Stopping functionals for Gaussian quadrature formulas

Ehrich, Sven

doi:10.1016/s0377-0427(00)00496-9

Cited by 12 publications

(6 citation statements)

References 53 publications

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“…Ehrich [5] makes a few remarks on the existence of Radau-Kronrod and Lobatto-Kronrod formulas, citing [1], but does not comment on their construction.…”

Section: Gander and Gautschi [9] Calculate The 7-point Extension Of Tmentioning

confidence: 99%

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Calculation of Radau–Kronrod and Lobatto–Kronrod quadrature formulas

Laurie

2007

Numer Algor

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We show how to apply routines from the software package OPQ by Walter Gautschi in order to compute the optimal extension of an n-point generalized Radau or Lobatto formula. The method is applicable to any weight function for which enough three-term recursion coefficients are known. The idea on which the method is based was first shown by Paola Baratella in 1979. Program code in the format of M-files conforming to the conventions of OPQ is given.Keywords Quadrature formula · Optimal extension · Kronrod · ORTHPOL Computations involving orthogonal polynomialsIn a series of papers starting in 1967, gaining momentum with the survey [11] and the landmark paper [12], and culminating in the monograph [17], Walter Gautschi deeply explored the computational aspects of polynomials p k generated by three-term recursion relations of the formwith p −1 (x) = 0, p 0 (x) = 1.Dedicated to Walter Gautschi, at 79 still a pioneer.

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“…Ehrich [5] makes a few remarks on the existence of Radau-Kronrod and Lobatto-Kronrod formulas, citing [1], but does not comment on their construction.…”

Section: Gander and Gautschi [9] Calculate The 7-point Extension Of Tmentioning

confidence: 99%

“…While preparing the first draft of this paper, I discovered the reference to Baratella's paper [1] in Sven Ehrich's survey [5]. Giovanni Monegato sent me a reprint of [1], after perusing which I decided that the key idea had been anticipated to the extent that my paper was not worth submitting.…”

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confidence: 99%

Calculation of Radau–Kronrod and Lobatto–Kronrod quadrature formulas

Laurie

2007

Numer Algor

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“…We now wish to compare the estimate obtained with a classical one. For that purpose, we follow the presentation given in the survey article [5], using the quadrature formula given in Theorem 1. (See also the relevant references in [5].…”

Section: Ostrowski-grüss Type Inequalitiesmentioning

confidence: 99%

Inequalities of Ostrowski–Grüss type and applications

Ujević

2002

Appl. Math. (Warsaw)

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Some new inequalities of Ostrowski-Grüss type are derived. They are applied to the error analysis for some Gaussian and Gaussian-like quadrature formulas. 1. Introduction. In this paper we derive some new inequalities of Ostrowski-Grüss type. They are applied to the error analysis for some Gaussian and Gaussian-like quadrature formulas. We consider Gauss-Legendre, Chebyshev, Radau and Lobatto quadratures. A similar error analysis for rules of Newton-Cotes type can be found in the literature. In particular, the mid-point, trapezoid and Simpson rules have been investigated recently ([1], [2], [4], [8], [11]) with a view to obtaining bounds on the quadrature rule in terms of a variety of norms involving, at most, the first derivative. The present work brings results for the above mentioned Gaussian and Gaussian-like quadrature rules giving explicit error bounds. We use Peano type kernels and an approach via inequalities. The error bounds are expressed in terms of first derivatives. The general approach used in the past required the assumption of bounded derivatives of degree higher than one. Our error bounds are in general (but not always) better than the Peano bounds. We also consider composite quadrature rules. Our analysis allows the determination of a partition that would assure a prescribed error tolerance. In Section 2 we establish some inequalities of Ostrowski-Grüss type. Such inequalities are considered in [2], [4], [8] and [11]. In Sections 3-6 we apply these inequalities to obtain explicit error bounds for the above mentioned quadrature rules. In Section 7 we give a general procedure for forming

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“…Functionals based on function evaluations that provide estimates for the quadrature error are called stopping functionals. For detailed information on the stopping functionals for Gaussian type quadrature rules, see the survey papers of Ehrich [5] and Monegato [15] and the references therein. Differences in quadrature formulas are important examples of stopping functionals.…”

Section: Introductionmentioning

confidence: 99%

Modified anti-Gauss and degree optimal average formulas for Gegenbauer measure

Hascelik

2008

Applied Numerical Mathematics

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Stopping functionals for Gaussian quadrature formulas

Cited by 12 publications

References 53 publications

Calculation of Radau–Kronrod and Lobatto–Kronrod quadrature formulas

Calculation of Radau–Kronrod and Lobatto–Kronrod quadrature formulas

Inequalities of Ostrowski–Grüss type and applications

Modified anti-Gauss and degree optimal average formulas for Gegenbauer measure

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