An Extrapolated PSS Iterative Method for Positive Definite Linear Systems

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 non-Hermitian 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.