改进的牛顿预测–校正格式

Predictor-corrector improvement of Newton method

中文关键词  牛顿算法  预测–校正格式  非线性方程  迭代方法

英文关键词  Newton method  predictor-corrector rule  nonlinear equation  iterative methods

中文摘要

      在数值分析领域中, 牛顿算法由于其形式的简单性及快速的收敛性而被广泛地应用于求解非线性方程问 题. 受一类求解方程的预测–校正技术的启示, 本文针对求解非线性方程单根的问题提出了一种牛顿预测–校正格 式, 并将其推广到多维向量值函数情况. 为此, 首先用图描述了这种新的预测–校正格式并导出了其收敛阶. 这种新 格式每步迭代仅需计算一次函数值和一次导函数值. 然后, 经过测试函数的检验, 并与牛顿算法及其他高阶算法 (1 + p2 阶、3阶、4阶、5阶、6阶)比较, 表明新算法具有较快的收敛性. 最后, 将这种新格式推广到多维向量值函数, 采用泰勒公式证明了其收敛性, 并给出了一个二维算例来验证其收敛的有效性.

英文摘要

      In numerical analysis, Newton method is the most commonly used iterative technique for determining a root of a nonlinear equation for its simplicity and fast rate of convergence. Motivated by a class of predictor-corrector technique for root-finding, we present a predictor-corrector modification for the standard Newton method in approximating the root of a univariate nonlinear function, and extend it to the multi-dimensional vector-valued functions. First, the predictor-corrector rule is described using a figure and its convergence order is analyzed. The modified method only requires evaluating one function and one first derivative in a step. Then, we use numerical examples to demonstrate the faster convergence achieved by this modification than the Newton method and other higher-order (1 2 order, 3rd order, 4th order, 5th order and 6th order) algorithms. Third, the predictor-corrector improvement is extended to multi-dimensional vector valued functions. The convergence is proved by using Taylor formula, and is illustrated by a two-dimensional example.