洞察 - Computational Algebra and Geometry - # Subresultant polynomials for multiple univariate polynomials in Newton basis

Formulating Subresultant Polynomials for Multiple Univariate Polynomials in Newton Basis

Q: How can the proposed formula for subresultant polynomials in Newton basis be extended to handle the case where the input polynomials are not all expressed in the same Newton basis?

The proposed formula for subresultant polynomials in Newton basis can be extended to accommodate input polynomials expressed in different Newton bases by introducing a transformation mechanism that aligns the bases. This involves identifying a common set of nodes or a reference Newton basis that can serve as a bridge between the various bases used for the input polynomials. To achieve this, one could utilize a basis transformation technique that maps each polynomial from its respective Newton basis to the common basis. This transformation can be performed using interpolation methods or by constructing a change of basis matrix that relates the coefficients of the polynomials in their original bases to those in the common basis. Once all polynomials are expressed in the same Newton basis, the existing formula for subresultant polynomials can be applied directly. Moreover, the formulation of the companion matrix can be adapted to account for the different bases by ensuring that the matrix operations respect the structure of the polynomials in their respective bases. This approach not only preserves the algebraic properties of the subresultants but also maintains the integrity of the computations, allowing for a seamless integration of polynomials from various Newton bases.

Q: What are the potential numerical stability advantages of computing subresultant polynomials directly in Newton basis compared to first transforming to power basis?

Computing subresultant polynomials directly in Newton basis offers several numerical stability advantages over the traditional approach of transforming to power basis first. One of the primary concerns with basis transformation is the potential for numerical instability, which can arise due to the ill-conditioning of the transformation matrices, especially when the nodes of the Newton basis are close together or poorly chosen. By performing computations directly in the Newton basis, one can avoid the intermediate step of transforming to power basis, which often involves significant numerical error due to rounding and loss of precision. The Newton basis is particularly advantageous in interpolation scenarios, where the polynomials are evaluated at specific nodes. This direct computation minimizes the risk of introducing errors that can accumulate during the transformation process. Additionally, the structure of the Newton basis allows for more stable evaluations of polynomials, as it is inherently designed to handle polynomial interpolation and can provide better conditioning for numerical algorithms. This stability is crucial in applications such as polynomial system solving and quantifier elimination, where accurate computation of subresultants is essential for the overall reliability of the results.

Q: Can the ideas developed in this work be applied to other types of non-standard polynomial bases beyond Newton basis?

Yes, the ideas developed in this work can be applied to other types of non-standard polynomial bases beyond Newton basis. The framework established for formulating subresultant polynomials in Newton basis, particularly the use of companion matrices and determinant polynomials, is versatile and can be adapted to various polynomial bases, such as Bernstein, Lagrange, or Chebyshev bases. To extend the methodology to other bases, one would need to define the appropriate companion matrix and determinant polynomial formulations that correspond to the specific properties of the new basis. For instance, the construction of the companion matrix would need to reflect the structure of the polynomials in the chosen basis, ensuring that the endomorphism defined by the matrix aligns with the multiplication operations relevant to that basis. Moreover, the generalization of the determinant polynomial concept, as introduced in the paper, allows for flexibility in accommodating different polynomial sets. By adjusting the definitions and properties of the matrices involved, researchers can derive subresultant polynomials that maintain the algebraic characteristics of the original polynomials, regardless of the basis used. This adaptability opens up new avenues for research and application in computational algebra, enabling the development of basis-preserving algorithms for a wider range of polynomial bases, thus enhancing the robustness and applicability of resultant theory in various mathematical and engineering contexts.

核心概念

A new formula is developed to express the subresultant polynomials of multiple univariate polynomials in the same Newton basis as the input polynomials.

摘要

The paper addresses the problem of formulating subresultant polynomials for several univariate polynomials when they are expressed in Newton basis, rather than the standard power basis.

