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

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By Walter Gautschi

Purdue University

32-digit values of the first 100 recurrence coefficients for the weight function w(x)=((1-om2*x^2)*(1-x^2))^(-1/2) on [-1,1], om2=1/2

Version 1.0 - published on 22 Nov 2016 doi:10.4231/R7HH6H1D - cite this Archived on 23 Dec 2016

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)=((1-om2*x^2)*(1-x^2))^(-1/2) on [-1,1],  om2=1/2, are computed by a modified-moment-based method using the routine sr_ellcheb(dig,32,100,1/2), where dig=36 has been determined by the routine dig_ellcheb(100,1/2,32,4,32), attesting to the high stability of the modified Chebyshev algorithm. (The output seems to suggest that the recurrence coefficients beta_k are exactly equal to 1/4 for k > =39, but this is only true in 32-digit precision, as computation in higher precisions will show.) For the modified moments, see Example 4.4 in Walter Gautschi, "On generating orthogonal polynomials", SIAM Journal on Statistical and Scientific Computing 3 (1982), 289-317. doi: 10.1137/0903018. The software provided in this dataset allows generating an arbitrary number N of recurrence coefficients for arbitrary parameter om2, 0< om2< 1, and for different precisions.

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

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