Error bounds for some quadrature rules with maximal trigonometric degree of exactness

2012 ◽  
Author(s):  
Marija P. Stanić ◽  
Aleksandar S. Cvetković ◽  
Tatjana V. Tomović
Keyword(s):  
Error Bounds ◽  
Quadrature Rules ◽  
Degree Of Exactness

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 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.

Unified error bounds for all Newton–Cotes quadrature rules

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Mohammad Masjed-Jamei
Keyword(s):  
Error Bound ◽  
Error Bounds ◽  
Order Derivative ◽  
Quadrature Rules ◽  
Stringent Condition ◽  
Large N ◽  
Almost All ◽  
Integrand Function

AbstractIt is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing an n-order derivative requires a lot of differentiation for large n, the main problem is that an error bound for an n-point Newton-Cotes quadrature is only relevant for a function that is n times differentiable, a rather stringent condition. In this paper, by defining two specific linear kernels, we resolve this problem and obtain new error bounds for all closed and open types of Newton-Cotes quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function and are very general such that they cover almost all existing results in the literature. Some illustrative examples are given in this direction.

scholarly journals A unified generalization of some quadrature rules and error bounds

2013 ◽  
Vol 219 (9) ◽  
pp. 4765-4774 ◽  
Author(s):  
Wenjun Liu ◽  
Yong Jiang ◽  
Adnan Tuna
Keyword(s):  
Error Bounds ◽  
Quadrature Rules
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