The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points

Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315{329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss- Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors.

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The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points

Journal of Computational and Applied Mathematics ◽

10.1016/s0377-0427(05)80007-x ◽

1990 ◽

Vol 33 (3) ◽

pp. 315-329 ◽

Cited By ~ 10

Author(s):

Walter Gautschi ◽

Shikang Li

Keyword(s):

Analytic Functions ◽

Remainder Term ◽

Quadrature Rules ◽

End Points

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Error bounds for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind

Filomat ◽

10.2298/fil1601231m ◽

2016 ◽

Vol 30 (1) ◽

pp. 231-239 ◽

Cited By ~ 1

Author(s):

Ljubica Mihic ◽

Aleksandar Pejcev ◽

Miodrag Spalevic

Keyword(s):

Weight Function ◽

Quadrature Formula ◽

Error Bounds ◽

Analytic Functions ◽

Remainder Term ◽

Applied Mathematics ◽

Maximum Modulus ◽

End Points ◽

The Third ◽

Fourth Kind

For analytic functions the remainder terms of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points -+1, for Gauss-Lobatto quadrature formula with multiple end points with Chebyshev weight function of the third and the fourth kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi and Li in paper [The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, Journal of Computational and Applied Mathematics 33 (1990) 315-329.]

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On Appel-Type Quadrature Rules

Georgian Mathematical Journal ◽

10.1515/gmj.2002.405 ◽

2002 ◽

Vol 9 (3) ◽

pp. 405-412

Author(s):

C. Belingeri ◽

B. Germano

Keyword(s):

Remainder Term ◽

Quadrature Rule ◽

Bernoulli Numbers ◽

Quadrature Rules ◽

Rule Based ◽

Numerical Example ◽

The Given ◽

Derivatives Of

Abstract The Radon technique is applied in order to recover a quadrature rule based on Appel polynomials and the so called Appel numbers. The relevant formula generalizes both the Euler-MacLaurin quadrature rule and a similar rule using Euler (instead of Bernoulli) numbers and even (instead of odd) derivatives of the given function at the endpoints of the considered interval. In the general case, the remainder term is expressed in terms of Appel numbers, and all derivatives appear. A numerical example is also included.

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