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Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations

Author

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  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Essam R. El-Zahar

    (Department of Mathematics, College of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
    Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt)

  • Chih-Wen Chang

    (Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan)

Abstract

This paper proposes a dynamical approach to determine the optimal values of the parameters used in each iteration of the symmetric successive over-relaxation (SSOR), accelerated over-relaxation (AOR), and symmetric accelerated over-relaxation (SAOR) methods for solving linear equation systems. When the optimal values of the parameters in the SSOR, AOR, and SAOR are hard to determine as some fixed values, they are obtained by minimizing the merit functions, which are based on the maximal projection technique between the left- and right-hand-side vectors, which involves the input vector, the previous step values of the variables, and the parameters. The novelty is a new concept of the dynamical optimal values of the parameters, instead of the fixed values and the maximal projection technique. In a preferred range, the optimal values of the parameters can be quickly determined by using the golden section search algorithm with a loose convergence criterion. Without knowing and having the theoretical optimal values in general, the new methods might provide an alternative and proper choice of the values of the parameters for accelerating the convergence speed. Numerical testings of the linear Poisson equation discretized to a matrix–vector form and a Lyapunov equation form were used to assess the performance of the DOSSOR, DOAOR, and DOSAOR dynamical optimal methods.

Suggested Citation

  • Chein-Shan Liu & Essam R. El-Zahar & Chih-Wen Chang, 2023. "Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations," Mathematics, MDPI, vol. 11(18), pages 1-21, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3828-:d:1234218
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    References listed on IDEAS

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    1. Huang, Zheng-Ge & Xu, Zhong & Lu, Quan & Cui, Jing-Jing, 2015. "Some new preconditioned generalized AOR methods for generalized least-squares problems," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 87-104.
    2. Li, Cheng-Liang & Ma, Chang-Feng, 2019. "An accelerated symmetric SOR-like method for augmented systems," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 408-417.
    3. Hadjidimos, A. & Yeyios, A., 1982. "Symmetric accelerated overrelaxation (SAOR) method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 24(1), pages 72-76.
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    Cited by:

    1. Chein-Shan Liu & Chih-Wen Chang, 2024. "Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations," Mathematics, MDPI, vol. 12(7), pages 1-21, March.
    2. Chein-Shan Liu & Essam R. El-Zahar & Chih-Wen Chang, 2024. "Optimal Combination of the Splitting–Linearizing Method to SSOR and SAOR for Solving the System of Nonlinear Equations," Mathematics, MDPI, vol. 12(12), pages 1-24, June.

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