Modified Hermitian-normal splitting iteration methods for a class of complex symmetric linear systems
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Modified Hermitian-normal splitting iteration methods for a class of complex symmetric linear systems Ya-Kun Du1 · Mei Qin1 Received: 25 December 2019 / Revised: 30 May 2020 / Accepted: 6 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, the modifications of the Hermitian-Normal splitting iteration methods for solving a class of complex symmetric linear systems are presented. Theoretical analysis shows that the modified iteration methods of Hermitian-normal splitting are unconditionally convergent; the coefficient matrices of the two linear systems solved in each iteration of the methods are real symmetric positive definite. Inexact version of the methods employs the Krylov subspace method as an internal iteration to accelerate. Numerical examples from two model problems are given to illustrate the effectiveness of the modified iteration methods. Keywords Complex symmetric matrix · Hermitian-normal splitting · Modified · Convergence Mathematics Subject Classification 65F10 · 65F50
1 Introduction Many problems in the field of scientific computation can be viewed as solving a large and space symmetric linear system (Axelsson and Kucherov 2000), including quantum mechanics, fluid dynamics, and electromagnetic problems. Now, we consider a large sparse system of linear equations as follows: Ax = b,
A ∈ C n×n x, b ∈ C n ,
(1.1)
Communicated by Zhong-Zhi Bai. This work was supported by National Natural Science Foundation of China (No. 11101282), by Shanghai Leading Academic Discipline Project (No. XTKX2012), and by Innovation Program of Shanghai Municipal Education Commission (No. 14YZ096).
B
Mei Qin [email protected] Ya-Kun Du [email protected]
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College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China 0123456789().: V,-vol
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Y.-K. Du, M. Qin
where A is symmetric and nonsingular. There are many iterative methods that need to effectively split the coefficient matrix A to solve this problem. For example, the classic Jacobi and Gauss–Seidel iterations method (Hageman and Young 1971; Saad 1996) and the generalized Lanczos method (Widlund 1978) split the matrix A into the Hermitian and skew-Hermitian parts which is formed as: A = W + iT,
(1.2)
√ where W = 21 (A + A∗ ) and i T = 21 (A − A∗ ), i = −1 is the imaginary unit, and A∗ is denoted the conjugate transpose of the matrix A. We assume T = 0, which implies that A is non-Hermitian matrix. Bai et al. (2003) presented the formation of the HSS iteration method: given an initial guess x (0) for k = 0, 1, . . . until {x (k) } converges, compute: (α I + W )x (k+1/2) = (α I − i T )x (k) + b, (α I + i T )x (k+1) = (α I − W )x (k+1/2) + b,
(1.3)
where α is a given positive constant. Bai et al. (2004) applied this method to the saddle point problem directly or as a preconditioner. Bai et al. (2006) analyzed the optimal parameter α ∗ that minimizes the spectral radius of the iteration matrix of the HSS method to accelerate convergence. Later, B
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