非線性方程的算法研究

1、非線性方程的算法研究    非線性方程的算法研究求解非線性方程的迭代算法研究     目錄：摘要 5-6ABSTRACT 6-7第1章緒論 10-171.1研究背景 10-111.2迭代法及其有關(guān)概念、定理 11-121.2.1有關(guān)迭代法的基本概念 111.2.2與迭代法相關(guān)的定義 111.2.3有關(guān)迭代法的收斂定理 11-121.3Newtonsmethod迭代法及其推導(dǎo) 12-141.3.1牛頓法(TheNewtonsme

2、thod)及其局部收斂定理 12-131.3.2牛頓迭代法(Newtonsmethod)的推導(dǎo) 13-141.3.3牛頓法的幾何意義 141.4三階收斂牛頓法的變式 14-161.5本文的主要工作及結(jié)構(gòu) 16-17第2章一個(gè)新的六階收斂牛頓法 17-222.1引言 172.2算法描述 17-182.3收斂性分析 18-202.4數(shù)值計(jì)算 20-212.5結(jié)論 21-22第3章一類求解非線性方程的算法 22-273.1引言 223.2算法描述及其收斂性分析 

3、22-243.2.1新算法導(dǎo)入 22-233.2.2收斂性分析 23-243.3Ostrowski算法的新推導(dǎo) 243.4數(shù)值計(jì)算 24-253.5結(jié)論 25-27第4章一族解非線性方程的三階或四階方法 27-334.1引言 27-284.2方法的提出及其收斂性分析 28-304.3與其它迭代式的關(guān)系 30-314.4數(shù)值計(jì)算 31-324.5結(jié)論 32-33第5章對非線性方程迭代算法的進(jìn)一步思考 33-375.1引言 335.2修正的-冪平均牛頓法 33

4、-345.3八個(gè)參數(shù)的牛頓迭代式 34-37第6章總結(jié)與展望 37-396.1主要結(jié)論 376.2后續(xù)工作中的展望 37-39致謝 39-40參考文獻(xiàn) 40-44附錄 44 【摘要】 在利用數(shù)學(xué)工具研究社會現(xiàn)象和自然現(xiàn)象,或解決工程技術(shù)等問題時(shí),很多問題都可以歸結(jié)為非線性方程f ( x ) = 0的求解問題,無論在理論研究方面還是在實(shí)際應(yīng)用中,求解非線性方程都占了非常重要的地位。迭代法是求解非線性方程f ( x ) = 0根的一種最重要的方法,而迭代法的優(yōu)劣對于非線性問題求解速度的快慢和結(jié)果的好壞都有很大的影響,所

5、以從實(shí)際出發(fā),進(jìn)行高計(jì)算效能迭代算法的研究具有重要的科學(xué)價(jià)值和實(shí)際意義。本文討論了求解非線性方程的迭代算法研究,這里所說的迭代算法是指在Newton法基礎(chǔ)上改進(jìn)的算法。主要討論基于Newton法的迭代函數(shù),通過增加迭代、近似代替或增加參數(shù),提出了一些新的牛頓法的變式,給出了實(shí)數(shù)范圍內(nèi)求解單根的迭代方法,并通過數(shù)值實(shí)驗(yàn)驗(yàn)證了新算法的有效性。全文共分為六章。第一章概述了相關(guān)的基礎(chǔ)理論,主要介紹了非線性方程的研究背景和求解非線性方程的常用方法迭代法,詳細(xì)回顧了Newton法及其研究現(xiàn)狀。第二章討論了通過結(jié)合經(jīng)典牛頓法與幾何平均牛頓法,提出了一個(gè)新的求解非線性方程的六階收斂算法。在每次迭代過程中只需計(jì)

