On computing the zeros of Laguerre–Sobolev polynomials
(2025) Numerical Algorithms — Vol. 100, n° 4, p. 1507-1526 (2025)
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- Abstract
- In this work, we focus on the computation of the zeros of a monic Laguerre-Sobolev orthogonal polynomial of degree n. Taking into account the associated four-term recurrence relation, this problem can be formulated as a generalized eigenvalue problem , involving a lower bidiagonal matrix and a 2-banded lower Hessenberg matrix of order n. Unfortunately, the considered generalized eigenvalue problem is very ill-conditioned, and classical balancing procedures do not improve it. Therefore, customary techniques for solving the generalized eigenvalue problem, like the Q Z method, yield unreliable results. Here, we propose a novel balancing procedure that drastically reduces the ill-conditioning of the eigenvalues of the involved matrix pencil. Moreover, we propose a fast and reliable algorithm, with O(n 2) computational complexity and O(n) memory, exploiting the structure of the considered matrix pencil .
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Citations
Laudadio, T., Mastronardi, N., Marcellán Español, F., Van Buggenhout, N., & Van Dooren, P. (2025). On computing the zeros of Laguerre–Sobolev polynomials. Numerical Algorithms, 100(4), 1507-1526. https://doi.org/10.1007/s11075-025-02021-z (Original work published 2025)