On preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting method for complex symmetric lin
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On preconditioned Euler‑extrapolated single‑step Hermitian and skew‑Hermitian splitting method for complex symmetric linear systems Xian Xie1 · Hou‑biao Li1 Received: 30 April 2020 / Revised: 30 September 2020 / Accepted: 20 October 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract In this paper, we introduce a preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting (PE-SHSS) iteration method for solving a class of complex symmetric system of linear equations. The convergence properties of the PE-SHSS iteration method are investigated under suitable restrictions. In addition, the spectral properties of the corresponding preconditioned matrix are discussed. Finally, three numerical examples are also used to verify the effectiveness of the PESHSS iteration method. Keywords Complex symmetric matrix · PE-SHSS method · Convergence analysis Mathematics Subject Classification 65F10 · 65F50
1 Introduction In this paper, we consider a class of complex symmetric system of the form
√
(1)
Ax ≡ (W + iT)x = b,
where i = −1 , W, T ∈ ℝn×n are symmetric positive semi-definite matrices. b ∈ ℂn is a given vector and x ∈ ℂn is the unknown vector. Here and in the sequel, we suppose W, T ≠ 0 , which means that A is non-Hermitian matrix. Complex linear systems of the form (1) arise from many practical problems in scientific computing and engineering applications, for instance, diffuse optical tomography [1], quantum mechanics [2], molecular scattering [3], structural dynamics [4] and so on. For more examples and additional references, we refer to [5–7]. In order to solve the complex linear systems (1), lots of iteration methods have been presented. For instance, Bai et al. proposed the Hermitian and skew-Hermitian * Hou‑biao Li [email protected] 1
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China
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splitting (HSS) iteration method [8]. To avoid the complex arithmetic, Bai et al. presented the modified HSS (MHSS) method [10] { 1 (𝛼I + W)x(k+ 2 ) = (𝛼I − iT)x(k) + b, (2) (𝛼I + T)x(k+1) = (𝛼I + iW)x(k) − ib, where 𝛼 is a given positive constant and I denotes the identity matrix. Moreover, Bai designed the preconditioned MHSS (PMHSS) method [11] { 1 (𝛼P + W)x(k+ 2 ) = (𝛼I − iT)x(k) + b, (3) (𝛼P + T)x(k+1) = (𝛼I + iW)x(k) − ib, where 𝛼 is a given positive constant and P ∈ ℝn×n is a prescribed symmetric positive definite matrix. In addition, Li et al. presented the lopsided PMHSS (LPMHSS) iteration method [12] based on LHSS method [9]. Other modified and generalized HSS iteration methods, we can refer to [13–20]. Note that all the iteration schemes mentioned above are two-step methods. Then, Li et al. introduced a single-step HSS (SHSS) iteration method [21]
(𝛼I + H)x(k+1) = (𝛼I − S)x(k+1) + b,
(4)
where 𝛼 is a given positive constant. To further improve the convergence rate, Wu et al. proposed a non-alternating precondition
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