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Numerical algorithms for the determinants of opposite-bordered and singly-bordered tridiagonal matrices

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Abstract

A recursive algorithm for the determinant evaluation of general opposite-bordered tridiagonal matrices has been proposed by Jia et al. (J Comput Appl Math 290:423–432, 2015). Since the algorithm is a symbolic algorithm, it never suffers from breakdown. However, it may be time-consuming when many symbolic names emerge during the symbolic computation. In this paper, without using symbolic computation, first we present a novel breakdown-free numerical algorithm for computing the determinant of an n-by-n opposite-bordered tridiagonal matrix, which does not require any extra memory storage for the implementation. Then, we present a cost-efficient algorithm for the determinants of opposite-bordered tridiagonal matrices based on the use of the combination of an elementary column operation and Sylvester’s determinant identity. Furthermore, we provide some numerical results with simulations in Matlab implementation in order to demonstrate the accuracy and efficiency of the proposed algorithms, and their competitiveness with other existing algorithms. The corresponding results in this paper can be readily obtained for computing the determinants of singly-bordered tridiagonal matrices.

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Notes

  1. The low rank structure of an opposite-bordered tridiagonal matrix means that all submatrices taken out of the lower and upper triangular part of the matrix have rank at most 3.

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Acknowledgements

The author wishes to thank anonymous referees for useful comments that enhanced the quality of this paper.

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  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, Shaanxi, China

    Ji-Teng Jia

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  1. Ji-Teng Jia
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Correspondence to Ji-Teng Jia.

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This work was supported by the Natural Science Foundation of China (NSFC) under Grant 11601408.

Appendix: The implementation of Algorithm 2

Appendix: The implementation of Algorithm 2

In the following, we show that the implementation of Algorithm 2 can be memory-efficient. More precisely, only five vectors with size n (or \(n-1\)) are needed during the whole computational process.

figure d

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Jia, JT. Numerical algorithms for the determinants of opposite-bordered and singly-bordered tridiagonal matrices. J Math Chem 58, 1828–1845 (2020). https://doi.org/10.1007/s10910-020-01157-8

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