Core Concepts

This paper presents a basis-preserving algorithm for constructing the Bezout matrix of two Newton polynomials, which avoids the need for transformation from Newton basis to monomial basis and significantly reduces the computational cost and numerical instability.

Abstract

The paper addresses the problem of constructing Bezout matrices for Newton polynomials in a basis-preserving approach. The key insights are:
The authors 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.
This approach avoids the need for transformation from Newton basis to monomial basis, which significantly reduces the computational cost and mitigates numerical instability caused by basis transformation.
The proposed algorithm takes two univariate polynomials in Newton basis as input and produces a Bezout matrix for the given polynomials, while ensuring that the resulting matrix is formulated in the given basis.
The authors also demonstrate an application of the proposed algorithm in the construction of the confederate resultant matrix for Newton polynomials, and experimental results show that the proposed algorithms perform significantly better than the basis-transformation-based ones.

Stats

The paper provides the following key metrics:
The computational complexity of the proposed algorithm BezNewton_preserving is O(n^2) for multiplications and additions, where n is the degree of the input polynomials.
The computational complexity of the basis-transformation-based algorithm BezNewton_trans is O(n^3) for both multiplications and additions.

Quotes

"The basis-preserving formulation of Bezout matrix can preserve the structure in the Bezout matrix better."
"The proposed algorithm does not require any basis transformation, thus avoiding the heavy computational cost and numerical instability caused by such transformations."

Key Insights Distilled From

by Jing Yang,We... at **arxiv.org** 04-30-2024

Deeper Inquiries

The basis-preserving Bezout matrix algorithm can be extended to handle multivariate polynomials in non-standard bases by considering the internal structure of the Bezout matrices in the given basis. When dealing with multivariate polynomials, the algorithm needs to account for the different variables and their corresponding bases. By formulating the problem in terms of the specific non-standard bases for each variable, the algorithm can be adapted to generate the Bezout matrix in the specified basis used to define the input polynomials. This extension would involve modifying the recursive steps and computations to accommodate the additional variables and bases present in multivariate polynomials.

The basis-preserving Bezout matrix algorithm has potential applications in various areas of computer algebra and symbolic computation. One key application is in the field of computational algebraic geometry, where the algorithm can be used to compute resultant matrices for systems of polynomial equations. These resultant matrices play a crucial role in solving polynomial systems, computing intersections of algebraic varieties, and determining the number of solutions to polynomial equations. Additionally, the algorithm can be applied in polynomial factorization, polynomial GCD computation, and solving systems of polynomial equations in non-standard bases. The efficiency and numerical stability of the basis-preserving algorithm make it a valuable tool in symbolic computation tasks involving polynomials.

The insights from the basis-preserving Bezout matrix algorithm can be applied to develop efficient algorithms for computing other matrix representations, such as the companion matrix or the Sylvester matrix, for polynomials in non-standard bases. By understanding the structure and properties of Bezout matrices in non-standard bases, similar algorithms can be designed for computing companion matrices, which are essential in polynomial factorization and solving polynomial equations. The recursive nature of the basis-preserving algorithm can be adapted to handle the computations required for companion matrices and Sylvester matrices in non-standard bases, providing a basis-preserving approach for these matrix representations. This approach can lead to improved efficiency and numerical stability in symbolic computation tasks involving polynomial matrices.

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