32-digit values of the first 100 recurrence coefficients for a logarithmic weight function with rational square-root argument

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By Walter Gautschi

Purdue University

32-digit values of the first 100 recurrence coefficients for the weight function w(x)=log((1+x^(1/2))/(1-x^(1/2))) on [0,1]

Version 1.0 - published on 23 Mar 2017 doi:10.4231/R7XG9P5S - cite this Archived on 24 Apr 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)=log((1+x^(1/2))/(1-x^(1/2))) on [0,1] are computed by a moment-based method using the routine sr_logratsq(dig,32,100), where dig=180 has been determined by the routine dig_slogratsq(100,172,4,32). The software provided in this dataset allows generating an arbitrary number N of recurrence coefficients for any desired precision. It requires Matlab release R2011b or later. The recurrence coefficients produced can be used to obtain the (2n)-point, n <= 100, Gaussian quadrature formula for the weighted integral over [-1,1] with the (variable-sign) weight function log((1+x)/(1-x)). The Gaussian nodes and weights are x_k=(+/-) sqrt(u_k) resp. w_k=(+/-) v_k/x_k, where u_k and 2v_k are the nodes and weights of the n-point Gauss formula for the weight function w(x); cf. Piessens, R., M.M. Chawla, and N. Jayarajan, "Gaussian quadrature formulas for the numerical calculation of integrals with logarithmic singularity", J. Computational Phys. 21 (1976), 356-360. This implemented in the routine test_Piess.m for n=1:9 in 32-digit arithmetic. The results agree with those in Table 1 of the cited reference to all 20 digits given there.

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Notes

The dataset consists of one text file and six Matlab scripts. The scripts require Matlab release R2011b or later.

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