The existence and construction of rational Gauss-type quadrature rules

Abstract

Abstract. Consider a hermitian positive-definite linear functional F, and assume we have m distinct nodes fixed in advance anywhere on the real line. In this paper we then study the existence and construction of nth rational Gauss-Radau (m = 1) and Gauss-Lobatto (m = 2) quadrature formulas that approximate F{f }. These are quadrature formulas with n positive weights and with the n−m remaining nodes real and distinct, so that the quadrature is exact in a (2n−m)-dimensional space of rational functions.

Rational Gauss Quadrature

“…Remark 3.15. Deckers and Bultheel [3,4] describe an approach for determining orthonormal rational functions and associated rational Gauss quadrature rules different from that of the present paper. They show that the orthonormal rational functions satisfy a three-term recursion relation in which the "coefficients" are linear fractional transformations.…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php generalized eigenvalue problem (6) in [4] can be replaced by the solution of a (standard) real symmetric eigenvalue problem when α n = ∞ and the poles in C\R among the set {α j } n−1 j=1 appear in complex conjugate pairs. We note that when the inner product is defined in terms of a discrete measure determined by a large matrix A and a vector v (cf.…”

“…(3.1)), the computation of an additional rational function with a new real pole α requires the solution of a linear system of equations with the matrix A − αI when the approach of section 3 is used. Application of the recursion formulas in [3,4] requires the solution of two linear system of equations with the matrix A − αI. …”

“…The nodes and weights of the associated rational Gauss quadrature rules are determined by solving a generalized eigenvalue problem with tridiagonal matrices. Deckers and Bultheel [4] do not require the poles to appear in complex conjugate pairs or the integrand to be real-valued. The present paper focuses on the integration of real-valued functions on a real interval.…”

Rational Gauss-Chebyshev Quadratures for Wireless Performance Analysis

“…For such integrands, it is more appropriate to consider the GCQ rule based on orthogonal rational functions, rather than orthogonal polynomials, with preassigned poles to simulate the singularities of the integrand [11], [12] (and references therein). Some important results on the rational GCQ rule for the real poles outside [−1, 1] are presented in the following section.…”