6、算兩個(gè)函數(shù)值和兩個(gè)一階導(dǎo)數(shù)值,而且無需計(jì)算二階導(dǎo)數(shù)。對一組普遍所采用的測試問題而言,數(shù)值計(jì)算表明該算法所需要的迭代次數(shù)和效率指數(shù)對大多數(shù)的問題都優(yōu)于經(jīng)典牛頓法和幾何平均牛頓法。第三章討論在已有算法的基礎(chǔ)上,提出了構(gòu)造解非線性方程新算法的一種通用的框架,即綜合利用各種不同插值方法的優(yōu)點(diǎn),通過令兩個(gè)同階的迭代式近似相等,將某一式子的近似值代入其它同階的迭代式中,可以導(dǎo)出同階收斂且具有自己特性的新的或已存在的算法,采用通用例子進(jìn)行的數(shù)值實(shí)驗(yàn)表明新算法能與經(jīng)典牛頓法媲美。而且,許多求解非線性方程的算法如著名的四階收斂Ostrowski算法也可在此框架下得到。第四章討論了將已有算法的存在形式進(jìn)行變形,可

7、以歸納為統(tǒng)一的形式,通過增加參數(shù)得到了更一般的算法,收斂性分析證明在參數(shù)滿足特定關(guān)系的條件下,將得到不同收斂階的新算法或已存在的算法。第五章從理論上闡述了兩個(gè)加參迭代式的收斂性。第六章總結(jié)了本文的主要結(jié)論,并對牛頓法研究的前景以及下一步的研究的動向進(jìn)行展望。  【Abstract】 When we make use of mathematical tools to research social phenomena and natural phenomena, or to resolve the engineering and other problems, a lot of pro

8、blems can be ended up with solving the equation f ( x ) = 0, the nonlinear equations have a very important role in research of both theory and practical application. The iterative method is an important method to slove the equation f ( x ) = 0, whether the nonlinear equations will be solved well or

9、not is directly affected by the choice of iterative method. Therefore, the research on iterative method with high efficiency means a lot in terms of both scientific research and practical application.              This paper discu

10、sses the research of iterative method for solving nonlinear equations, here refers to the iterative method is based on the improved algorithm of Newtons method. This paper discusses the method based on Newtons iteration function, by increasing the iteration, similar to replace or increase the parame

11、ters, thus put forward some new variants of Newtons method, and give the iterative method for solving simple root in R, and through numerical examples are given to illustrate the effectiveness of the new method. This paper consists of six chapters.Chapter 1 we summarize mainly the associated basic t

12、heory, mainly introduces the research background of nonlinear equations and the common methods for solving nonlinear equations - iterative methods, a detailed review Newtons method and the study status.Chapter 2 discusses through a combination of classical Newtons method and the geometric mean Newto

13、ns method, and proposed a new sixth order convergence of Newons method. Which only requires two evaluations of the function and two evaluations of the derivative per iteration, but does not require the second derivative. For a group of widely used testing questions, numerical conclusions show that t

14、he efficiency of the method is better than Classical Newtons method and Geometric mean Newtons method for most questions.Chapter 3 presents a common framework of new algorithm for solving nonlinear equations, which based on the existing algorithm. That is, comprehensive utilization of the advantages

15、 of different interpolation methods, let two methods of the same order approximately equate, the approximation of a particular statement holds on behalf of the other into the same order of the iterative, we can export the same order convergence and has its own characteristics of new or existing meth

16、ods. Some common numerical examples show that the new method can compete with the classical Newtons method. Moreover, many algorithms for solving nonlinear equations such as the famous fourth-order convergence Ostrowskis algorithm can also be obtained within this framework.Chapter 4 discusses the ex

17、istence of the existing algorithms form deformation, can be summarized as a unified form, by adding parameters to get a more general algorithm, convergence analysis shows that in the relations between the parameters satisfy certain conditions, and will be obtain different convergence order of the new algorithms or pre-existing algorithms.Chapter 5 from the theory illustrated the convergence of two added parameters iterative.Chapter 6 sum