Practical convergent splittings and acceleration methods for non-Hermitian positive definite linear systems
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Practical convergent splittings and acceleration methods for non-Hermitian positive definite linear systems Chuan-Long Wang · Guo-Yan Meng · Yan-Hong Bai
Received: 11 January 2012 / Accepted: 20 July 2012 / Published online: 12 August 2012 © Springer Science+Business Media, LLC 2012
Abstract We present two practical convergent splittings for solving a nonHermitian positive definite system. By these new splittings and optimization models, we derive three new improved Chebyshev semi-iterative methods and discuss convergence of these methods. Finally, the numerical examples show that the acceleration methods can reduce evidently the amount of work in computation. Keywords Non-Hermitian · Convergent splitting · Acceleration methods Mathematics Subject Classifications (2010) 65F10 · 15A06
1 Introduction and preliminaries Consider a large sparse system of linear equations Ax = b , A = (aij) ∈ Cn×n and b ∈ Cn ,
(1.1)
Communicated by: Raymond H. Chan. This work is supported by NSF of China (11071184) and NSF of Shanxi Province (2010011006). C.-L. Wang (B) · Y.-H. Bai Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, Shanxi Province, People’s Republic of China e-mail: [email protected] G.-Y. Meng Department of Computer Science, Xinzhou Normal University, Xinzhou 034000, Shanxi Province, People’s Republic of China
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where the coefficient matrix A is assumed to be non-Hermitian positive definite. The partition A = M − N is called a splitting if M is nonsingular. The splitting iterative method is popular in solving a linear systems (1.1). For a monotone matrix, an H-matrix, or an Hermitian positive definite (or semidefinite) matrix, the splitting A = M − N is convergent if it is a weak regular splitting, an H-compatible splitting or a P-regular splitting. The standard idea to establish an efficient iterative method is to fully use the specific properties of the matrix involved in developing appropriate way of splitting. For details, we refer to [1, 3, 8–10, 17, 19] and the references therein. For a non-Hermitian positive definite matrix A, Wang and Bai [2, 20] derived several sufficient conditions to guarantee the convergence of a single splitting iterative method. For the same system Bai et al. [4–6] also established some alternatives called HSS and PSS iterative methods. For a non-Hermitian positive definite linear system it has been proved that the HSS and PSS methods converge unconditionally to the unique solution of the system (1.1). However, a drawback of these methods is that an Hermitian or a skewHermitian system of linear equations has to be solved at each iteration, even though various methods for solving the skew-Hermitian system have been developed recently [13–16, 21, 22]. In this paper, we develop some new convergent splittings for a non-Hermitian positive definite matrix and improve the Chebyshev semi-iterative method [11, 12] by using the specific properties of the new splitting and optimization models. Finally, we examine the advantages of our methods by carrying
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