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.