Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices
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Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices Xin Qi1 · Hui-Ting Wu1 · Xiao-Yong Xiao1 Received: 1 January 2020 / Revised: 15 May 2020 / Accepted: 20 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, a new method for solving complex nonlinear systems with symmetric Jacobian matrices is proposed—modified Newton generalized successive overrelaxation (MN-GSOR) iteration method. The new MN-GSOR method helps us to solve nonlinear systems using the GSOR method as an inner iterative approximation for solving the Newton equations. Next, we use the Hölder continuous condition instead of the Lipschitz assumption to analyze and prove the convergence properties of the MN-GSOR method. Numerical experiments are conducted to show the superiority and effectiveness of MN-GSOR method, and comparisons are made to see the advantages over some other recently proposed methods. Furthermore, when the imaginary part of the Jacobian matrix is symmetric but non-definite, some recently proposed methods are not applicable, but the MN-GSOR method still works well. Keywords Complex systems of nonlinear equations · Convergence analysis · Modified Newton-PMHSS · Modified Newton-GSOR · Modified Newton-HSS Mathematics Subject Classification 60F10 · 60F50
1 Introduction We want to solve a class of large sparse complex systems of nonlinear equations F(u) ˆ = 0,
(1)
with F : D ⊂ Cn → Cn being nonlinear and continuously differentiable.
Communicated by Zhong-Zhi Bai.
B B 1
Xin Qi [email protected] Xiao-Yong Xiao [email protected] Department of Mathematics, School of Science, Nanchang University, 999 Xuefu Road, Nanchang 330031, People’s Republic of China 0123456789().: V,-vol
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First of all, we have a review on solving the linear system Auˆ = b,
ˆ b ∈ Cn . A ∈ Cn×n , u,
(2)
Here A is a complex matrix having the following form: A = W + iT, (3) √ where i = −1 represents the imaginary unit, throughout the paper. The matrices W , T ∈ Rn×n are real, symmetric and positive semi-definite, but at least one of them is positive definite. Hereafter, for simplicity but without loss of generality, we suppose that W is symmetric positive definite. This system has numerous important applications in scientific computing, Chemical reactions and Mechanical engineering. For more applications on complex symmetric systems, we suggest the readers refer to Karlsson (1995); Zhang and Sun (2011) and Papp and Vizvari (2006). At the earliest, the Hermitian/skew-Hermitian splitting (HSS) method and preconditioned HSS (PHSS) method has been introduced by Bai et al. Bai et al. (2003, 2004). They can efficiently and quickly solve positive definite linear systems. This method based on the following Hermitian/skew-Hermitian (HS) splitting of the matrix A = H + S, where H=
1 1 (A + A∗ ) = W and S = (A − A∗ ) = i T , 2 2
with A∗ being its conjugate transpose. Given an initial approximation uˆ 0 ∈ Cn and a positive constant α, t
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