【摘要】：隨著科學(xué)技術(shù)的發(fā)展、電子信息技術(shù)的日益更新,科學(xué)研究及工程計(jì)算中的許多問(wèn)題都可以通過(guò)數(shù)學(xué)模型的構(gòu)建,將實(shí)際問(wèn)題轉(zhuǎn)化為方程(包括線性或者非線性代數(shù)方程等)的求解問(wèn)題.本文主要研究非線性方程f(x)=0的迭代法求解.總共分為三章:第一章為緒論部分.介紹了所研究的非線性方程的實(shí)際背景,回顧了迭代法的發(fā)展與研究現(xiàn)狀,概述了本文所需的一些基本概念.第二章提出了一種改進(jìn)的三步六階迭代法.此方法以O(shè)strowski四階迭代法和M.Grau六階迭代法為基礎(chǔ)進(jìn)行構(gòu)造.新迭代法每迭代一次,需要計(jì)算三個(gè)函數(shù)值和一個(gè)一階導(dǎo)數(shù)值,其效率指數(shù)為6~(1/4)≈1.565.在章末通過(guò)數(shù)值算例驗(yàn)證了新方法的收斂階.第三章提出了改進(jìn)的兩族八階迭代法.首先介紹Hermite插值擬合法,利用已知函數(shù)值對(duì)原函數(shù)進(jìn)行擬合,近似代替新增導(dǎo)數(shù)值,減少新迭代法中函數(shù)導(dǎo)數(shù)的計(jì)算量.再在W.Bi的八階迭代法和王霞的八階迭代法的基礎(chǔ)上,改進(jìn)得到的一族改進(jìn)的三步八階迭代法,其范圍更加廣泛,當(dāng)對(duì)未知量取定值時(shí)可以得到王霞的八階迭代法.另一族八階迭代法是根據(jù)已知函數(shù)值引入實(shí)值函數(shù),對(duì)新增導(dǎo)數(shù)值近似替換修改得到,當(dāng)取不同實(shí)值函數(shù)時(shí)可以得到不同的八階迭代法.兩族新迭代法每迭代一次,都需要計(jì)算三個(gè)函數(shù)值和一個(gè)一階導(dǎo)數(shù)值,效率指數(shù)都為8~(1/4)≈1.682.在章節(jié)末通過(guò)數(shù)值算例驗(yàn)證了兩族新迭代法的收斂階.
[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，

本文編號(hào)：2221976