On a parameterized inversion-free iterative algorithm for solving the nonlinear matrix equation $ X+\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = I $

Changzhou Li; Boyu Wang; Chao Yuan; Shiliang Chen; Changzhou Li; Boyu Wang; Chao Yuan; Shiliang Chen

doi:10.3934/math.2025850

AIMS Mathematics

2025, Volume 10, Issue 8: 19018-19032. doi: 10.3934/math.2025850

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Research article

On a parameterized inversion-free iterative algorithm for solving the nonlinear matrix equation $ X+\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = I $

1.
School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong, 723001, China
2.
School of Sciences, Northeast Electric Power University, Jilin, 132012, China
3.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

Received: 25 June 2025 Revised: 07 August 2025 Accepted: 18 August 2025 Published: 21 August 2025
MSC : 15A24, 47H10, 65H05

In this paper, we propose a parameterized inversion-free iterative algorithm to compute the positive definite solution of the nonlinear matrix equation $ X+\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = I $. Then, we prove that the proposed algorithm converges. Finally, in comparison with three well-known existing algorithms, the accuracy and effectiveness of the proposed algorithm are demonstrated by some numerical examples.
- nonlinear matrix equations,
- positive definite solution,
- inversion-free iteration methods,
- convergence analysis
Citation: Changzhou Li, Boyu Wang, Chao Yuan, Shiliang Chen. On a parameterized inversion-free iterative algorithm for solving the nonlinear matrix equation $ X+\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = I $[J]. AIMS Mathematics, 2025, 10(8): 19018-19032. doi: 10.3934/math.2025850

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Abstract

In this paper, we propose a parameterized inversion-free iterative algorithm to compute the positive definite solution of the nonlinear matrix equation $ X+\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = I $. Then, we prove that the proposed algorithm converges. Finally, in comparison with three well-known existing algorithms, the accuracy and effectiveness of the proposed algorithm are demonstrated by some numerical examples.

References

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