Related papers: Generalized companion matrix for approximate GCD
We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common…
We consider the problem of computing the partition function $\sum_x e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\Bbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$. In the case of a quadratic polynomial $f$,…
The Generalized Parton Distributions (GPDs) of the nucleon are analyzed within the relativistic constituent quark model formulated on the light-front. It is shown that the matrix elements of the plus component of the one-body vector current…
We prove that, over a field $\mathbb{F}$ of odd characteristic $p$, a companion matrix $C$ is the sum of $E$ and $N$, with $E$ $p$-potent (i.e. $E^p = E$,) and $N$ nilpotent, if and only if the trace of $C$ is an integer multiple of unity…
Variational approximation methods have proven to be useful for scaling Bayesian computations to large data sets and highly parametrized models. Applying variational methods involves solving an optimization problem, and recent research in…
We give an effective solution of the conjugacy problem for two by two matrices over the polynomial ring in one variable over a finite field.
Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution $G$. In many queueing models,…
The link between the nucleon generalized parton distributions and the non-diagonal one-body density matrix in momentum space is studied. Attention is focussed on the region where quark generalized parton distributions (GPD's) describe…
We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group…
Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full…
The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite difference like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using…
We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the…
Matrix multiplication is a fundamental building block for large scale computations arising in various applications, including machine learning. There has been significant recent interest in using coding to speed up distributed matrix…
We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be…
The proximal generalized alternating direction method of multipliers (p-GADMM) is substantially efficient for solving convex composite programming problems of high-dimensional to moderate accuracy. The global convergence of this method was…
In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we get the majorization relationship between the zeros of convex combinations of incomplete polynomials and an…
Approximation of entire functions by their pad\'e approximants has been examined in the past. It is true that generically such an approximation holds. However, examining this problem from another viewpoint, we obtain stronger generic…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
In this short note, we prove a G-equivariant generalisation of McDuff-Segal's group-completion theorem for finite groups G. A new complication regarding genuine equivariant localisations arises and we resolve this by isolating a simple…
This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the…