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

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 231-239 ◽  
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.]

scholarly journals A generalized Birkhoff–Young quadrature formula

2016 ◽  
Vol 32 (2) ◽  
pp. 203-213
Author(s):  
GRADIMIR V. MILOVANOVIC ◽  
◽  
MILOLJUB ALBIJANIC ◽  
Keyword(s):  
Numerical Integration ◽  
Explicit Form ◽  
Quadrature Formula ◽  
Analytic Functions ◽  
Maximal Degree ◽  
Special Cases

A generalized (4n + 1)-point Birkhoff–Young quadrature of interpolatory type with the maximal degree of precision for numerical integration of analytic functions is derived. An explicit form of the node polynomial of such kind of quadratures is obtained. Special cases and an example are presented.

scholarly journals Error bounds of Micchelli–Rivlin quadrature formula for analytic functions

2013 ◽  
Vol 169 ◽  
pp. 23-34 ◽  
Author(s):  
Aleksandar V. Pejčev ◽  
Miodrag M. Spalević
Keyword(s):  
Quadrature Formula ◽  
Error Bounds ◽  
Analytic Functions

scholarly journals A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130571 ◽  
Author(s):  
Mohsin Javed ◽  
Lloyd N. Trefethen
Keyword(s):  
Error Bound ◽  
Quadrature Formula ◽  
Analytic Functions ◽  
Trapezoidal Rule ◽  
Contour Integral ◽  
Periodic Functions ◽  
Midpoint Rule ◽  
Geometric Convergence ◽  
Maclaurin Formula

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic functions in another. Similar results are also given for the midpoint rule.

A generalized Birkhoff–Young–Chebyshev quadrature formula for analytic functions

2011 ◽  
Vol 218 (3) ◽  
pp. 944-948 ◽  
Author(s):  
Gradimir V. Milovanović ◽  
Aleksandar S. Cvetković ◽  
Marija P. Stanić
Keyword(s):  
Quadrature Formula ◽  
Analytic Functions ◽  
Chebyshev Quadrature
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