[Paper Review] Solution of Interpolation Problems via the Hankel Polynomial Construction

Alexei Yu. Uteshev, Ivan Baravy|arXiv (Cornell University)|Mar 29, 2016

Mathematical functions and polynomials4 citations

TL;DR

This paper presents a novel method for solving polynomial and rational interpolation problems using Hankel polynomials derived from weighted sums of interpolation data. By leveraging Jacobi's 19th-century determinant identities and the Berlekamp-Massey algorithm, it enables efficient computation of interpolants and provides a robust framework for error correction and resultant computation in polynomial systems.

ABSTRACT

We treat the interpolation problem $ \\{f(x_j)=y_j\\}_{j=1}^N $ for polynomial and rational functions. Developing the approach by C.Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences $ \\{\\sum_{j=1}^N x_j^ky_j/W^{\\prime}(x_j) \\}_{k\\in \\mathbb N} $ and $ \\{\\sum_{j=1}^N x_j^k/(y_jW^{\\prime}(x_j)) \\}_{k\\in \\mathbb N} $; here $ W(x)=\\prod_{j=1}^N(x-x_j) $. The obtained results are applied for the error correction problem, i.e. the problem of reconstructing the polynomial from a redundant set of its values some of which are probably erroneous. The problem of evaluation of the resultant of polynomials $ p(x) $ and $ q(x) $ from the set of values $ \\{p(x_j)/q(x_j) \\}_{j=1}^N $ is also tackled within the framework of this approach.

Motivation & Objective

• To develop a systematic approach to polynomial and rational interpolation using Hankel polynomials generated from interpolation data.
• To enable efficient computation of interpolants via recursive algorithms based on Jacobi's determinant identities.
• To address the error correction problem by reconstructing polynomials from redundant, possibly corrupted, data sets.
• To compute the resultant of two polynomials using Hankel determinants derived from rational function values.
• To extend classical interpolation methods to handle symbolic parameters and degenerate cases in polynomial systems.

Proposed method

• Construct Hankel polynomials using sequences derived from interpolation nodes and values: $\tau_k = \sum_{j=1}^N y_j \frac{x_j^k}{W'(x_j)}$ and $\widetilde{\tau}_k = \sum_{j=1}^N \frac{1}{y_j} \frac{x_j^k}{W'(x_j)}$, where $W(x) = \prod_{j=1}^N (x - x_j)$.
• Utilize the recursive identity $A_k \mathcal{H}_{k-2}(x) + (B_k - x)\mathcal{H}_{k-1}(x) + \frac{1}{A_k}\mathcal{H}_k(x) \equiv 0$ for efficient computation of Hankel polynomials.
• Apply the Berlekamp-Massey algorithm to compute minimal polynomials and interpolate rational functions with varying numerator and denominator degrees.
• Represent the resultant of two polynomials $p(x)$ and $q(x)$ as a Hankel determinant $H_n(\{\widetilde{\tau}\})$ of order $n$, enabling symbolic parameter analysis.
• Use subresultant-like properties of Hankel minors to recover common roots of $p(x)$ and $q(x)$ via rational expressions involving Hankel coefficients.

Experimental results

Research questions

• RQ1 How can Hankel polynomials be systematically constructed from interpolation data to solve polynomial and rational interpolation problems?
• RQ2 Can the recursive structure of Hankel determinants be exploited to efficiently compute interpolants, especially in the presence of redundant or corrupted data?
• RQ3 How can the resultant of two polynomials be computed directly from function values without explicit factorization?
• RQ4 What conditions allow for the recovery of a polynomial from a corrupted, redundant set of values using Hankel-based methods?
• RQ5 How can common roots of two polynomials be reconstructed using Hankel determinants and their minors in symbolic parameter settings?

Key findings

• The Hankel polynomial construction provides a unified framework for both polynomial and rational interpolation, with explicit formulas derived from Jacobi’s determinant identities.
• The recursive computation of Hankel polynomials via the Berlekamp-Massey algorithm enables efficient solution of interpolation problems even for large data sets.
• For the error correction problem, the method successfully reconstructs a polynomial of degree $n < N-1$ from $N$ values when some are corrupted, assuming the $x$-nodes are correct.
• The resultant of two polynomials $p(x)$ and $q(x)$ is expressed as a Hankel determinant $H_3(\{\widetilde{\tau}\})$, which vanishes precisely when $p(x)$ and $q(x)$ share a common root.
• In Example 6.2, the resultant is computed symbolically, and the value $\alpha = -9$ is found to yield a common zero $\lambda = 2 - \sqrt{3}$, demonstrating the method’s utility in parameter-dependent systems.