摘要最小二乘法最早由高斯提出，主要用于天文学和地测学，在早期数理统计方法的发展中，最小二乘法起了很大的作用，因此丹麦统计学家霍尔把它称为“数理统计学的母亲”。随着现代电子计算机的飞速发展，不同的最小二乘问题的解法也相继被提出，并逐步建立了较为完善的理论，这使得最小二乘法的运用变得更加普及,并且许多其他学科也以最小二乘法为基础，可运用于信号处理、图像处理以及通信工程等方面。87385

本课题主要是研究最小二乘问题的数值解法，分为线性和非线性两大类。首先介绍解线性最小二乘算法的消元法以及正交化方法。其次给出解非线性最小二乘问题的连续极小化算法和牛顿迭代法。最后通过实例比较得出在处理不同问题时不同的最小二乘解法各有特点。

毕业论文关键词：最小二乘解法；线性；非线性；正交化；牛顿迭代法

Abstract    Least squares method is firstly put forward by Gaussian, mainly used for astronomy and geodesy。 In the development of early mathematical statistics method, least square method played a big role, so Danish statistician hall to it is called "the mother of mathematical statistics"。 With the rapid development of the modern computer, least squares problems of different solutions have been proposed, and gradually established a relatively perfect theory, which makes the use of least square method became more popular, and many other subjects are based on the least square method，and can be used in signal processing, image processing and communication engineering and so on。

This paper mainly studies the numerical solution of the least squares problem, which is pided into two categories: linear and nonlinear。 Firstly, the elimination method and the method of orthogonal design to solve linear least square algorithm are introduced。 Then it introduces the continuous minimization algorithm and Newton iteration methods for solving the nonlinear least squares problem。 Finally, it is concluded that the different least square solutions of different problems have different characteristics。

Keywords: least square method；linear；nonlinear；orthogonal；newton iterative method

目  录

第一章 绪论 1来-自+优=尔;论.文:网www.youerw.com +QQ7520.18766

1。1 研究背景及意义 1

1。2 最小二乘法的研究现状 1

1。3 本文主要内容 2

第二章 最小二乘简介 3

2。1基本概念 3

2。2 基本定理 4

2。3 算法简介 6

第三章 线性最小二乘问题解法 7

3。1 正交化方法的思想及步骤 7

3。1。1 算法思想 7

3。1。2 算法的步骤 8

3。2 消元法的思想及步骤 9

3。1。1 算法思想 9

3。1。2 算法的步骤 10

3。2 本章小结 10

第四章 非线性最小二乘问题解法 11

4。1 牛顿迭代算法思想及步骤 11

4。1。1 算法思想 11

4。1。2 算法的步骤 12

4。2 连续极小化算法思想及步骤