弦截法及Steffensen方法的收敛速度及计算效率

弦截法及Steffensen方法的收敛速度及计算效率

论文编号:XXLW103 论文字数:5544,页数:18

摘要
在数学中的许多问题，我们进行分析和归纳后为解函数方程,其中为线性或非线性的方程，而求解方程最有效的方法之一便是迭代法。本文对几种常用迭代法即Newton法、弦截法及Steffensen方法的收敛速度及计算效率进行分析比较，并对Steffensen方法进行编程实例演示。为能更加深刻地理解和把握Newton法、弦截法及Steffensen方法这3种迭代方法，通过对比分析它们各方面的优劣，我们能选用一个较好的迭代方法解决给定的问题。Steffensen方法是一种很好的求解非线性方程组的方法，它有和Newton法一样的计算效率，而没有导数的计算，有更广的使用范围。通过研究，我们更加全面地理解各种迭代法的使用条件和效率，让我们能更好地用迭代法解决相关问题。

关键词：Newton法弦截法 Steffensen方法收敛速度计算效率

Abstract
Many problems in mathematics could be summarized as a linear or nonlinear function equation:. Iterative method is one of the most efficient methods for solving the equation. In this paper, we study the convergent rate and computational efficiency of some familiar iterative methods, such as Newton method、secant method and Steffensen method, and some numerical experiments as well. By comparing their advantages and shortcomings, we could choose a more suitable method to solve the equation. Among others, Steffensen method is a very good method of solving nonlinear equations, which has the same computational efficiency as Newton method and avoids computing derivative. Through the study, we could know the conditions and efficiency of the three iterative methods better and apply these methods efficiently.

Keywords: Newton method; secant method; Steffensen method; convergence rate; computational efficiency

中文摘要 .......................................................................i
英文摘要 ......................................................................ii
目录 .........................................................................iii
第一章    三种常见数值解法及其原理 .............................................1
1.1 前言 .................................................................1
1.2 Newton法求解非线性方程 ...............................................2
1.3 弦截法求解非线性方程 ..................................................4
1.4 Steffensen法求解非线性方程 ...........................................5
第二章   三种常见数值解法收敛速度比较 ...........................................7
     2.1 收敛速度的定义 ........................................................7
     2.2 Newton法、弦截法、Steffensen方法收敛速度的分析 .......................7
     2.3 结论 ..................................................................8
第三章   三种常见数值解法计算效率的比较 .........................................9
     3.1 计算效率的定义 ........................................................9
     3.2 Newton法、弦截法、Steffensen方法计算效率的分析 .......................9
     3.3 实例演示及结论 ........................................................9
第四章   Steffensen迭代法在MATLAB上的实现 ....................................11
第五章   总结 ..................................................................13
致谢 ...........................................................................14
参考文献 .......................................................................15

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