一种基于Lagrange插值的新高阶三步迭代算法

A New High-Order Three-Step Iterative Method Based on Lagrange Interpolation Method

摘要: 本文基于一个求解非线性方程的三阶迭代法，引入Lagrange插值思想来估计其导数，构造出一种新的高阶迭代法。并给出收敛性分析。可以发现只需要增加一个函数值计算，能够将一个四阶的迭代法的收敛阶提高到五阶。我们给出了五阶收敛性的理论分析，进一步通过数值实验验证了新的迭代法的有效性，对非线性方程问题的求解具有一定的实际应用价值。

Abstract: This paper primarily discusses iterative methods for solving nonlinear equations. Based on a third-order iterative method for solving nonlinear equations, we introduce an Lagrange interpolation approach to estimate its derivative, thereby constructing a new high-order iterative method. A convergence analysis is also provided. It is found that by adding just one function evaluation, the convergence order of a fourth-order iterative method can be increased to fifth order. We provide a theoretical proof of the fifth-order convergence and further verify the effectiveness of the new iterative method through numerical experiments, demonstrating its potential value in solving nonlinear equations.