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Asymptotic expansion of generalized Jacobi polynomials


JACOBI is a collection of programmes for computing the asymptotic expansions of generalized Jacobi polynomials in the complex plane. The Matlab and Julia versions also compute the plots given in the main reference article, test correctness, give an indication for heuristics and contain a simple application for generalised Gaussian quadrature.


The code is distributed under the BSD license and this is a compressed version of all files below.


The Maple sheet includes a derivation of the formulas found in the reference article. GeneralOrderv10.mw, GeneralOrderv10.pdf. (Tested with Maple version 17.)


alphan.m, asy_left.m, asy_lens.m, asy_outer.m, asy_right.m, betan.m, computePlotsFromArticle.m, contour_integrals.m, exactPolys.m, gamman.m, , gauss.m, illustrateFunctionality.m, plotConv.m, README, r_jacobi.m, trap_rule.m, , UExplicit.m, UQ.m, WV.m. (Tested with MATLAB R2013b.)


asy_expansions.jl, computePlotsFromArticle.jl, illustrateFunctionality.jl, precomputations.jl, README, UExplicit.jl, validation.jl (Tested with Julia 0.4.0.)


This code is based on the article [technical report, DOI]

  1. Alfredo Deano, Daan Huybrechs and Peter Opsomer. Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials.. To appear in Advances in Computational Mathematics DOI.

It uses the Riemann-Hilbert analysis from

  1. A. B. J. Kuijlaars, K. T.-R. McLaughlin, W. Van Assche, and M. Vanlessen. The Riemann-Hilbert approach to strong asymptotics of orthogonal polynomials on [-1, 1]. Adv. Math. , Vol. 188 pp. 337-398. 2004.