摘要不管是在工程应用，还是科学计算中，我们常常都会遇到许多关于求解非线性方程的问题。而什么是非线性方程呢？就是因变量与自变量之间的关系，而不是线性的非线性方程关系，这类方程很多，例如平方关系、三角函数关系、指数关系等等。86250

我们在求解这一类的方程时，经常会用到迭代法，且这些迭代法大多是有收敛条件的。对于迭代过程的收敛性，只要有足够的迭代次数，我们就可以使结果达到任意的精度。但有时迭代过程收敛缓慢，从而使计算量变得很大，由此可知，迭代过程的加速是个非常重要的过程。因此，在本文总将会涉及非线性方程迭代法，如不动点迭代法、Newton迭代法，及它们的收敛条件；同时本文也会涉及加速收敛法，比如Aitken加速收敛法、 Steffensen加速收敛法，将为大家讲述它们的加速原理。

其中，Newton迭代法是求解非线性方程的主要方法之一，其优点就是收敛速度较快，但其也存在不少的缺点。因此本文将会为大家讲述一些关于Newton迭代法的改进方法，来提高Newton迭代法的计算精度和减少计算的。

毕业论文关键字：非线性方程、迭代法、Aitken加速收敛法

Abstract Both in engineering application and in scientific computing，we often encounter many problems about the solution of nonlinear equations。The so-called nonlinear equations contain the relationship between the dependent variable and independent variable and the relationship between these variables are not linear 。These equations have many different forms such as square,exponential, trigonometric functions and so on。

The iterative methods are used for solving these type of equations and they are not always converged for them。 For the convergence of the iterative process, as long as you have to do sufficient number of iterations, the accuracy of solution can achieve any arbitrary precision。 But sometimes iterative process converges slowly, so the amount of calculation becomes large。 Therefore, to accelerate the iterative process is an important method。 This involves iterative methods of nonlinear equations, such as fixed point iteration method, Newton iterative method, and their convergence condition。 This article will involve in the accelerating convergence method, such as Aitken accelerating convergence method, Steffensen method to accelerate convergence。These methods will tell you about acceleration principle。

Wherein, Newton iterative method is one of the main methods for solving nonlinear equations, and its advantage is that its convergence is fast, but it also has many disadvantages。 Therefore, this article will tell you something about the improvement Newton iteration method to improve the accuracy of Newton interative and reduce the calculation of Newton iterative method。

Key words：Nonlinear equations, Iterative method, Aitken method 

目 录

第一章  绪论 1

1。1  研究背景 1

1。2  非线性方程的概述 1

第二章  不动点迭代法与收敛性 3

2。1  不动点迭代法 3

2。2  收敛性 4

2。3  收敛速度 7

第三章  Newton 迭代法与其收敛性 9

3。2  Newton迭代法的收敛性 11

3。3  Newton迭代法的总结 13

3。3。1  Newton迭代法可以应用于开方公式