Key highlights:

The authors extend the concept of companion matrix from power basis to Newton basis, which allows them to construct a matrix that can be used to formulate the subresultant polynomials in the given Newton basis.
They generalize the concept of determinant polynomial from power basis to Newton basis, enabling the expression of subresultant polynomials in the provided Newton basis.
The newly developed formula for subresultant polynomials in Newton basis is shown to be equivalent to the subresultant polynomials in power basis after expansion.
As an application, the authors devise a method for computing the GCD of several numerical Newton polynomials without involving basis transformation.
The formula generalizes previous work on subresultant polynomials in roots by allowing an arbitrary choice of nodes for the Newton basis, rather than restricting to the roots of one of the input polynomials.

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Subresultants of Several Univariate Polynomials in Newton Basis

by Weidong Wang... 在 arxiv.org 09-11-2024

https://arxiv.org/pdf/2212.03422.pdf

Subresultants of Several Univariate Polynomials in Newton Basis

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How can the proposed formula for subresultant polynomials in Newton basis be extended to handle the case where the input polynomials are not all expressed in the same Newton basis?

The proposed formula for subresultant polynomials in Newton basis can be extended to accommodate input polynomials expressed in different Newton bases by introducing a transformation mechanism that aligns the bases. This involves identifying a common set of nodes or a reference Newton basis that can serve as a bridge between the various bases used for the input polynomials.
To achieve this, one could utilize a basis transformation technique that maps each polynomial from its respective Newton basis to the common basis. This transformation can be performed using interpolation methods or by constructing a change of basis matrix that relates the coefficients of the polynomials in their original bases to those in the common basis. Once all polynomials are expressed in the same Newton basis, the existing formula for subresultant polynomials can be applied directly.
Moreover, the formulation of the companion matrix can be adapted to account for the different bases by ensuring that the matrix operations respect the structure of the polynomials in their respective bases. This approach not only preserves the algebraic properties of the subresultants but also maintains the integrity of the computations, allowing for a seamless integration of polynomials from various Newton bases.

What are the potential numerical stability advantages of computing subresultant polynomials directly in Newton basis compared to first transforming to power basis?

Computing subresultant polynomials directly in Newton basis offers several numerical stability advantages over the traditional approach of transforming to power basis first. One of the primary concerns with basis transformation is the potential for numerical instability, which can arise due to the ill-conditioning of the transformation matrices, especially when the nodes of the Newton basis are close together or poorly chosen.
By performing computations directly in the Newton basis, one can avoid the intermediate step of transforming to power basis, which often involves significant numerical error due to rounding and loss of precision. The Newton basis is particularly advantageous in interpolation scenarios, where the polynomials are evaluated at specific nodes. This direct computation minimizes the risk of introducing errors that can accumulate during the transformation process.
Additionally, the structure of the Newton basis allows for more stable evaluations of polynomials, as it is inherently designed to handle polynomial interpolation and can provide better conditioning for numerical algorithms. This stability is crucial in applications such as polynomial system solving and quantifier elimination, where accurate computation of subresultants is essential for the overall reliability of the results.

Can the ideas developed in this work be applied to other types of non-standard polynomial bases beyond Newton basis?

Yes, the ideas developed in this work can be applied to other types of non-standard polynomial bases beyond Newton basis. The framework established for formulating subresultant polynomials in Newton basis, particularly the use of companion matrices and determinant polynomials, is versatile and can be adapted to various polynomial bases, such as Bernstein, Lagrange, or Chebyshev bases.
To extend the methodology to other bases, one would need to define the appropriate companion matrix and determinant polynomial formulations that correspond to the specific properties of the new basis. For instance, the construction of the companion matrix would need to reflect the structure of the polynomials in the chosen basis, ensuring that the endomorphism defined by the matrix aligns with the multiplication operations relevant to that basis.
Moreover, the generalization of the determinant polynomial concept, as introduced in the paper, allows for flexibility in accommodating different polynomial sets. By adjusting the definitions and properties of the matrices involved, researchers can derive subresultant polynomials that maintain the algebraic characteristics of the original polynomials, regardless of the basis used.
This adaptability opens up new avenues for research and application in computational algebra, enabling the development of basis-preserving algorithms for a wider range of polynomial bases, thus enhancing the robustness and applicability of resultant theory in various mathematical and engineering contexts.