Modified Stieltjes polynomials and Gauss-Kronrod quadrature rules
type: Item
instance of: scholarly article
title: Modified Stieltjes polynomials and Gauss-Kronrod quadrature rules
zbMATH Open document ID: 1384.41022
published in: Numerische Mathematik
publication date: 2018-01-26T00:00:00Z
review text: Whereas \(n\)-point Gaussian quadrature formulas exist for all \(n \in \mathbb N\) and all nonnegative and integrable weight functions, it is well known that the corresponding Kronrod extensions exist only for a subset of this class of weights. In order to find a remedy for this deficiency, the authors develop a class of modified Kronrod quadrature formulas where, in addition to the usual \(n\) Gaussian nodes, \(r\) further nodes are prescribed, with \(r \leq n\). One then seeks the remaining \(n+1-r\) nodes and suitable weights such that the associated \((2n+1)\)-point quadrature formula has degree of exactness \(3n+1-r\). Clearly, the case \(r=0\) amounts to the standard Kronrod construction. The main goal of the paper is to exploit the additional freedom obtained when choosing \(r > 0\), with a particular emphasis on the cases \(r=1\) with the additional prescribed node being at one end point of the interval of integration, and \(r=2\) with both end points being prescribed. The main result for these cases is that, for a very large class of weight functions and sufficiently large \(n\), the associated modified Kronrod quadrature formulas exist and have the desirable properties that (a) all nodes are simple, (b) all nodes are inside the interval of integration, (c) the free nodes interlace with the prescribed nodes, and (d) all weights are positive. A detailed investigation of the associated nodal polynomials (i.e.\ the generalized Stieltjes polynomials) shows that their asymptotic behaviour is very similar to the behaviour of classical orthogonal polynomials. The work seems to have some relations to the results of \textit{S. Ehrich} [ISNM, Int. Ser. Numer. Math. 112, 67--76 (1993; Zbl 0799.65024)].
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 6830163
zbMATH Keywords: numerical integration
zbMATH Keywords: Kronrod method
zbMATH Keywords: Stieltjes polynomial
author: B. de la Calle Ysern
author: Miodrag M. Spalević
describes a project that uses: EXTEND
describes a project that uses: QUADPACK
describes a project that uses: OPQ
MaRDI profile type: Publication
full work available at URL: s00211-017-0901-y
OpenAlex ID: W2679179110
cites work: Q4199433
cites work: Q5338374
cites work: Suboptimal Kronrod extension formulae for numerical quadrature
cites work: Q5733898
cites work: Quadrature formulas for Fourier coefficients
cites work: A connection between quadrature formulas on the unit circle and the interval \([-1,1]\)
cites work: Computation of Gauss-Kronrod quadrature rules
cites work: Optimal extension of the Szeg quadrature
cites work: Stieltjes-type polynomials on the unit circle
cites work: Ultraspherical Stieltjes Polynomials and Gauss–Kronrod Quadrature Behave Nicely for $\lambda< 0$
cites work: Truncated generalized averaged Gauss quadrature rules
cites work: Internality of generalized averaged Gauss rules and their truncations for Bernstein-Szegő weights
cites work: Q3032744
cites work: Q4820343
cites work: A historical note on Gauss-Kronrod quadrature
cites work: Q3752135
cites work: Generalized averaged Szegő quadrature rules
cites work: Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle
cites work: Addition of Points to Gauss–Laguerre Quadrature Formulas
cites work: Q5339721
cites work: Q5533728
cites work: Anti-Gaussian quadrature formulas
cites work: Calculation of Gauss-Kronrod quadrature rules
cites work: Calculation of Radau-Kronrod and Lobatto-Kronrod quadrature formulas
cites work: Stieltjes Polynomials and Related Quadrature Rules
cites work: An overview of the computational aspects of Kronrod quadrature rules
cites work: Gauss-Kronrod quadrature formulae. A survey of fifty years of research
cites work: The Optimum Addition of Points to Quadrature Formulae
cites work: An algorithm for generating interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions
cites work: Algorithm 672: generation of interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions
cites work: Modified optimal quadrature extensions
cites work: On the asymptotic behaviour of functions of the second kind and Stieltjes polynomials and on the Gauss-Kronrod quadrature formulas
cites work: Stieltjes polynomials and functions of the second kind
cites work: Ultraspherical Gauss--Kronrod Quadrature Is Not Possible for $\lambda > 3$
cites work: Stieltjes polynomials and Gauss-Kronrod quadrature for Jacobi weight functions
cites work: Q4746719
cites work: New block quadrature rules for the approximation of matrix functions
cites work: Generalized averaged Gauss quadrature rules for the approximation of matrix functionals
cites work: On generalized averaged Gaussian formulas
cites work: A note on generalized averaged Gaussian formulas
cites work: On generalized averaged Gaussian formulas. II
DOI: 10.1007/S00211-017-0901-Y
description: scientific article; zbMATH DE number 6830163