32-digit values of the first 100 recurrence coefficients for the bimodal weight function with parameter ε=.1

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By Walter Gautschi

Purdue University

32-digit values of the first 100 recurrence coefficients for the weight function w(x)=exp[-(x^2-1)^2/(4*ε)] on [-Inf,Inf], ε=.1

Version 1.0 - published on 03 Jan 2017 doi:10.4231/R76T0JNX - cite this Archived on 04 Feb 2017

Licensed under Attribution 3.0 Unported

Description

32-digit values of the first 100 recurrence coefficients for orthogonal polynomials relative to the weight function w(x)=exp[-(x^2-1)^2/(4*ε)] on [-Inf,Inf], ε=.1, are computed by a multicomponent discretization procedure using the routine sr_OPbimod(32,100) with dig=34, epsi=.1 entered at the prompt. The value dig=34 has been determined by the routine dig_sOPbimod(100,32,2,32), attesting to the high stability of the procedure. (Both routines may take several hours to run.) For details, see Exercise 2.38(c) in Walter Gautschi, "Orthogonal polynomials in MATLAB: Exercises and solutions", SIAM, Philadelphia, PA (2016). Auxiliary routines muOPbimod_gp.m and explore_mu.m are intended to help determine suitable values for the parameter μ needed when ε < 1/10. (This requires a minor temporary change in the routine smcdis.m as explained on p.130 of the cited reference.) The value of μ should be taken to be at least equal to 1; when N = 100, other selected values of μ for ε = .008:-.001:.001 are found to be μ = 1.7, 4.7, 8.0, 15.3, 26.2, 44.4, 84.2, 201.3. The software provided in this dataset allows generating an arbitrary number N of recurrence coefficients for arbitrary ε > 0 as well as for different precisions.

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The dataset consists of one text file and nine Matlab scripts.

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