
Mathematics Mathematics-General Mathematics
CiteScore
4.00
自引率
16.70%
发文量
4032
审稿时长
21.9 days
期刊介绍: Mathematics (ISSN 2227-7390) is an international, open access journal which provides an advanced forum for studies related to mathematical sciences. It devotes exclusively to the publication of high-quality reviews, regular research papers and short communications in all areas of pure and applied mathematics. Mathematics also publishes timely and thorough survey articles on current trends, new theoretical techniques, novel ideas and new mathematical tools in different branches of mathematics.
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