scholarly journals Error bounds for interpolatory quadrature rules on the unit circle

2000 ◽  
Vol 70 (233) ◽  
pp. 281-297 ◽  
Author(s):  
J. C. Santos-León
Keyword(s):  
Unit Circle ◽  
Error Bounds ◽  
Quadrature Rules ◽  
Interpolatory Quadrature

scholarly journals New error bounds for Gauss-Legendre quadrature rules

Filomat ◽  
2014 ◽  
Vol 28 (6) ◽  
pp. 1281-1293 ◽  
Author(s):  
Mohammad Masjed-Jamei
Keyword(s):  
Quadrature Formula ◽  
Error Bounds ◽  
Quadrature Rule ◽  
Order Derivative ◽  
Quadrature Rules ◽  
Linear Kernel ◽  
Large N ◽  
Interpolatory Quadrature ◽  
Integrand Function ◽  
Legendre Quadrature

It is well-known that the remaining term of any n-point interpolatory quadrature rule such as Gauss-Legendre quadrature formula depends on at least an n-order derivative of the integrand function, which is of no use if the integrand is not smooth enough and requires a lot of differentiation for large n. In this paper, by defining a specific linear kernel, we resolve this problemand obtain new bounds for the error of Gauss-Legendre quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function. Some illustrative examples are given in this direction.

scholarly journals Revisited Optimal Error Bounds for Interpolatory Integration Rules

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
François Dubeau
Keyword(s):  
Error Bounds ◽  
Error Term ◽  
Quadrature Rules ◽  
Optimal Error ◽  
Integration By Parts ◽  
Integration Rules

We present a unified way to obtain optimal error bounds for general interpolatory integration rules. The method is based on the Peano form of the error term when we use Taylor’s expansion. These bounds depend on the regularity of the integrand. The method of integration by parts “backwards” to obtain bounds is also discussed. The analysis includes quadrature rules with nodes outside the interval of integration. Best error bounds for composite integration rules are also obtained. Some consequences of symmetry are discussed.

scholarly journals Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circle

2019 ◽  
Vol 83 (3) ◽  
pp. 1029-1061
Author(s):  
Cleonice F. Bracciali ◽  
Junior A. Pereira ◽  
A. Sri Ranga
Keyword(s):  
Unit Circle ◽  
Recurrence Relation ◽  
Quadrature Rules

scholarly journals Errors of Gauss-Radau and Gauss-Lobatto quadratures with double end point

2019 ◽  
Vol 13 (2) ◽  
pp. 463-477
Author(s):  
Aleksandar Pejcev ◽  
Ljubica Mihic
Keyword(s):  
Explicit Expression ◽  
Error Bounds ◽  
Analytic Functions ◽  
Remainder Term ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
End Points ◽  
End Point ◽  
Appl Math

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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