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.

Researchers should cite this work as follows:

• Gautschi, W. (2017). 32-digit values of the first 100 recurrence coefficients for a logarithmic weight function with rational square-root argument. Purdue University Research Repository. doi:10.4231/R7XG9P5S

  BibTex | EndNote

1. Computer Science
2. Computer Science (Dept)
3. Gaussian quadrature
4. Mathematics
5. Modification algorithms for orthogonal polynomials
6. Orthogonal polynomials
7. Walter Gautschi Archives
8. weight functions

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