求解非线性方程的高阶迭代法研究
发布时间：2018-09-04 11:35
【摘要】：随着科学技术的发展、电子信息技术的日益更新,科学研究及工程计算中的许多问题都可以通过数学模型的构建,将实际问题转化为方程(包括线性或者非线性代数方程等)的求解问题.本文主要研究非线性方程f(x)=0的迭代法求解.总共分为三章:第一章为绪论部分.介绍了所研究的非线性方程的实际背景,回顾了迭代法的发展与研究现状,概述了本文所需的一些基本概念.第二章提出了一种改进的三步六阶迭代法.此方法以Ostrowski四阶迭代法和M.Grau六阶迭代法为基础进行构造.新迭代法每迭代一次,需要计算三个函数值和一个一阶导数值,其效率指数为6~(1/4)≈1.565.在章末通过数值算例验证了新方法的收敛阶.第三章提出了改进的两族八阶迭代法.首先介绍Hermite插值拟合法,利用已知函数值对原函数进行拟合,近似代替新增导数值,减少新迭代法中函数导数的计算量.再在W.Bi的八阶迭代法和王霞的八阶迭代法的基础上,改进得到的一族改进的三步八阶迭代法,其范围更加广泛,当对未知量取定值时可以得到王霞的八阶迭代法.另一族八阶迭代法是根据已知函数值引入实值函数,对新增导数值近似替换修改得到,当取不同实值函数时可以得到不同的八阶迭代法.两族新迭代法每迭代一次,都需要计算三个函数值和一个一阶导数值,效率指数都为8~(1/4)≈1.682.在章节末通过数值算例验证了两族新迭代法的收敛阶.
[Abstract]:With the development of science and technology and the updating of electronic information technology, many problems in scientific research and engineering calculation can be constructed by mathematical model. The practical problem is transformed into the solving problem of the equation (including linear or nonlinear algebraic equation etc.). In this paper, the iterative method for solving the nonlinear equation f (x) _ 0 is studied. A total of three chapters: the first chapter is the introduction. This paper introduces the practical background of the nonlinear equations studied, reviews the development and research status of iterative methods, and summarizes some basic concepts needed in this paper. In chapter 2, an improved three-step and six-order iterative method is proposed. This method is constructed on the basis of Ostrowski fourth order iterative method and M.Grau sixth order iteration method. The new iteration method needs to calculate three function values and a first order derivative value, and its efficiency exponent is 6 ~ (1 / 4) 鈮，
本文编号：2221976
[Abstract]:With the development of science and technology and the updating of electronic information technology, many problems in scientific research and engineering calculation can be constructed by mathematical model. The practical problem is transformed into the solving problem of the equation (including linear or nonlinear algebraic equation etc.). In this paper, the iterative method for solving the nonlinear equation f (x) _ 0 is studied. A total of three chapters: the first chapter is the introduction. This paper introduces the practical background of the nonlinear equations studied, reviews the development and research status of iterative methods, and summarizes some basic concepts needed in this paper. In chapter 2, an improved three-step and six-order iterative method is proposed. This method is constructed on the basis of Ostrowski fourth order iterative method and M.Grau sixth order iteration method. The new iteration method needs to calculate three function values and a first order derivative value, and its efficiency exponent is 6 ~ (1 / 4) 鈮，
本文编号：2221976
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