Software Repository for Gauss quadrature and Christoffel function

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By Walter Gautschi

Purdue University

This series contains sets of MATLAB scripts related to Gauss quadrature and Christoffel function.

Version 1.0 - published on 29 May 2020 doi:10.4231/78GE-EA31 - cite this Archived on 29 Jun 2020

Licensed under CC0 1.0 Universal

Description

This series is a software repository for Gaussian quadrature and the related Christoffel function. Two major objectives are

  1. to make Gauss quadrature rules, for a large variety of weight functions, easily accessible;
  2. to document and illustrate approximations available for Gauss quadrature weights and Christoffel functions.

With regard to 1., the number n of quadrature points and the desirable precision (number of digits) can be arbitrary, and so are the values of parameters possibly present in the weight function. With regard to 2., there are known approximations for quadrature weights when the support interval of the weight function is finite. We conjecture them, and provide evidence for them, to be valid also for the corresponding Christoffel functions. The case of half-infinite and infinite support intervals is more intriguing: there are known approximations for quadrature weights in the case of Laguerre and Hermite weight functions. They are conjectured here to apply also to Christoffel functions. Indeed, if multiplied by appropriate constants (depending on n), they all seem to be valid for essentially arbitrary weight functions.

The repository contains over fifty datasets, each dealing with a particular weight function. We distinguish between classical, quasi-classical, and non-classical weight functions and associated orthogonal polynomials. The classical ones include the well-established polynomials associated with such names as Jacobi, Laguerre, Hermite, Meixner, and Pollaczek. Quasi-classical orthogonal polynomials share with the classical ones the fact that their three-term recurrence relation is known in closed form, whereas for non-classical ones, the recurrence relation is not known explicitly and must be generated by one of several known methods.

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