基于Thiele-连分式逼近的改进迭代算法及收敛性分析

Improved Iterative Formula and Convergence Analysis Based on Thiele’s Continued Fraction for Solving Nonlinear Equations

基于Thiele-连分式逼近是有理函数逼近的重要组成部分,在很多领域实现了应用.本文通过对Thiele-连分式的前三项多项式截断和泰勒级数展开得到3个新的求解非线性方程的迭代格式,通过分析其收敛性验证了其在第n项截断多项式的收敛阶数随n值的增加而增高,并通过数值实例验证该方法的收敛速度和效率指数优于Newton迭代.

Thiele’s continued fraction approximation is an important aspect of rational function approximation,and has realized important applications in many fields.In this paper,three new iterative schemes for solving nonlinear equations are obtained by truncating the first three polynomial terms of Thiele’s continued fraction and expanding Taylor series.By analyzing its convergence,it is proved that the convergence order of the NTH term truncated polynomial increases with the increase of n value,and the convergence speed and efficiency index of the proposed method are better than those of Newton iteration by numerical examples.