Shanks变换及其应用+文献综述
本文介绍了Shanks变换的定义及形式,并使用Shanks变换分析讨论了线性方程组的迭代求解问题.在一定条件下将原本发散的迭代序列改变为收敛序列,并加速求解.本文还讨论了Shanks变换在级数加速求解的应用.33226
毕业论文关键词: 线性方程组 级数 迭代 加速 收敛 Shanks变换
Shanks Transformation and its Application
Abstract
The iterative method of linear algebraic equations is an extreme method is an effective method for the solution of large sparse matrix equation s. The basic idea is to a certain limit process to gradually approach the exact solution of linear equations, a step-by-step approximation method. The iterative method will be iterative formula for a deformation of n linear equations. For any given iteration of the initial value, by an iterative scheme can generate a vector sequence, our aim is the solution to solving the equations, so we will want to limit approximation the solution of equations of vector sequences.
This paper introduced the definition and form of the Shanks Transformation, The iterative method of linear equations is discussed and analysed by using Shanks Transformation. The pergent iterative sequence is changed into a convergent one under some conditions, and accelerated the solutions. The application of the Shanks Transformation in the acceleration of the solution of the series has been studied.
Key words: linear equations, series, iteration, acceleration, convergence, Shanks Transformation
目录
1 Shanks变换介绍 1
1.1 Shanks变换简介 1
1.2 Shanks变换举例 1
1.3 Shanks变换推导 2
2 Shanks变换的应用 3
2.1 Shanks变换在线性方程组迭代解法的收敛与加速中的应用 3
2.1.1数值例子 9
2.2 Shanks变换在级数收敛中的加速作用 12
2.2.1 Shanks变换在Floquet矢量模级数的混合加速算法中的应用 12
2.2.2 数值算例 14
2.2.3 Shanks变换在加速高速电路中周期结构电容参数计算中的作用 16
2.2.3 算例 16
3 结论 18
1 Shanks变换介绍
1.1 Shanks变换简介[1-3]
在数值分析中,Shanks变换是一种增加数列收敛速率的非线性加速方法[4].该方法是以在1955年重新发现该数列变换的Daniel Shanks的名字来命名的,但其最初是由R.schmidt在1941提出并且发表的.
对于一个数列 来说,级数
的值是确定的,我们把部分和 定义为
并且构成了一个新的数列 .当级数收敛时, 将会趋近于n趋向于无穷大时的极限A,我们把数列 的Shanks变换S( )定义为
并且构成了一个新的数列,数列S( )总是比数列 更快的收敛,通过计算 , 等等来重复的使用Shanks变换将会使收敛进一步加快.
Shanks变换是一种加速收敛技术,应用于级数的处理,使得收敛更快.也应用于微分方程解的处理.
1.2 Shanks变换举例
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