A fast elementary algorithm for computing the determinant of toeplitz matrices

In recent years, a number of fast algorithms for computing the determinant of a Toeplitz matrix were developed. The fastest algorithm we know so far is of order $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrix and $k$ is the bandwidth size. This is possible because such a determinant can be expressed as the determinant of certain parts of $n$-th power of a related $k \times k$ companion matrix. In this paper, we give a new elementary proof of this fact, and provide various examples. We give symbolic formulas for the determinants of Toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when $k$ is small.Comment: 12 pages. The article is rewritten completely. There are major changes in the title, abstract and references. The results are generalized to any Toeplitz matrix, but the formulas for Pentadiagonal case are still include

A fast algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems

In this paper, we develop a new algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems. Numerical experiments are given in order to illustrate the validity and efficiency of our algorithm.The authors would like to thank the supports of the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

An elementary algorithm for computing the determinant of pentadiagonal Toeplitz matrices

AbstractOver the last 25 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≥4 is an integer that needs to be chosen freely at the beginning of the algorithm. For example, we can compute det(Tn) in n+O(logn) and 82n+O(logn) operations if we choose k as 56 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

Determinants of some pentadiagonal matrices

In this paper we consider pentadiagonal ((n+1)times(n+1)) matrices with two subdiagonals and two superdiagonals at distances (k) and (2k) from the main diagonal where (1le k < 2kle n). We give an explicit formula for their determinants and also consider the Toeplitz and “imperfect” Toeplitz versions of such matrices. Imperfectness means that the first and last (k) elements of the main diagonal differ from the elements in the middle. Using the rearrangement due to Egerváry and Szász we also show how these determinants can be factorized