An Aitken method for accelerating the solution of the non–symmetric algebraic Riccati equation in transport theory

Zhichun Xie; Changfeng Ma; Shihai Li; Zhichun Xie; Changfeng Ma; Shihai Li

doi:10.3934/math.2026971

AIMS Mathematics

2026, Volume 11, Issue 8: 24090-24105. doi: 10.3934/math.2026971

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Research article

An Aitken method for accelerating the solution of the non–symmetric algebraic Riccati equation in transport theory

1.
School of Artificial Intelligence, Xiamen Institute of Technology, Xiamen 361021, China
2.
School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China

Received: 29 May 2026 Revised: 18 July 2026 Accepted: 27 July 2026 Published: 06 August 2026
MSC : 15A24, 65F10

In this paper, we study the accelerated solution of a class of non–symmetric algebraic Riccati equations arising from transport theory. Based on the simple iterative method, the Aitken method is introduced to solve the above special type of Riccati equation quickly, which improves the convergence speed to some extent. However, in some cases $ (\alpha, c) = (0, 1) $, the convergence speed decreases. Therefore we construct new methods with Aitken acceleration structures to preserve or improve the convergence speed based on the shift technique. Theoretical analysis and numerical experiments show the effectiveness of the method. Finally, we conclude that the generality of Aitken's method is that it can improve the convergence speed of iterative methods for other fixed-point equations.
- non–symmetric algebraic Riccati equation,
- Aitken method,
- Shift technique
Citation: Zhichun Xie, Changfeng Ma, Shihai Li. An Aitken method for accelerating the solution of the non–symmetric algebraic Riccati equation in transport theory[J]. AIMS Mathematics, 2026, 11(8): 24090-24105. doi: 10.3934/math.2026971

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Abstract

In this paper, we study the accelerated solution of a class of non–symmetric algebraic Riccati equations arising from transport theory. Based on the simple iterative method, the Aitken method is introduced to solve the above special type of Riccati equation quickly, which improves the convergence speed to some extent. However, in some cases $ (\alpha, c) = (0, 1) $, the convergence speed decreases. Therefore we construct new methods with Aitken acceleration structures to preserve or improve the convergence speed based on the shift technique. Theoretical analysis and numerical experiments show the effectiveness of the method. Finally, we conclude that the generality of Aitken's method is that it can improve the convergence speed of iterative methods for other fixed-point equations.

References

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