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

Jing Yang, Wei Yang

TL;DR

This work introduces a basis-preserving algorithm for constructing Bézout matrices of two Newton polynomials directly in the Newton basis, thereby avoiding costly basis transformations and reducing numerical instability. The core contribution is a recurrence for Bézout entries and the ${\sf BezNewton\_preserving}$ algorithm, which achieves $\mathcal{O}(n^2)$ time compared to $\mathcal{O}(n^3)$ for naive or transformation-based methods. The paper also applies the method to confederate resultant matrices for Newton polynomials and demonstrates substantial practical gains over basis-transformation approaches, both in theory and Maple experiments. The approach provides a clearer, structure-preserving view of Bézout matrices in Newton basis and opens avenues for efficient Newton-basis subresultants and related resultants computations.

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.

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

TL;DR

This work introduces a basis-preserving algorithm for constructing Bézout matrices of two Newton polynomials directly in the Newton basis, thereby avoiding costly basis transformations and reducing numerical instability. The core contribution is a recurrence for Bézout entries and the algorithm, which achieves time compared to for naive or transformation-based methods. The paper also applies the method to confederate resultant matrices for Newton polynomials and demonstrates substantial practical gains over basis-transformation approaches, both in theory and Maple experiments. The approach provides a clearer, structure-preserving view of Bézout matrices in Newton basis and opens avenues for efficient Newton-basis subresultants and related resultants computations.

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.
Paper Structure (18 sections, 5 theorems, 31 equations, 1 figure, 6 tables)

This paper contains 18 sections, 5 theorems, 31 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Main Results
  5. Proofs
  6. Proof of Theorem \ref{['thm:main']}
  7. Proof of Proposition \ref{['prop:algorithm']}
  8. Comparison
  9. A description of the algorithm ${\sf BezNewton\_trans}$
  10. Complexity
  11. Performance
  12. Application
  13. Problem formulation
  14. Three approaches for computing confederate resultant matrices
  15. Comparison
  16. ...and 3 more sections

Key Result

Theorem 7

Given $\boldsymbol{\lambda}\in\mathbb{F}^n$ which determines a Newton basis $\boldsymbol{N}_{\boldsymbol{\lambda}}(x)$ of $\mathbb{F}_{n}[x]$ and $F,G\in\mathbb{F}_{n}[x]$ be as in eq:F+G, let $\boldsymbol{B}_{\tilde{\boldsymbol{N}}_{\boldsymbol{\lambda}}}=(c_{i,j})_{n\times n}$ be the Bézout matrix where

Figures (1)

  • Figure 1: Two approaches for computing the Bézout matrix in Newton basis

Theorems & Definitions (15)

  • Definition 1: Bézout matrix 2007_Aruliah_Corless_Gonzalez_Vega_Shakoori
  • Remark 2
  • Definition 3
  • Example 4
  • Theorem 7
  • Example 8: Continued from Example \ref{['ex:comparison_Bez']}
  • Example 9: Continued from Example \ref{['ex:comparison_Bez']}
  • Proposition 10
  • proof
  • proof
  • ...and 5 more