求解正定线性方程组的外推的PSS迭代方法

An Extrapolated PSS Iterative Method for Positive Definite Linear Systems

为了更高效地求解大型稀疏正定线性方程组,提出了一种外推的正定和反Hermitian迭代方法。新方法首先对系数矩阵进行正定和反Hermitian分裂(PSS),再构造出了一种新的非对称二步迭代格式,同时理论分析了新方法的收敛性,并给出了新方法收敛的充要条件。数值实验表明,通过参数值的选择,新方法比PSS迭代方法和外推的Hermitian和反Hermitian分裂(EHSS)迭代方法具有更快的收敛速度和更小的迭代次数,选择合适的参数值时新方法的收敛效率可以大大提高。

Positive definite linear systems arise in many areas of scientific computing and engineering applications, such as solid mechanics, dynamics, nonlinear programming and partial differential equations. This paper proposes an extrapolated positive definite and skew-Hermitian(EPSS) iterative method for solving large sparse positive definite linear systems. The new method splits the coefficient matrix into positive definite matrix and skew-Hermitian matrix, and then constructs a new non-symmetric two-step iterative scheme. The new method can not only solve nonHermitian positive definite linear equations, but also be used for solving Hermitian positive definite linear equations,which greatly accelerates the convergence speed of the iterative method. The theoretical analysis shows that the new method is convergent. The necessary and sufficient conditions for the convergence of the new method are given. The spectral radius of the iterative matrix of the new method is smaller than that of the iterative matrix of the positive definite and skew-Hermitian(PSS) iterative method when selecting appropriate variables. After that numerical experiments are given to show that the new method is more competitive than the PSS iteration method and the extrapolated Hermitian and skew-Hermitian(EHSS) iterative method. Finally, numerical experiments analyze the sensitivity of the parameters in the EPSS iterative method and find the approximate optimal parameters.