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

Hascelik, A. Ihsan

doi:10.1016/j.apnum.2006.11.006

Cited by 8 publications

(4 citation statements)

References 18 publications

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“…This section aims to provide an error estimate for the anti-Gauss rule in weighted Sobolev spaces. Such an estimate, which will be useful for our aims, is similar to inequality (15); see (28) in Proposition 1. To prove it, we need two additional properties of the nodes and weights appearing in (16), which are stated in the following lemma.…”

Section: Convergence Results For the Anti-gauss Rule In Weighted Spacesmentioning

confidence: 91%

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Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

2020

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This paper is concerned with the numerical approximation of Fredholm integral equations of the second kind. A Nyström method based on the anti-Gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces. The Nyström interpolants corresponding to the Gauss and the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation, under suitable assumptions which are easily verified for a particular weight function. Hence, an error estimate is available, and the accuracy of the solution can be improved by approximating it by an averaged Nyström interpolant. The effectiveness of the proposed approach is illustrated through different numerical tests.

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Section: Convergence Results For the Anti-gauss Rule In Weighted Spacesmentioning

confidence: 91%

“…In particular, it has been proved that for some weight functions the averaged formula has a higher degree of exactness [21,23,34]. Several researchers investigated and generalized the anti-Gauss formula in relation to the approximation of integrals; see [1,3,15,19,22,28,33].…”

Section: Introductionmentioning

confidence: 99%

Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

2020

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“…A more reliable stopping criterion may be using the difference of Gauss and anti-Gauss rules, since the exact value for the integral is bracketed by the N -point Gauss rule and the (N + 1)-point anti-Gauss rule for many integrands. For anti-Gauss and related average quadrature rules, see [19,5,4,16] and the references therein. The correct digits are underlined.…”

Section: A Mathematicamentioning

confidence: 99%

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

Hascelik

2009

Journal of Computational and Applied Mathematics

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With existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the formwhere f is a sufficiently smooth function on [0, 1]. Gautschi has developed software (as scripts in Matlab) for computing these integrals for the special case r = ω = 1. In this paper, an algorithm (as a Mathematica program) is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range. Numerical examples are given of testing the performance of the algorithm/program.

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“…Ehrich [6] showed that Q GF 2 +1 is the optimal stratified extension for Gauss-Laguerre and Gauss-Hermite q.f., in the corresponding cases. See also [10] for the Gauss-Gegenbauer q.f..…”

Section: Introductionmentioning

confidence: 99%

A note on generalized averaged Gaussian formulas

Spalević

2007

Numer Algor

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We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss-Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss-Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.

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

Cited by 8 publications

References 18 publications

Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

A note on generalized averaged Gaussian formulas

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