Abstract
In the present paper, we mainly consider the direct solution of cyclic tridiagonal linear systems. By using the specific low-rank and Toeplitz-like structure, we derive a structure-preserving factorization of the coefficient matrix. Based on the combination of such matrix factorization and Sherman–Morrison–Woodbury formula, we then propose a cost-efficient algorithm for numerically solving cyclic tridiagonal linear systems, which requires less memory storage and data transmission. Furthermore, we show that the structure-preserving matrix factorization can provide us with an explicit formula for n-th order cyclic tridiagonal determinants. Numerical examples are given to demonstrate the performance and efficiency of our algorithm. All of the experiments are performed on a computer with the aid of programs written in MATLAB.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
L.K. Bieniasz, High order accurate, one-sided finite-difference approximations to concentration gradients at the boundaries, for the simulation of electrochemical reaction-diffusion problems in one-dimensional space geometry. Computational Biology and Chemistry 27(3), 315–325 (2003)
R. Baronas, F. Ivanauskas, J. Kulys, Mathematical Modeling of Biosensors (Springer, Dordrecht, 2010)
D. Britz, J. Strutwolf, Digital Simulationin Electrochemistry (Springer, Berlin, 2016)
L.K. Bieniasz, A specialised cyclic reduction algorithm for linear algebraic equation systems with quasi-tridiagonal matrices. Journal of Mathematical Chemistry 55, 1793–1807 (2017)
A. Martin, J.Y. Trépanier, M. Reggio, X.Y. Guo, Transient ablation regime in circuit breakers. Plasma Science and Technology 9(6), 653–656 (2007)
A.J. Amar, B.F. Blackwell, J.R. Edwards, One-dimensional ablation using a full Newton’s method and finite control volume procedure. Journal of Thermophysics and Heat Transfer 22(1), 71–82 (2008)
T.Y. Gou, A. Sandu, Continuous versus discrete advection adjoints in chemical data assimilation with CMAQ. Atmospheric Environment 45, 4868–4881 (2011)
G.H. Golub, C.V. Loan, Matrix Computations, 3rd edn. (The Johns Hopkins University Press, Baltimore, 1996)
J.T. Jia, Q.X. Kong, A symbolic algorithm for periodic tridiagonal systems of equations. Journal of Mathematical Chemistry 52(8), 2222–2233 (2014)
J.T. Jia, A breakdown-free algorithm for computing the determinants of periodic tridiagonal matrices. Numerical Algorithms 83(1), 149–163 (2020)
T. Sogabe, New algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems. Applied Mathematics & Computation 202(2), 850–856 (2008)
T. Sogabe, F. Yilmaz, A note on a fast breakdown-free algorithm for computing the determinants and the permanents of \(k\)-tridiagonal matrices. Applied Mathematics and Computation 249, 98–102 (2014)
Y.L. Wei, Y.P. Zheng, Z.L. Jiang, S. Shon, A study of determinants and inverses for periodic tridiagonal Toeplitz matrices with perturbed corners involving mersenne numbers. Mathematics 7(10), 893–903 (2019)
J.T. Jia, J. Wang, T.F. Yuan, K.K. Zhang, B.M. Zhong, An incomplete block-diagonalization approach for evaluating the determinants of bordered \(k\)-tridiagonal matrices. Journal of Mathematical Chemistry 60(8), 1658–1673 (2022)
J.T. Jia, J. Wang, Q. He, Y.C. Yan, A division-free algorithm for numerically evaluating the determinant of a specific quasi-tridiagonal matrix. Journal of Mathematical Chemistry 60(9), 1695–1706 (2022)
K.H. Rosen, Discrete Mathematics and its Applications, 6th edn. (McGraw-Hill, New York, 2007)
A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd edn. (Cambridge University Press, Cambridge, 1996)
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
Ji-Teng Jia: Conceived, designed and performed the analysis; Prepared the manuscript. Fu-Rong Wang: Performed the numerical analysis and wrote the manuscript. Rong **e: Performed the numerical experiments. Yi-Fan Wang: Contributed analysis tools.
Corresponding author
Ethics declarations
Ethical approval
Not applicable.
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jia, JT., Wang, FR., **e, R. et al. An efficient numerical algorithm for solving linear systems with cyclic tridiagonal coefficient matrices. J Math Chem (2024). https://doi.org/10.1007/s10910-024-01631-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10910-024-01631-7
Keywords
- Cyclic tridiagonal matrices
- Tridiagonal matrices
- Linear systems
- Matrix factorization
- Sherman–Morrison–Woodbury formula
- Determinants