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A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

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  • Malik Zaka Ullah

    (Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

Abstract

The goal of this article is to investigate a new solver in the form of an iterative method to solve X + A ∗ X − 1 A = I as an important nonlinear matrix equation (NME), where A , X , I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:2994-:d:685559
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    References listed on IDEAS

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    1. Jing Li, 2013. "Solutions and Improved Perturbation Analysis for the Matrix Equation," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, June.
    2. 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.
    3. Sourav Shil & Hemant Kumar Nashine & Ali Jaballah, 2021. "Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations," Journal of Mathematics, Hindawi, vol. 2021, pages 1-22, August.
    4. Engwerda, J.C., 1993. "On the existence of a positive definite solution of the matrix equation X = ATX-1A = I," Other publications TiSEM 9d762863-0dfe-4aeb-8a13-5, Tilburg University, School of Economics and Management.
    5. F. Soleymani & M. Sharifi & S. Shateyi & F. Khaksar Haghani, 2014. "An Algorithm for Computing Geometric Mean of Two Hermitian Positive Definite Matrices via Matrix Sign," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-6, August.
    6. Yunbo Tian & Chao Xia & Fazlollah Soleymani, 2021. "On the Low-Degree Solution of the Sylvester Matrix Polynomial Equation," Journal of Mathematics, Hindawi, vol. 2021, pages 1-4, July.
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    Cited by:

    1. 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.

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