Applications

The connection between orthogonal polynomials and quadrature rules has already been elucidated in §1.4. The centerpieces were the Gaussian quadrature rule and its close relatives—the Gauss–Radau and Gauss–Lobatto rules (§1.4.2). There are, however, a number of further extensions of Gauss’s approach to numerical quadrature. Principal among them are Kronrod’s idea of extending an n-point Gauss rule to a (2n + 1)-point rule by inserting n + 1 additional nodes and choosing all weights in such a way as to maximize the degree of exactness (cf. Definition 1.44), and Turán’s extension of the Gauss quadrature rule allowing not only function values, but also derivative values, to appear in the quadrature sum. More recent extensions relate to the concept of accuracy, requiring exactness not only for polynomials of a certain degree, but also for rational functions with prescribed poles. Gauss quadrature can also be adapted to deal with Cauchy principal value integrals, and there are other applications of Gauss’s ideas, for example, in combination with Stieltjes’s procedure or the modified Chebyshev algorithm, to generate polynomials orthogonal on several intervals, or, in comnbination with Lanczos’s algorithm, to estimate matrix functionals. The present section is to discuss these questions in turn, with computational aspects foremost in our mind. We have previously seen in Chapter 2 how Gauss quadrature rules can be effectively employed in the context of computational methods; for example, in computing the absolute and relative condition numbers of moment maps (§2.1.5), or as a means of discretizing measures in the multiple-component discretization method for orthogonal polynomials (§2.2.5) and in Stieltjes-type methods for Sobolev orthogonal polynomials (§2.5.2). It is time now to discuss the actual computation of these, and related, quadrature rules.

Computational Methods

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.

Basic Theory

This introductory chapter is to present a quick review of material on orthogonal polynomials that is particularly relevant to computation. Proofs of most results are included; for those requiring more extensive analytic treatments, references are made to the literature.