相关论文: Efficient Computation of the Characteristic Polyno…
We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an $n\times n$ matrix over a finite field that requires $O(n^3)$ field operations and O(n) random vectors, and is well suited for successful practical…
We analyze the precision of the characteristic polynomial of an $n\times n$ p-adic matrix A using differential precision methods developed previously. When A is integral with precision O(p^N), we give a criterion (checkable in time…
We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of…
A polynomial-time algorithm for computing the permanent in any field of characteristic 3 is presented in this article. The principal objects utilized for that purpose are the Cauchy and Vandermonde matrices, the discriminant function and…
There is a digraph corresponding to every square matrix over $\mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
New algorithms are presented for computing annihilating polynomials of Toeplitz, Hankel, and more generally Toeplitz+ Hankel-like matrices over a field. Our approach follows works on Coppersmith's block Wiedemann method with structured…
In this paper we use Euler-Seidel matrices method to find out some properties of exponential and geometric polynomials and numbers. Some known results are reproved and some new results are obtained.
In this paper, we investigate characteristic polynomials of matrices in min-plus algebra. Eigenvalues of min-plus matrices are known to be the minimum roots of the characteristic polynomials based on tropical determinants which are designed…
We use recent results on algorithms for Markov decision problems to show that a canonical form for a generalized P-matrix can be computed, in some important cases, by a strongly polynomial algorithm.
Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper…
The paper addresses the calculation of correlation functions of permanental polynomials of matrices with random entries. By exploiting a convenient contour integral representation of the matrix permanent some explicit results are provided…
We survey three methods for proving that the characteristic polynomial of a finite lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on…
Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…
In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F}\in\mathbb{K}[x]^{n\times n}$ over a field $\mathbb{K}$, we give a deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm…
We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written…
In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and real…
We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t…
We give several new algorithms for dense polynomial multiplication based on the Kronecker substitution method. For moderately sized input polynomials, the new algorithms improve on the performance of the standard Kronecker substitution by a…