Year: 2022
Type: preprint
Abstract: In this paper we included some basic properties for weighted greatest common divisors, and discuss how to speed up computing the weighted greatest common divisor. By ordering the 'weights' we are able to significantly shorten the operations to computing wgcd. In the absence of an efficient algorithm for computing wgcd by ordering the weights, and using $\gcd$, we significantly reduce the numbers for which we want to compute wgcd. As a final result in this paper we prove that: If $\mathbf{x} = (x_{0},\dots ,x_{n})\in \mathbb{Z}^{n+1}$, with weights $\mathfrak{w}=(q_{0},\dots ,q_{n})$ and $q_{0}\leq \cdots \leq q_{n}$, then ${\rm wgcd}_w(\mathbf{x}) = {\rm wgcd}_w(y_0,y_1,\dots,y_n)$, where $y_i = \gcd(x_i,\dots,x_n)$, and $y_0\leq y_1 \leq\dots \leq y_n$.
Source: arXiv (Cornell University)
Author: Zaka, Orgest
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4306707367