An elementary algorithm for computing the determinant of pentadiagonal Toeplitz matrices

Over the last 2525 years, various fast algorithms for computing the determinant of a pentadiagonal Toeplitz matrices were developed. In this paper, we give a new kind of elementary algorithm requiring 56⋅⌊n−4k⌋+30k+O(logn) operations, where k≥4k≥4 is an integer that needs to be chosen freely at the beginning of the algorithm. For example, we can compute det(Tn)det(Tn) in n+O(logn)n+O(logn) and 82n+O(logn) operations if we choose kk as 5656 and ⌊2815(n−4)⌋, respectively. For various applications, it will be enough to test if the determinant of a pentadiagonal Toeplitz matrix is zero or not. As in another result of this paper, we used modular arithmetic to give a fast algorithm determining when determinants of such matrices are non-zero. This second algorithm works only for Toeplitz matrices with rational entries.