求解线性方程组的一种新方法

将线性方程组的一般系数矩阵转化为对称正定矩阵,从而把原线性方程组的求解问题转化为一个等价变分问题的极少值点寻优问题,借助对分寻优法进行求解。算例结果表明,本文方法不仅对于良态线性方程组的求解问题是有效的,而且对于病态线性方程组的求解问题同样是有效的。

New method in solving linear system of equations

At the two sides of a system of linear equations, simultaneously left multiplying the conjugate matrix corresponding its coefficient matrix, the general coefficient matrix is changed into a symmetric positive one. Based on the variational principle, the solving problem for the original group of linear equation is transformed into an equivalent no constrained optimization programming. Then a half-division optimization method can be used to solve the problem. The results of the given examples prove that the method is effective for good-conditioned or ill-conditioned group of linear equations. Compared with the steepest descent method, Newton method and conjugate gradient method etc. show that the method provided has the following characteristics, such as wide suiting range, high convergence rate, high convergence precision, no beginning iteration point, simple algorithm, easy programming, strong ill conditioned-resistant. At last, the shortcomings of the method are also discussed.