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A note on the structured perturbation analysis for the inversion formula of Toeplitz matrices

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Abstract

The invertibility of a Toeplitz matrix can be assessed based on the solvability of two standard equations. The inverse of the nonsingular Toeplitz matrix can then be represented as the sum of products of circulant and skew-circulant (CS) matrices. In this note, we provide a new structured perturbation analysis for the CS representation of Toeplitz inversion and the new upper bound is just half as large as the existing upper bound proposed by Wu et al. (Numer Linear Algebra Appl 22(4):777–792, 2015) and Feng et al. (East Asian J Appl Math 5(2):160–175, 2015). Meanwhile, some practical issues and numerical experiments involving the numerical solutions of fractional partial differential equations are reported to support our theoretical findings.

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Notes

  1. Since the CPU time required for once GSF-CS (or GSF) is very short, it will be not easy to observe the time differences. Therefore, here we record the (total) CPU time required for GSF-CS (or GSF) to execute 2000 runs. Such situations often occur in numerical fPDEs; see e.g. [9, 10] and also Example 2.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their insightful comments and suggestions that greatly improved the quality of this paper. This research is supported by NSFC (No. 11801463) and the Applied Basic Research Program of Sichuan Province (No. 2020YJ0007). The last author is a member of Gruppo Nazionale per il Calcolo Scientifico (GNCS) of Istituto Nazionale di Alta Matematica (INdAM), and this work was partially supported by INdAM-GNCS Project CUP_E55F22000270001.

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Authors and Affiliations

  1. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, People’s Republic of China

    Jing Wu & Xian-Ming Gu

  2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK, Groningen, The Netherlands

    Xian-Ming Gu

  3. School of Mathematical Sciences, Sichuan Normal University, Chengdu, 610068, Sichuan, People’s Republic of China

    Yong-Liang Zhao

  4. School of Business Administration, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, People’s Republic of China

    Yu-Yun Huang

  5. Faculty of Computer Science, Free University of Bozen-Bolzano, piazza Domenicani, 3, 39100, Bozen-Bolzano, Italy

    Bruno Carpentieri

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  1. Jing Wu
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  2. Xian-Ming Gu
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  3. Yong-Liang Zhao
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Correspondence to Xian-Ming Gu.

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Wu, J., Gu, XM., Zhao, YL. et al. A note on the structured perturbation analysis for the inversion formula of Toeplitz matrices. Japan J. Indust. Appl. Math. 40, 645–663 (2023). https://doi.org/10.1007/s13160-022-00543-w

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