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A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

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  • Sadek, Lakhlifa
  • Bataineh, Ahmad Sami
  • Isik, Osman Rasit
  • Alaoui, Hamad Talibi
  • Hashim, Ishak

Abstract

In this paper, we apply the Bernstein collocation method to construct the solution set of the Sylvester matrix differential equation (Sy-MDE) which involves the Lyapunov matrix differential equation. The method depends on the collocation method and Bernstein polynomials. The main advantage of the proposed method is that by using this method Sy-MDE reduces to a linear system of algebraic equations which can be solved by using an appropriate iterative method. We analyze the error and give a theorem that bounds the error. We also give the residual correction procedure to estimate the error. By using the procedure, we obtain a new approximate solution, namely a corrected Bernstein collocation solution. To illustrate how the proposed method is applied, several examples are given. Numerical experiments show the effectiveness and accuracy of the method for solving such types of Sy-MDE.

Suggested Citation

  • Sadek, Lakhlifa & Bataineh, Ahmad Sami & Isik, Osman Rasit & Alaoui, Hamad Talibi & Hashim, Ishak, 2023. "A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 475-488.
    • Handle: RePEc:eee:matcom:v:212:y:2023:i:c:p:475-488
    DOI: 10.1016/j.matcom.2023.05.011
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    References listed on IDEAS

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    1. Ahmad Sami Bataineh & Osman Rasit Isik & Moa’ath Oqielat & Ishak Hashim, 2021. "An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs," Mathematics, MDPI, vol. 9(4), pages 1-15, February.
    2. de Klerk, E. & den Hertog, D. & Elabwabi, G., 2008. "On the complexity of optimization over the standard simplex," European Journal of Operational Research, Elsevier, vol. 191(3), pages 773-785, December.
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