Jacobi与Gauss—Seidel迭代法求解线性方程组收敛性比较与研究

A Comparative Study of Astringency of Coupled Linear Equation Solved With Jacobi and Gauss-Seidel Methods

Jacobi迭代法和Gauss-Seidel迭代法是计算机求解线性方程组常用的两种迭代法，但是这两种方法对方程的收敛性要求很严，大部分方程组均不能用以求解．给出一些基本技巧：对于简单的2阶方程组，若Jacobi法与Gauss-Seidel法均发散，可交换其两行求得其解；对一般性方程，给出一个应用性较强的定理，将方程Aχ=b→A^rAχ=A^Tb，可以用Gauss-Seidel求得任何|A|≠O方程组的解．

Jacobi and Gauss-Seidel are two iterative methods for solving linear equation sets. Most of coupled linear equations can't be solved through these methods because of the strict astringency they require of the equation.The solution of the equation set of two ranks can be obtained through swapping two rows, with divergence of the above two methods. An available theorem is proved aiming at common equations, that is, transfiguring Ax=bA~T Ax=A~Tb , if |A|≠0 ,solutions can be obtained with Gauss-Seidel iterative method.