S. Sheikhi and H. Esmaeili
Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran.
Received 25 October, 2024; accepted in revised form 20 February, 2026
Abstract: In this paper, we present a novel inverse-free iterative method to compute the maximal positive definite solution for the matrix equation X + A∗X−1A = I. The proposed method generates a rapidly converging sequence directed towards the inverse matrix, using a fixed point method formed by a specific type of polynomials, thereby facilitating the estimation of the maximal positive definite solution while adhering to a defined stopping criterion. We provide a thorough proof of convergence for the proposed methodology, outlining the necessary conditions and the convergence rate required to effectively solve the matrix equation. We show that the sequence converging quadratically to the inverse matrix plays a significant role in enhancing the convergence rate of the method, thereby improving its overall effectiveness in solving the matrix equation. In the numerical results section, we consider several matrices of varying sizes and scenarios, paying particular attention to computational cost and execution time. Furthermore, the numerical evaluations demonstrate that our method outperforms alternative approaches, especially in reducing the total number of matrix-matrix multiplications and minimizing execution time.
© European Society of Computational Methods in Sciences and Engineering
Keywords: Matrix equation, Iterative method, Inverse-free, Maximal positive definite solution,
Matrix-matrix multiplications.
Mathematics Subject Classification: 65F10 , 65F30, 65H10, 15A24, 93B40