LAGUERRE
Asymptotic expansions of generalized Laguerre polynomials
Introduction
LAGUERRE is a collection of programmes for computing the asymptotic expansions of generalized Laguerre polynomials in the complex plane. The Matlab and Sage versions also compute plots like in the main reference article, give an indication for heuristics, allow printing source or LaTeX code for higher order terms and contain a simple application for generalised Gaussian quadrature.
Availability
The code is distributed under the BSD license and Laguerre3.tar.gz is a compressed version of the files explained below.
Sage
Two Sage worksheets (http://www.sagemath.org/) include a derivation of the formulas found in the reference article. PlotMonomialQ.sws contains code to make convergence plots of the standard Laguerre polynomial and SymbolicUQ.sws computes symbolic results for the higher order terms and prints source or LaTeX code of them. (Tested with SageMath Version 6.9.)
Matlab
The main script is ComputePlotsFromArticle.m: it checks the asymptotic expansions for correctness (using the results from compRecLagBF.jl) and makes plots using plotConv.m. The script GaussLaguerre.m contains a proof of concept that generalized Gauss-Laguerre quadrature rules can be constructed in this framework. The script WriteUQ.m automatically writes out higher order terms to a file and Heuristics.m enables a comparison between expansions in different regions.
s is a structure containing the asymptotic information: s.alpha and s.q are the inputs above, s.betan(n,T) is the expansion of the MRS number. s.pa(n,x,z) is the expansion of the orthonormal polynomial in the outer region, s.pb(n,x,z) the expansion in the inner region (0,\beta_n), s.pc near the right disk and s.pd near the left disk: when appended with 'woq', e^(n V_n(z)/2 is not included. s.bnm1 and s.an are the expansions of the recurrence coefficients, s.gamman is the expansion of the normalizing constants. When isnumeric(q), s.m is length(q)-1; else, s.np is the input above. The other fields of s contain temporary variables.
This variable is constructed by getAsy.m. It needs the U-matrices from UQ.m, which calls for WV.m, carrying out the convolutions. These functions require the (maximum) number of terms in the asymptotic expansions. exactPolys computes the exact polynomials we compare against. We make a distinction between non-polynomial Q(x), general polynomial Q(x), 'Monomial' Q(x) = q_m x^m + q_0 and linear Q(x). (Tested with MATLAB R2016b.)
Julia
The Julia script compRecLagBF.jl computes the recurrence coefficients for the nonstandard weight functions considered in the paper to be used in the Matlab script. (Tested with Julia version 0.3.8.)Publications
This code is based on the article
- Daan Huybrechs and Peter Opsomer. Construction and implementation of asymptotic expansions for Laguerre-type orthogonal polynomials.. In preparation at this point in time (December 2016).
See also the included technical report TW676.pdf and
http://www.cs.kuleuven.be/publicaties/rapporten/tw/TW676.abs.html. It is based on the Riemann-Hilbert analysis from
-
M. Vanlessen.
Strong asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory.
Constr. Approx. , Vol. 25 pp. 125-175.
2007.
http://dx.doi.org/10.1007/s00365-005-0611-z
