英语翻译在自然科学和工程技术中,很多问题的解决最后都归结为解线性代数方程组.随着科学技术的不断深入发展,所需要求解问题的复杂性不断提高.因此,大型稀疏方程组的高效求解在科学计算中具有普遍意义.线性方程组的数值解法一般分为直接法和迭代法.本论文主要分析解线性方程组的直接法中的列主元高斯消去法、LU分解法,进而给出求解对称正定矩阵的平方根算法,并且还分析迭代法中的雅克比迭代法、高斯-赛德尔迭代法和逐次超松弛迭代法,最后在分析逐次超松弛迭代法法的基础上讨论共轭梯度法.我还运用已学的数值分析的知识对上述方法给出比较详尽的推导过程,并编制了相应的MATLAB程序,还给出了实例,进行求解分析.关键字：线性方程组 直接法 平方根算法 迭代法 共轭梯度法

In natural science and engineering technology, the resolutions of many problems are reduced to solving linear algebraic equations. With the development of science and technology, the complexity of a problem to be solved is increasing. Therefore, highly efficient solution of large scale sparse matrices is of universal meaning in scientific computation. General numerical solutions of linear equations are direct methods and iterative methods. This article mainly analyzes such direct methods as Gaussian Elimination and LU-Factorization, then proproses a square-rooting method to find the symmetrical positive definite matrices. The iterative methods, Jacobi method iterative method, Gauss-Seidel Method, and Successive Over Relaxation (SOR) method, are also analyzed. We finally discuss the Conjugate Gradient method based on the analysis of SOR method. We also provide detailed derivation of these methods using mathematical analysis. Matlab programs are developed and tested for solving linear equations.
Keywords: linear equations, direct methods, square-rooting, iterative methods, conjugate gradient method.