幂法和反幂法的matlab实现

幂法和反幂法的matlab实现

幂法求矩阵主特征值及对应特征向量

摘要

矩阵特征值的数值算法，在科学和工程技术中很多问题在数学上都归结为矩阵的特征值问题，所以说研究利用数学软件解决求特征值的问题是非常必要的。实际问题中，有时需要的并不是所有的特征根，而是最大最小的实特征根。称模最大的特征根为主特征值。

幂法是一种计算矩阵主特征值（矩阵按模最大的特征值）及对应特征向量的迭代方法，它最大的优点是方法简单，特别适用于大型稀疏矩阵，但有时收敛速度很慢。

用java来编写算法。这个程序主要分成了四个大部分：第一部分为将矩阵转化为线性方程组；第二部分为求特征向量的极大值；第三部分为求幂法函数块；第四部分为页面设计及事件处理。其基本流程为幂法函数块通过调用将矩阵转化为线性方程组的方法，再经过一系列的验证和迭代得到结果。

关键字：主特征值；特征向量；线性方程组；幂法函数块

POWER METHOD FOR FINDING THE EIGENVALUES AND CORRESPONDING EIGENVECTORS OF THE

MATRIX

ABSTRACT

Numerical algorithm for the eigenvalue of matrix, in science and engineering technology, a

lot of problems in mathematics are attributed matrix characteristic value problem, so that studies using mathematical software to solve the eigenvalue problem is very necessary. In practical problems, sometimes need not all eigenvalues, but the maximum and minimum eigenvalue of real. The characteristic value of the largest eigenvalue of the modulus maximum.

Power method is a calculation of main features of the matrix values (matrix according to the characteristics of the largest value) and the corresponding eigenvector of iterative method. It is the biggest advantage is simple method, especially for large sparse matrix, but sometimes the convergence speed is very slow.

Using java to write algorithms. This program is divided into three parts: the first part is the matrix is transformed into linear equations; the second part for the sake of feature vector of the maximum; the third part is

the exponentiation function block. The fourth part is the page design and event

processing .The basic process is a power law function block by calling the matrix is transformed into linear equations method, after a series of validation and iteration results.

Power method for finding the eigenvalues and corresponding eigenvectors of the matrix

Key words: Main eigenvalue; characteristic vector; linear equations; power function block

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目录

1幂法......................................................... . (1)

1.1幂法的基本理论和推导 (1)

1.2幂法算法的迭代向量规范化 (2)

2概要设计........................................................ (3)

2.1设计背景 (3)

2.2运行流程........................................... . (3)

2.3运行环境........................................... (3)

3程序详细设计 (4)

3.1矩阵转化为线性方程组……..………………………………………. .

4

3.2特征向量的极大值 (4)

3.3求幂法函数块............….....…………...…......…………………………

3.4界面设计与事件处理..........................................................................4运行过程及结果................................................ (6)

4.1 运行过程....................................... ..................………………………………………. .6

4.2 运行结果................................................ .. (6)