Description
32-digit values of the first 100 beta coefficients for orthogonal polynomials relative to the weight function w(x)=x^μ*exp(-x^ν) on [-Inf,Inf], μ=0, ν=4 are computed by a moment-based method using the routine sr_freud(dig,32,100,0,4), where dig=84 has been determined by the routine dig_freud(100,0,4,76,4,32). For the respective moments, see Exercise 2.23(a) in Walter Gautschi, "Orthogonal polynomials in MATLAB: Exercises and Solutions", Software, Environments, and Tools, SIAM, Philadelphia, PA, 2016. The software provided in this dataset allows generating an arbitrary number N of recurrence coefficients with arbitrary exponents μ > -1, ν > 0, as well as for different precisions. In applications, the related weight function exp(-x^{^4}/4) on [-Inf,Inf] is often used. Its recurrence coefficients can be obtained from ours simply by dividing all alpha-coefficients and the first beta-coefficient by √2 and dividing the remaining beta-coefficients by 2.
Cite this work
Researchers should cite this work as follows:
- Walter Gautschi (2016). 32-digit values of the first 100 recurrence coefficients for the Freud weight function with exponent 4. (Version 2.0). Purdue University Research Repository. doi:10.4231/R7VX0DHD
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Notes
The version 2.0 of this dataset was supplemented by four Matlab scripts and an updated text file.