IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i7p1578-d1106495.html
   My bibliography  Save this article

Newton’s Iteration Method for Solving the Nonlinear Matrix Equation X + ∑ i = 1 m A i * X − 1 A i = Q

Author

Listed:
  • Chang-Zhou Li

    (School of Mathematics, Jilin University, Changchun 130012, China)

  • Chao Yuan

    (School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China)

  • An-Gang Cui

    (School of Mathematics and Statistics, Yulin University, Yulin 719000, China)

Abstract

In this paper, we study the nonlinear matrix equation (NME) X + ∑ i = 1 m A i * X − 1 A i = Q . We transform this equation into an equivalent zero-point equation, then we use Newton’s iteration method to solve the equivalent equation. Under some mild conditions, we obtain the domain of approximation solutions and prove that the sequence of approximation solutions generated by Newton’s iteration method converges to the unique solution of this equation. In addition, the error estimation of the approximation solution is given. Finally, the comparison of two well-known approaches with Newton’s iteration method by some numerical examples demonstrates the superiority of Newton’s iteration method in the convergence speed.

Suggested Citation

  • Chang-Zhou Li & Chao Yuan & An-Gang Cui, 2023. "Newton’s Iteration Method for Solving the Nonlinear Matrix Equation X + ∑ i = 1 m A i * X − 1 A i = Q," Mathematics, MDPI, vol. 11(7), pages 1-11, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1578-:d:1106495
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/7/1578/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/7/1578/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Engwerda, J.C. & Ran, A.C.M. & Rijkeboer, A.L., 1992. "Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A*X-1A=Q," Other publications TiSEM cbc6bc1e-3bbf-49a4-8222-d, Tilburg University, School of Economics and Management.
    2. Malik Zaka Ullah, 2021. "A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation," Mathematics, MDPI, vol. 9(23), pages 1-7, November.
    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. Li, Lei & Wang, Qing-Wen & Shen, Shu-Qian, 2015. "On positive definite solutions of the nonlinear matrix equations X±A*XqA=Q," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 556-566.
    2. Sbrana, Giacomo & Poloni, Federico, 2013. "A closed-form estimator for the multivariate GARCH(1,1) model," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 152-162.
    3. Nashine, Hemant Kumar & Bose, Snehasish, 2019. "Solution of a class of cross-coupled nonlinear matrix equations," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    4. Malik Zaka Ullah, 2021. "A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation," Mathematics, MDPI, vol. 9(23), pages 1-7, November.

    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:11:y:2023:i:7:p:1578-:d:1106495. 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.