A preconditioned SSOR iteration method for solving complex symmetric system of linear equations

Tahereh Salimi Siahkolaei,,Faculty Of Mathematical Sciences, University Of Guilan, Rasht, Iran ,Davod Khojasteh Salkuyeh

doi:10.3934/naco.2019033

Tahereh Salimi Siahkolaei, ,Faculty Of Mathematical Sciences, University Of Guilan, Rasht, Iran + Show 1 more

Open Access

https://doi.org/10.3934/naco.2019033

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Abstract

We present a preconditioned version of the symmetric successive overrelaxation (SSOR) iteration method for a class of complex symmetric linear systems. The convergence results of the proposed method are established and conditions under which the spectral radius of the iteration matrix of the method is smaller than that of the SSOR method are analyzed. Numerical experiments illustrate the theoretical results and depict the efficiency of the new iteration method.

Highlights

Many scientific and engineering applications often require the solution of large and sparse complex symmetric system of linear equationsAu ≡ (W + iT )u = b, A ∈ Cn×n, u, b ∈ Cn, (1)where W, T ∈ Rn×n
Several effective iteration methods have been proposed in the literature for solving the system (1)
We present the iteration numbers (IT) and CPU time (CPU) for the preconditioned MHSS (PMHSS), generalized successive overrelaxation (GSOR), successive overrelaxation (SSOR), accelerated variant of the SSOR (ASSOR) and preconditioned version of the symmetric successive overrelaxation (PSSOR) iteration methods for different grids

Summary

Introduction

We study conditions under which the spectral radius of the PSSOR iteration matrix becomes smaller than that of the SSOR method. The optimal value of the iteration parameters ω and α which minimize the spectral radius of the PSSOR method are given by ωopt = 1 ±. In each iteration of the PMHSS, GSOR, SSOR, ASOR and PSSOR iteration methods, we use the Cholesky factorization of the coefficient matrices to solve the sub-systems. We take CH = μK with μ a damping coefficient, M = I, CV = 10I, and K the five-point centered difference matrix approximating the negative Laplacian operator with homogeneous Dirichlet boundary conditions, on a uniform mesh in the unit square [0, 1] × [0, 1] with the mesh-size h = 1/(m + 1).

Method

Conclusion