复代数方程组几种迭代法的比较

本文从二维非线性Schrödinger方程出发,推导出五对角的复代数方程组,并应用高斯-赛德尔迭代法、SOR迭代法、复双共轭梯度法以及预处理复双共轭梯度法等对求解的计算量进行了比较。同时,又将复代数方程组化成七对角的实代数方程组,用高斯-赛德尔迭代法、SOR迭代法以及PCG法(预处理共轭梯度法)等进行了比较。结果表明,PCG法在上述几种方法中是最有效的。本文还对SOR松弛因子的选择进行了讨论。

A NUMERICAL COMPARISON FOR ITERATIVE METHODS OF COMPLEX ALGEBRAIC EQUATION SYSTEMS

From the 2-D nonlinear Schrödinger equation, a complex algebraic equation system is obtained. This paper uses Gauss-Seidel, SOR, Complex BI-CG and complex BI-PCG to solve the system and compares the total costs of iterations of these iterative methods. Meanwhile, the complex equation system is also transformed into a real system whose coefficient matrix is hepta-diagonal. Gauss-Seidel, SOR and PCG methods are then used to solve it and the total costs of iterations are also compared. The result shows that the PCG method is most effective comparing with the others. It is discussed as well that how to select the optimal relaxation factor of SOR method for the systems considered.