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

Exact particle and kinetic energy densities for one-dimensional confined gases of non-interacting fermions

We propose a new method for the evaluation of the particle density and kinetic pressure profiles in inhomogeneous one-dimensional systems of non-interacting fermions, and apply it to harmonically confined systems of up to N=1000 fermions. The method invokes a Green's function operator in coordinate space, which is handled by techniques originally developed for the calculation of the density of single-particle states from Green's functions in the energy domain. In contrast to the Thomas-Fermi (local density) approximation, the exact profiles under harmonic confinement show negative local pressure in the tails and a prominent shell structure which may become accessible to observation in magnetically trapped gases of fermionic alkali atoms.Comment: 8 pages, 3 figures, accepted for publication in Phys. Rev. Let

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

One-dimensional non-interacting fermions in harmonic confinement: equilibrium and dynamical properties

We consider a system of one-dimensional non-interacting fermions in external harmonic confinement. Using an efficient Green's function method we evaluate the exact profiles and the pair correlation function, showing a direct signature of the Fermi statistics and of the single quantum-level occupancy. We also study the dynamical properties of the gas, obtaining the spectrum both in the collisionless and in the collisional regime. Our results apply as well to describe a one-dimensional Bose gas with point-like hard-core interactions.Comment: 11 pages, 5 figure