A Unified Calculus-Based Analysis of Polynomial Interpolation and Least-Squares Regression: Theory, Conditioning, and a Novel Stability Result

A Unified Calculus-Based Analysis of Polynomial Interpolation and Least-Squares Regression: Theory, Conditioning, and a Novel Stability Result | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 27 February 2026 V1 Latest version Share on A Unified Calculus-Based Analysis of Polynomial Interpolation and Least-Squares Regression: Theory, Conditioning, and a Novel Stability Result Author : Sourish Balaji Ramakrishnan 0009-0006-4051-7436 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.177222680.00185315/v1 164 views 61 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper presents a mathematically rigorous comparison between polynomial interpolation (Lagrange form) and least-squares approximation through the framework of calculus and classical approximation theory. From first principles, interpolation polynomials and the normal equations are derived via stationary conditions on the sum-of-squared-residual functional, highlighting interpolation's exactness at data nodes versus regression's minimization of global L 2-error. Stability and error behavior are contrasted: Runge's phenomenon and ill-conditioned Vandermonde systems for interpolation versus the stabilizing role of orthogonal polynomials in least squares. A new theorem is proved showing that, with quasi-uniform nodes, the discrete Gram matrix in a continuous orthogonal basis approximates the identity with error O(1/n 2), explaining the enhanced stability of least-squares under such distributions. Numerical experiments complement the theory, documenting condition numbers and errors across node choices. The paper concludes by noting how these principles extend to modern statistical and machine-learning settings. Supplementary Material File (paper-1.pdf) Download 388.23 KB Information & Authors Information Version history V1 Version 1 27 February 2026 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords partial least squares regression polynomial interpolation preconditioning vandermonde matrix Authors Affiliations Sourish Balaji Ramakrishnan 0009-0006-4051-7436 [email protected] PSBB Siruseri High School Student View all articles by this author Metrics & Citations Metrics Article Usage 164 views 61 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Sourish Balaji Ramakrishnan. A Unified Calculus-Based Analysis of Polynomial Interpolation and Least-Squares Regression: Theory, Conditioning, and a Novel Stability Result. Authorea . 27 February 2026. DOI: https://doi.org/10.22541/au.177222680.00185315/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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