Numerical algorithms for the determinants of opposite-bordered and singly-bordered tridiagonal matrices
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Numerical algorithms for the determinants of opposite‑bordered and singly‑bordered tridiagonal matrices Ji‑Teng Jia1 Received: 23 April 2020 / Accepted: 20 June 2020 © Springer Nature Switzerland AG 2020
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 costefficient 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. Keywords Opposite-bordered tridiagonal matrices · Singly-bordered tridiagonal matrices · Determinants · Breakdown-free algorithm · Cost-efficient algorithm Mathematics Subject Classification 05A17 · 05A19 · 15A06
This work was supported by the Natural Science Foundation of China (NSFC) under Grant 11601408. * Ji‑Teng Jia [email protected] 1
School of Mathematics and Statistics, Xidian University, Xi’an 710071, Shaanxi, China
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Vol.:(0123456789)
Journal of Mathematical Chemistry
1 Introduction and objectives From a theoretical point of view, the determinant provides important information about a matrix of coefficient of a linear system since the value tells whether the corresponding system has a unique solution or not. Also, the determinant is often used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. From a geometrical point of view, the determinant can be viewed as the scaling factor of the linear transformation described by the matrix. In this case, the transformation has an inverse operation exactly when the determinant is nonzero. We are concerned in this paper with the determinant of an n-by-n oppositebordered tridiagonal matrix An whose entries are defined as follows
⎧b, ⎪ i ⎪ ai , ⎪c, [An ]ij ∶= ⎨ i h, ⎪ i ⎪ gi , ⎪ 0, ⎩
for i = j, 2 ≤ i ≤ n − 1, for j = i − 1, 3 ≤ i ≤ n, for j = i + 1, 1 ≤ i ≤ n − 2, for j = 1, 1 ≤ i ≤ n, for j = n, 1 ≤ i ≤ n, else .
(1)
Throughout the paper, we assume that n ≥ 3 . The opposite-bordered tridiagonal matrix includes some important classes of matrices such as tridiagonal matrix [1, 2], periodic tridiagonal mat
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