非线性方程的算法研究

1、非线性方程的算法研究    非线性方程的算法研究求解非线性方程的迭代算法研究     目录：摘要 5-6ABSTRACT 6-7第1章绪论 10-171.1研究背景 10-111.2迭代法及其有关概念、定理 11-121.2.1有关迭代法的基本概念 111.2.2与迭代法相关的定义 111.2.3有关迭代法的收敛定理 11-121.3Newtonsmethod迭代法及其推导 12-141.3.1牛顿法(TheNewtonsme

2、thod)及其局部收敛定理 12-131.3.2牛顿迭代法(Newtonsmethod)的推导 13-141.3.3牛顿法的几何意义 141.4三阶收敛牛顿法的变式 14-161.5本文的主要工作及结构 16-17第2章一个新的六阶收敛牛顿法 17-222.1引言 172.2算法描述 17-182.3收敛性分析 18-202.4数值计算 20-212.5结论 21-22第3章一类求解非线性方程的算法 22-273.1引言 223.2算法描述及其收敛性分析 

3、22-243.2.1新算法导入 22-233.2.2收敛性分析 23-243.3Ostrowski算法的新推导 243.4数值计算 24-253.5结论 25-27第4章一族解非线性方程的三阶或四阶方法 27-334.1引言 27-284.2方法的提出及其收敛性分析 28-304.3与其它迭代式的关系 30-314.4数值计算 31-324.5结论 32-33第5章对非线性方程迭代算法的进一步思考 33-375.1引言 335.2修正的-幂平均牛顿法 33

4、-345.3八个参数的牛顿迭代式 34-37第6章总结与展望 37-396.1主要结论 376.2后续工作中的展望 37-39致谢 39-40参考文献 40-44附录 44 【摘要】 在利用数学工具研究社会现象和自然现象,或解决工程技术等问题时,很多问题都可以归结为非线性方程f ( x ) = 0的求解问题,无论在理论研究方面还是在实际应用中,求解非线性方程都占了非常重要的地位。迭代法是求解非线性方程f ( x ) = 0根的一种最重要的方法,而迭代法的优劣对于非线性问题求解速度的快慢和结果的好坏都有很大的影响,所

5、以从实际出发,进行高计算效能迭代算法的研究具有重要的科学价值和实际意义。本文讨论了求解非线性方程的迭代算法研究,这里所说的迭代算法是指在Newton法基础上改进的算法。主要讨论基于Newton法的迭代函数,通过增加迭代、近似代替或增加参数,提出了一些新的牛顿法的变式,给出了实数范围内求解单根的迭代方法,并通过数值实验验证了新算法的有效性。全文共分为六章。第一章概述了相关的基础理论,主要介绍了非线性方程的研究背景和求解非线性方程的常用方法迭代法,详细回顾了Newton法及其研究现状。第二章讨论了通过结合经典牛顿法与几何平均牛顿法,提出了一个新的求解非线性方程的六阶收敛算法。在每次迭代过程中只需计

6、算两个函数值和两个一阶导数值,而且无需计算二阶导数。对一组普遍所采用的测试问题而言,数值计算表明该算法所需要的迭代次数和效率指数对大多数的问题都优于经典牛顿法和几何平均牛顿法。第三章讨论在已有算法的基础上,提出了构造解非线性方程新算法的一种通用的框架,即综合利用各种不同插值方法的优点,通过令两个同阶的迭代式近似相等,将某一式子的近似值代入其它同阶的迭代式中,可以导出同阶收敛且具有自己特性的新的或已存在的算法,采用通用例子进行的数值实验表明新算法能与经典牛顿法媲美。而且,许多求解非线性方程的算法如著名的四阶收敛Ostrowski算法也可在此框架下得到。第四章讨论了将已有算法的存在形式进行变形,可

7、以归纳为统一的形式,通过增加参数得到了更一般的算法,收敛性分析证明在参数满足特定关系的条件下,将得到不同收敛阶的新算法或已存在的算法。第五章从理论上阐述了两个加参迭代式的收敛性。第六章总结了本文的主要结论,并对牛顿法研究的前景以及下一步的研究的动向进行展望。  【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