消去法等解线性方程组一论文.docx

消去法等解线性方程组（一） 消去法等解线性方程组（一） 中文摘要 Gauss消去法是目前求解中小规模线性方程组（即阶数不要太高，例如不超过1000）常用的方法，它一般用于系数矩阵稠密（即矩阵的绝大多数元素都是非零的）而没有任何特殊结构的线性方程组。本文对GAUSS、列主元及完全主元消去法这几种方法进行了比较。求解n阶线性方程组Ax=b的基本思想是在逐步消元的过程中，把方程组的系数矩阵化为上三角矩阵，从而将原方程组约化为容易求解的等价三角方程组，然后进行回代求解。 关键词：Gauss消去法； C++编程  ELIMINATION METHOD FOR SOLVING LINEAR EQUATIONS (一) ENGLISH ABSTRACT Gauss elimination method is the small size of solving linear equations (i.e. the order is not too high, for example, not more than 1000) commonly used method, it is generally used for the coefficient matrix dense (ie, most of the matrix elements are nonzero) without any special structure of linear equations. In this paper, GAUSS, the main element and completely Elimination PCA these methods were compared. Solving n linear equations Ax = b is the basic idea of the gradual elimination process, the equations of the coefficient matrix into upper triangular matrix, thus reduced to the original equations easy to solve trigonometric equations equivalence group then back substitution to solve. Keywords: Gauss elimination method; C + + Programming  目录 TOC \o 1-3 \h \z \u HYPERLINK \l _Toc358488043 消去法等解线性方程组（一） PAGEREF _Toc358488043 \h 1 HYPERLINK \l _Toc358488044 目录 PAGEREF _Toc358488044 \h 3 HYPERLINK \l _Toc358488045 一、问题背景 PAGEREF _Toc358488045 \h 4 HYPERLINK \l _Toc358488046 二、问题提出 PAGEREF _Toc358488046 \h 5 HYPERLINK \l _Toc358488047 三、数学原理与算法分析 PAGEREF _Toc358488047 \h 5 HYPERLINK \l _Toc358488048 3.1不选主元Gauss消去法 PAGEREF _Toc358488048 \h 5 HYPERLINK \l _Toc358488049 3.1.1不选主元Gauss消去法数学原理 PAGEREF _Toc358488049 \h 5 HYPERLINK \l _Toc358488050 3.1.2不选主元Gauss消去法算法步骤 PAGEREF _Toc358488050 \h 6 HYPERLINK \l _Toc358488051 3.2列主元Gauss消去法 PAGEREF _Toc358488051 \h 7 HYPERLINK \l _Toc358488052 3.2.1列主元Gauss消去法数学原理 PAGEREF _Toc358488052 \h 7 HYPERLINK \l _Toc358488053 3.2.1列主元Gauss消去法算法描述 PAGEREF _Toc358488053 \h 8 HYPERLINK \l _Toc358488054 3.3完全主元Gauss消去法 PAGEREF _Toc358488054 \h 8 HYPERLINK \l _Toc