Abstract
This paper is concerned with the generalized Sylvester equation \(AXB+CXD=E\), where A, B, C, D, E are infinite size matrices with a quasi Toeplitz structure, that is, a semi-infinite Toeplitz matrix plus an infinite size compact correction matrix. Under certain conditions, an equation of this type has a unique solution possessing the same structure as the coefficient matrix does. By separating the analysis on the Toeplitz part with that on the correction part, we provide perturbation results that cater to the particular structure in the coefficient matrices. We show that the Toeplitz part is well-conditioned if the whole problem, without considering the structure, is well-conditioned. Perturbation results that are illustrated through numerical examples are applied to equations arising in the analysis of a Markov process and the 2D Poisson problem.
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Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1I1A1A01062548) and the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2017R1A5A1015722). The authors thank the anonymous referees for providing very useful suggestions for improving this paper.
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Communicated by Lothar Reichel.
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Kim, HM., Meng, J. Structured perturbation analysis for an infinite size quasi-Toeplitz matrix equation with applications. Bit Numer Math 61, 859–879 (2021). https://doi.org/10.1007/s10543-021-00847-2
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DOI: https://doi.org/10.1007/s10543-021-00847-2