A Basis-preserving Algorithm for Computing the Bezout Matrix of Newton Polynomials

TL;DR

This paper introduces a basis-preserving algorithm for computing Bezout matrices directly in the Newton basis, reducing computational costs and numerical instability compared to traditional basis transformation methods.

Contribution

The paper presents a novel basis-preserving algorithm for Bezout matrices in Newton basis, avoiding basis transformation and improving efficiency and stability.

Findings

The proposed algorithm reduces computational cost.

It mitigates numerical instability from basis transformation.

Experimental results show superior performance over traditional methods.

Abstract

This paper tackles the problem of constructing Bezout matrices for Newton polynomials in a basis-preserving approach that operates directly with the given Newton basis, thus avoiding the need for transformation from Newton basis to monomial basis. This approach significantly reduces the computational cost and also mitigates numerical instability caused by basis transformation. For this purpose, we investigate the internal structure of Bezout matrices in Newton basis and design a basis-preserving algorithm that generates the Bezout matrix in the specified basis used to formulate the input polynomials. Furthermore, we show an application of the proposed algorithm on constructing confederate resultant matrices for Newton polynomials. Experimental results demonstrate that the proposed methods perform superior to the basis-transformation-based ones.