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An Efficient Iterative Approach for Hermitian Matrices Having a Fourth-Order Convergence Rate to Find the Geometric Mean

Author

Listed:
  • Tao Liu

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China)

  • Ting Li

    (School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China)

  • Malik Zaka Ullah

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

  • Abdullah Khamis Alzahrani

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

  • Stanford Shateyi

    (Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa)

Abstract

The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth order of convergence. The type of convergence is also discussed, which is global under an appropriate choice of the initial matrix. Numerical experiments are reported based on input matrices of different sizes as well as various stopping termination levels with comparisons to methods of the same nature and same number of matrix–matrix multiplications. The simulation results confirm the efficiency of the proposed scheme in contrast to its competitors of the same nature.

Suggested Citation

  • Tao Liu & Ting Li & Malik Zaka Ullah & Abdullah Khamis Alzahrani & Stanford Shateyi, 2024. "An Efficient Iterative Approach for Hermitian Matrices Having a Fourth-Order Convergence Rate to Find the Geometric Mean," Mathematics, MDPI, vol. 12(11), pages 1-12, June.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:11:p:1772-:d:1410286
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    References listed on IDEAS

    as
    1. F. Soleymani & P. S. Stanimirović & S. Shateyi & F. Khaksar Haghani, 2014. "Approximating the Matrix Sign Function Using a Novel Iterative Method," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, July.
    2. 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.
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