Computational Methods

Abstract

The fundamental problem is to compute the first n recursion coefficients αk (dλ), Βk(dλ), k = 0, 1, . . . , n − 1 (cf. §1.3.1), where n ≥ 1 is a (typically large) integer and dλ a positive measure given either implicitly via moment information or explicitly. In the former case, an important aspect is the sensitivity of the problem with respect to small perturbations in the data (the first 2n moments or modified moments); this is the question of conditioning. In principle, there is a simple algorithm, essentially due to Chebyshev, that produces the desired recursion coefficients from given moment information. The effectiveness of this algorithm, however, depends critically on the conditioning of the underlying problem. If the problem is ill-conditioned, as it often is, recourse has to be made either to symbolic computation or to the explicit form of the measure. A procedure applicable in the latter case is discretization of the measure and subsequent approximation of the desired recursion coefficients by those relative to a discrete measure. Other problems calling for numerical methods are the evaluation of Cauchy integrals of orthogonal polynomials and the problem of passing from the recursion coefficients of a measure to those of a modified measure—the original measure multiplied by a rational function. Finally, Sobolev orthogonal polynomials present their own problems of calculating recursion coefficients and zeros. Orthogonal polynomials as well as their recursion coefficients are expressible in determinantal form in terms of the moments of the underlying measure. Indeed, much of the classical theory of orthogonal polynomials is moment-oriented. This is true, in particular, of a classical algorithm due to Chebyshev, which generates the recursion coefficients directly from the moments, bypassing determinants. The use of moments, unfortunately, is numerically problematic inasmuch as they give rise to severe ill-conditioning. In many cases, particularly for measures with bounded support, it is possible, however, to work with the so-called “modified moments,” which lead to better conditioned problems and a more stable analog of the Chebyshev algorithm.