Abstract
In this paper, first we present an elementary approach for evaluating the determinant of an n-by-n periodic tridiagonal matrix with Toeplitz structure, which is based on the use of a certain type of matrix reformulation and linear transformation. Then, we propose a more efficient numerical algorithm with the cost of \( 12\lfloor \frac{n-4}{k}\rfloor +12k+O(\log n) \) for computing n-th order periodic tridiagonal Toeplitz determinants, where k is an integer that needs to be chosen at the beginning of the algorithm. Moreover, for periodic tridiagonal Toeplitz matrices with rational entries, we derive a fast and reliable algorithm to determine nonzero determinants via modular arithmetic. Some illustrative examples are provided, and the numerical results are compared with the ones obtained by Gaussian elimination algorithm and MATLAB built-in function.
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The authors would like to thank the anonymous referees for their valuable comments and suggestions that substantially enhanced the quality of the manuscript.
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Jia, JT., Wang, FR. On the efficient and accurate determinant evaluation of periodic tridiagonal Toeplitz matrices. J Math Chem 61, 1504–1521 (2023). https://doi.org/10.1007/s10910-023-01474-8
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DOI: https://doi.org/10.1007/s10910-023-01474-8