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.
Gauss-type Quadrature Rules for Rational Functions