IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i12p1808-d1412538.html
   My bibliography  Save this article

Optimal Combination of the Splitting–Linearizing Method to SSOR and SAOR for Solving the System of Nonlinear Equations

Author

Listed:
  • 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

The symmetric successive overrelaxation (SSOR) and symmetric accelerated overrelaxation (SAOR) are conventional iterative methods for solving linear equations. In this paper, novel approaches are presented by combining a splitting–linearizing method with SSOR and SAOR for solving a system of nonlinear equations. The nonlinear terms are decomposed at two sides through a splitting parameter, which are linearized around the values at the previous step, obtaining a linear equation system at each iteration step. The optimal values of parameters are determined to minimize the reciprocal of the maximal projection, which are sought in preferred ranges using the golden section search algorithm. Numerical tests assess the performance of the developed methods, namely, the optimal splitting symmetric successive over-relaxation (OSSSOR), and the optimal splitting symmetric accelerated over-relaxation (OSSAOR). The chief advantages of the proposed methods are that they do not need to compute the inverse matrix at each iteration step, and the computed orders of convergence by OSSSOR and OSSAOR are between 1.5 and 5.61; they, without needing the inner iterations loop, converge very fast with saving CPU time to find the true solution with a high accuracy.

Suggested Citation

  • 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.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1808-:d:1412538
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/12/1808/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/12/12/1808/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ullah, Malik Zaka & Serra-Capizzano, Stefano & Ahmad, Fayyaz, 2015. "An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 249-259.
    2. 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.
    3. R. H. AL-Obaidi & M. T. Darvishi & Predrag S. Stanimirović, 2022. "A Comparative Study on Qualification Criteria of Nonlinear Solvers with Introducing Some New Ones," Journal of Mathematics, Hindawi, vol. 2022, pages 1-20, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. R. H. Al-Obaidi & M. T. Darvishi, 2022. "Constructing a Class of Frozen Jacobian Multi-Step Iterative Solvers for Systems of Nonlinear Equations," Mathematics, MDPI, vol. 10(16), pages 1-13, August.
    3. Fayyaz Ahmad & Shafiq Ur Rehman & Malik Zaka Ullah & Hani Moaiteq Aljahdali & Shahid Ahmad & Ali Saleh Alshomrani & Juan A. Carrasco & Shamshad Ahmad & Sivanandam Sivasankaran, 2017. "Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs," Complexity, Hindawi, vol. 2017, pages 1-30, May.
    4. Chein-Shan Liu & Chung-Lun Kuo & Chih-Wen Chang, 2024. "Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse," Mathematics, MDPI, vol. 12(11), pages 1-31, June.
    5. Howk, Cory L., 2016. "A class of efficient quadrature-based predictor–corrector methods for solving nonlinear systems," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 394-406.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:12:p:1808-:d:1412538. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.