关于一种推广形式的Ehrlich迭代法  

On a Generalization of Ehrlich's Method

作　　者：黄清龙[1] 

机构地区：[1]常州大学怀德学院,江苏靖江214500

出　　处：《常州大学学报（自然科学版）》2016年第6期100-103,共4页Journal of Changzhou University：Natural Science Edition

基　　金：江苏省靖江市科技局与常州大学怀德学院产学研合作项目(CDHJZ1509008)

摘　　要：讨论Ehrlich迭代法的一种推广形式。这个推广的Ehrlich迭代法适用于同时求解高次代数方程的所有不同重数的复根。构造出迭代公式并给出收敛性定理,借助数学归纳法提供了收敛性和收敛阶的简洁证明。推广的Ehrlich迭代法和Newton迭代法的效率比较表明:当代数方程的根全为单根时,若代数方程次数不超过8则Newton迭代法的效率更高,若代数方程次数超过8则推广的Ehrlich迭代法效率更高;当高次代数方程的根不全为单根时,推广形式的Ehrlich迭代法的计算效率总是高于Newton迭代法的计算效率。A generalized Ehrlich＇s method is discussed.This generalized Ehrlich＇s method can be used to find all multiple complex roots of a high-order algebraic equation.The iterative formula is constructed and a version of its convergence theorem is given;by mathematical induction a more concise proof of the convergence and convergence order is proposed.Finally,By comparing the computational efficiency of the generalized Ehrlich＇s method and that of Newton method,the conclusions show that when the roots of algebraic equation are all single,the Newton iterative method is more efficient if the order of algebraic equations does not exceed 8,on the contrary the generalized Ehrlich iterative method is more efficient if the order of algebraic equation exceeds 8.The efficiency analysis also shows that the computational efficiency of the generalized Ehrlich＇s method is always higher than the computational efficiency of Newton iterative method when the roots of algebraic equation are not all single.

关 键 词：Ehrlich迭代法 推广 收敛性 计算效率 

分 类 号：O241.7[理学—计算数学]