On sensitivity of Gauss–Christoffel quadrature

Abstract

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss-Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. 123 148 D. P. O'Leary et al.several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss-Christoffel quadrature estimates. We then discuss conditions under which the Gauss-Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems.

Gene H. Golub and Gérard Meurant: Matrices, Moments and Quadrature with Applications

“…345-358]. As a suggestion for a further possible extension, the material could be complemented by demonstrating the sensitivity of the Gauss quadrature to small changes of the distribution function; see [67]. When the support of the distribution function (i.e., the set of points of its increase) is enlarged, the results of the Gauss quadrature for a fixed number of nodes may dramatically change.…”

“…There is a vast amount of literature on the problem, with the book by Gautschi [25] giving a thorough overview written from the orthogonal polynomial perspective; see also the literature in [67,Section 2]. The exposition in the book includes the generalization of the modified Chebyshev algorithm for indefinite weight functions (Section 5.4).…”

“…Here it is perhaps worth mentioning, as a little addition, that there is a direct derivation of the Lanczos and CG algorithms from the matrix (operator) moment problem, given by Vorobyev in [94,Chapter 3; in particular Section 4, pp. [65][66][67][68][69][70][71][72].…”

The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data

“…. , 1; see [12,31,37,52]. Depending on the noise level, for the smaller nodes of the distribution function ω the weights are completely dominated by noise, i.e., there exists an index J noise such that for j ≥ J noise…”

Model reduction using the Vorobyev moment problem

“…The basic result about the Gauss-Christoffel quadrature states, see also [35,Section 4] and the references given there, that the nodes of the n-point quadrature are equal to the roots of p n+1 (λ), i.e. the eigenvalues of T n .…”