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

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

Inverse properties of a class of seven-diagonal (near) Toeplitz matrices

This paper presents the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence explicit inverse formula is derived using the Sherman-Morrison formula. Related to the fixed-point iteration used to solve the differential equation, we show the positivity of the inverse matrix and construct an upper bound for the norms of the inverse matrix, which can be used to predict the convergence of the method

Algebraic, Block and Multiplicative Preconditioners based on Fast Tridiagonal Solves on GPUs

This thesis contributes to the field of sparse linear algebra, graph applications, and preconditioners for Krylov iterative solvers of sparse linear equation systems, by providing a (block) tridiagonal solver library, a generalized sparse matrix-vector implementation, a linear forest extraction, and a multiplicative preconditioner based on tridiagonal solves. The tridiagonal library, which supports (scaled) partial pivoting, outperforms cuSPARSE's tridiagonal solver by factor five while completely utilizing the available GPU memory bandwidth. For the performance optimized solving of multiple right-hand sides, the explicit factorization of the tridiagonal matrix can be computed. The extraction of a weighted linear forest (union of disjoint paths) from a general graph is used to build algebraic (block) tridiagonal preconditioners and deploys the generalized sparse-matrix vector implementation of this thesis for preconditioner construction. During linear forest extraction, a new parallel bidirectional scan pattern, which can operate on double-linked list structures, identifies the path ID and the position of a vertex. The algebraic preconditioner construction is also used to build more advanced preconditioners, which contain multiple tridiagonal factors, based on generalized ILU factorizations. Additionally, other preconditioners based on tridiagonal factors are presented and evaluated in comparison to ILU and ILU incomplete sparse approximate inverse preconditioners (ILU-ISAI) for the solution of large sparse linear equation systems from the Sparse Matrix Collection. For all presented problems of this thesis, an efficient parallel algorithm and its CUDA implementation for single GPU systems is provided

Bond-Propagation Algorithm for Thermodynamic Functions in General 2D Ising Models

Recently, we developed and implemented the bond propagation algorithm for calculating the partition function and correlation functions of random bond Ising models in two dimensions. The algorithm is the fastest available for calculating these quantities near the percolation threshold. In this paper, we show how to extend the bond propagation algorithm to directly calculate thermodynamic functions by applying the algorithm to derivatives of the partition function, and we derive explicit expressions for this transformation. We also discuss variations of the original bond propagation procedure within the larger context of Y-Delta-Y-reducibility and discuss the relation of this class of algorithm to other algorithms developed for Ising systems. We conclude with a discussion on the outlook for applying similar algorithms to other models.Comment: 12 pages, 10 figures; submitte