Description
32-digit values of the first 200 recurrence coefficients for orthogonal polynomials relative to the weight function w(x)=x^a* [log(1/x)]^b on [0,1], a=0, b=1, are computed by a moment-based method using the routine sr_l_alglog(dig,32,200,0,1), where dig=328 has been determined by the routine dig_l_alglog(200,0,1,320,4,32). The results are in agreement, except for occasional endfigure errors of 1 unit, with the first 20 recurrence coefficients given to 12 digits in Table 1 of Bernard Danloy, "Numerical construction of Gaussian quadrature formulas for int_0^1 (-Log x)*x^a*f(x)dx and int_0^Inf E_m(x)f(x)dx, Math. Comp. 27 (1973), 861-869. Use of the routine sgauss.m in the dataset doi:10.4231/R72805KQ also verified to the same accuracy the 10- and 20-point Gaussian quadrature formulae given in Table 2 of this reference.The software provided in this dataset allows generating an arbitrary number N of recurrence coefficients for arbitrary a > -1, b > -1, as well as for different precisions. If the singularities, with the same exponents, occur at the right endpoint, that is, if w(x)=(1-x)^a*[log(1/(1-x))]^b on [0,1], then the alpha-coefficients must be replaced by 1 minus the present ones, whereas the beta-coefficients remain the same.
Cite this work
Researchers should cite this work as follows:
- Walter Gautschi (2017). 32-digit values of the first 200 recurrence coefficients for the weight function w(x)=log(1/x) on [0,1]. (Version 2.0). Purdue University Research Repository. doi:10.4231/R7QN64RH
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Notes
The dataset consists of one text file and four Matlab scripts. This versions enables 32-digit values of the first 200 recurrence coefficients for the weight function w(x)=log(1/x) on [0,1].