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Open AccessArticle

On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

1
School of Mathematics and Finance, Putian University, Putian 351100, China
2
School of Mathematics and Statistic, Lanzhou University, Lanzhou 730000, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(2), 208; https://doi.org/10.3390/math8020208
Received: 31 December 2019 / Revised: 3 February 2020 / Accepted: 4 February 2020 / Published: 6 February 2020
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) iteration method for solving a class of large sparse and complex symmetric weakly nonlinear equations. At each inner process iterative step of the new methods, one only needs to solve the real subsystems with the same symmetric positive and definite coefficient matrix. Therefore, the computational workloads and computational storage will be saved in actual implements. The conditions for guaranteeing the local convergence are studied in detail. The quasi-optimal parameters are also proposed for both the C-to-R-based Picard iteration method and the NC-to-R iteration method. Numerical experiments are performed to show the efficiency of the new methods.
Keywords: weakly nonlinear equations; C-to-R preconditioner; local convergence; Picard iteration; complex symmetric matrix. weakly nonlinear equations; C-to-R preconditioner; local convergence; Picard iteration; complex symmetric matrix.
MDPI and ACS Style

Zeng, M.-L.; Zhang, G.-F. On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations. Mathematics 2020, 8, 208.

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