矩阵QR分解方法及其应用及C++源程序
关键词 QR分解 Householder变换 Givens变换 Schmidt正交化 Hessenberg矩阵 特征值 最小二乘法
毕业论文设计说明书外文摘要
Title QR decomposition of the matrix and its application
Abstract
The matrix decomposition is an integral part of matrix theory, has been a particular application in the fields of technology, especially in the computer widely used. Which is defined as a matrix decomposition into a special matrix (such as triangular matrix) and the product or form. There matrix decomposition method, many of them common with QR decomposition, triangular decomposition, full rank decomposition and singular value decomposition and the like. The QR decomposition is often used to solve the linear least squares problem, is also currently seeking the most effective general all eigenvalues of matrix and widely used method, and has great significance in practical applications. It has been successfully used in image processing and analysis, face recognition, watermarking and other practical applications. The present paper on matrix QR decomposition expanded summary exposition introduces the ideas, principles, procedures, and use of the method, and examples presented in detail and accompanied with a QR decomposition algorithm for solving eigenvalues and C language.
Keywords QR decomposition Householder Transformation Givens Transformation
Eigenvalues Linear least square method
1 绪论 1
1.1 研究背景及现状 1
1.2 线性代数的一些基本概念 1
1.2.1 范数、线性赋范空间 1
1.2.2 Hessenberg矩阵 2
1.2.3 正定矩阵 2
1.3 Householder变换 2
1.3.1 Householder变换定义 2
1.3.2 Householder变换性质 2
1.4 Givens变换 4
1.4.1 Givens变换定义 4
1.4.2 Givens变换性质 4
2 QR分解 6
2.1 QR分解定义 6
2.2 QR定理及QR分解方法 6
2.3 分解唯一性定理 8
3 QR算法求特征值 11
3.1 QR算法基本思想 11
3.2 QR算法的收敛性 12
3.3 QR算法迭代过程 13
3.4 QR算法迭代控制 20
3.5 带原点位移的QR算法 21
3.6 QR分解求特征值算法源程序 21
3.6.1 各函数功能实现 21
3.6.2 数据实验 22