Volume 50, pp. 71-97, 2018.

Computation of induced orthogonal polynomial distributions

Akil Narayan

Abstract

We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, encompassing those associated to classical orthogonal polynomial families, which is stable for polynomial degrees up to at least 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.

Full Text (PDF) [779 KB], BibTeX

Key words

orthogonal polynomials, induced distributions, sampling

AMS subject classifications

33C45, 65D15

Links to the cited ETNA articles

[5]	Vol. 13 (2002), pp. 119-147 Walter Gautschi: The interplay between classical analysis and (numerical) linear algebra - a tribute to Gene H. Golub

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