Volume 65, pp. 517-532, 2026.

Fast and accurate computation of the zeros of Gegenbauer-Sobolev orthogonal polynomials

Teresa Laudadio, Francisco José Marcellán Español, Nicola Mastronardi, Niel Van Buggenhout, and Paul Van Dooren

Abstract

Gegenbauer–Sobolev polynomials are related to the classical Gegenbauer polynomials, and they satisfy orthogonality relations with respect to a Sobolev inner product. In this paper, we focus on the study of the location and computation of the zeros of Gegenbauer–Sobolev polynomials. In particular, some interlacing properties between the zeros of the Gegenbauer–Sobolev polynomials and those of the Gegenbauer polynomials are shown. Moreover, two variants of a Newton-like method, known as the Ehrlich–Aberth method, are applied to compute the zeros of the considered polynomials. The Ehrlich–Aberth method is iterative, requiring the evaluation of the polynomial and its derivative at all the approximations of the zeros. This is accomplished by exploiting a relation that links the sequence of Gegenbauer–Sobolev polynomials with the sequence of Gegenbauer polynomials. Furthermore, in order to ensure fast convergence, the proposed algorithms exploit the derived interlacing properties to construct the initial guesses of the zeros required by the Ehrlich–Aberth method. The algorithms are shown to be accurate and very fast, with a computational complexity of the order of $\mathcal{O}(n^2) $ for a Gegenbauer–Sobolev polynomial of degree $ n. $

Full Text (PDF) [312 KB], BibTeX , DOI: 10.1553/etna_vol65s517

Key words

Sobolev orthogonal polynomials, Gegenbauer polynomials, zeros of polynomials

AMS subject classifications

33C47, 65D32, 65F15

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