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Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials

Deaño, Alfredo, Huybrechs, Daan, Opsomer, Peter (2016) Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials. Advances in Computational Mathematics, 42 . pp. 791-822. ISSN 1019-7168. E-ISSN 1572-9044. (doi:10.1007/s10444-015-9442-z) (KAR id:51339)


We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [−1, 1] with weight function: w(x)=(1−x)α(1+x)βh(x),α,β>−1 and h(x) a real, analytic and strictly positive function on [−1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in O(n) operations, rather than O(n2) based on the recurrence relation.

Item Type: Article
DOI/Identification number: 10.1007/s10444-015-9442-z
Projects: Orthogonality and Approximation. Theory and Applications in Science and Technology, Numerical and asymptotic methods for the evaluation of mathematical functions and associated
Additional information: Full text upload complies with journal requirements
Uncontrolled keywords: Asymptotic analysis; Jacobi polynomials; Riemann–Hilbert problems; Numerical software
Subjects: Q Science > QA Mathematics (inc Computing science) > QA297 Numerical analysis
Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Q Science > QA Mathematics (inc Computing science) > QA351 Special functions
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: Organisations -1 not found.
Organisations -1 not found.
Organisations -1 not found.
Depositing User: Alfredo Deano Cabrera
Date Deposited: 02 Nov 2015 11:12 UTC
Last Modified: 09 Dec 2022 07:25 UTC
Resource URI: (The current URI for this page, for reference purposes)

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