Recursive construction of biorthogonal polynomials for handling polynomial regression

Abstract

An adaptive procedure for constructing polynomials which are biorthogonal to the basis of monomials in the same finite-dimensional inner product space is proposed. By taking advantage of available orthogonal polynomials, the proposed methodology reduces the well-known instability problem arising from the matrix inversion involved in classical polynomial regression. The recurrent generation of the biorthogonal basis facilitates the upgrading of all its members to include an additional one. Moreover, it allows for a natural downgrading of the basis. This convenient feature leads to a straightforward approach for reducing the number of terms in the polynomial regression approximation. The merit of this approach is illustrated through a series of examples where the resulting biorthogonal basis is derived from Legendre, Laguerre, and Chebyshev orthogonal polynomials.

Publication DOI: https://doi.org/10.1016/j.amc.2025.129578
Divisions: College of Engineering & Physical Sciences > School of Computer Science and Digital Technologies > Applied Mathematics & Data Science
College of Engineering & Physical Sciences
Aston University (General)
Additional Information: Copyright © 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (https://creativecommons.org/licenses/by/4.0/).
Uncontrolled Keywords: Biorthogonal polynomials,Biorthogonal representation of orthogonal projections,Polynomial regression,Computational Mathematics,Applied Mathematics
Publication ISSN: 1873-5649
Data Access Statement: Software for generating the data and reproducing the examples has been made available on http://www.nonlinear-approx.info/examples/node018.html.
Last Modified: 30 Jun 2025 07:25
Date Deposited: 10 Jun 2025 14:50
Full Text Link:
Related URLs: https://www.sci ... 3042?via%3Dihub (Publisher URL)
http://www.scop ... tnerID=8YFLogxK (Scopus URL)
PURE Output Type: Article
Published Date: 2025-12-15
Published Online Date: 2025-06-10
Accepted Date: 2025-05-28
Authors: Rebollo-Neira, Laura (ORCID Profile 0000-0002-7420-8977)
Laurie, Jason (ORCID Profile 0000-0002-3621-6052)

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