A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function

Xiaofeng Wang; Ying Cao; Xiaofeng Wang; Ying Cao

doi:10.3934/math.2023625

AIMS Mathematics

2023, Volume 8, Issue 5: 12456-12471. doi: 10.3934/math.2023625

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A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function

Xiaofeng Wang ^,,
Ying Cao

School of Mathematical Sciences, Bohai University, Jinzhou 121000, Liaoning, China

Received: 07 January 2023 Revised: 23 February 2023 Accepted: 09 March 2023 Published: 27 March 2023
MSC : 65B99, 65H05

A new eighth-order Chebyshev-Halley type iteration is proposed for solving nonlinear equations and matrix sign function. Basins of attraction show that several special cases of the new method are globally convergent. It is analytically proven that the new method is asymptotically stable and the new method has the order of convergence eight as well. The effectiveness of the theoretical results are illustrated by numerical experiments. In numerical experiments, the new method is applied to a random matrix, Wilson matrix and continuous-time algebraic Riccati equation. Numerical results show that, compared with some well-known methods, the new method achieves the accuracy requirement in the minimum computing time and the minimum number of iterations.
- Chebyshev-Halley family,
- iterative method,
- basin of attraction,
- matrix sign function,
- stability
Citation: Xiaofeng Wang, Ying Cao. A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function[J]. AIMS Mathematics, 2023, 8(5): 12456-12471. doi: 10.3934/math.2023625

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Abstract

A new eighth-order Chebyshev-Halley type iteration is proposed for solving nonlinear equations and matrix sign function. Basins of attraction show that several special cases of the new method are globally convergent. It is analytically proven that the new method is asymptotically stable and the new method has the order of convergence eight as well. The effectiveness of the theoretical results are illustrated by numerical experiments. In numerical experiments, the new method is applied to a random matrix, Wilson matrix and continuous-time algebraic Riccati equation. Numerical results show that, compared with some well-known methods, the new method achieves the accuracy requirement in the minimum computing time and the minimum number of iterations.

References

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