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, respectively, 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.
Cite this work
Researchers should cite this work as follows:
- Gautschi, W. (2016). 32-digit values of the first 100 recurrence coefficients for the bimodal weight function with parameter ε=1. Purdue University Research Repository. doi:10.4231/R7BK19B6
The dataset consists of one text file and nine Matlab scripts.