The evaluation of the error term in some numerical quadrature formulae: An addendum

Some selected Ostrowski type inequalities and a connection with numerical integration are studied in this survey paper, which is dedicated to the memory of Professor D. S. Mitrinovic, who left us 25 years ago. His significant inuence to the development of the theory of inequalities is briefly given in the first section of this paper. Beside some basic facts on quadrature formulas and an approach for estimating the error term using Ostrowski type inequalities and Peano kernel techniques, we give several examples of selected quadrature formulas and the corresponding inequalities, including the basic Ostrowski's inequality (1938), inequality of Milovanovic and Pecaric (1976) and its modifications, inequality of Dragomir, Cerone and Roumeliotis (2000), symmetric inequality of Guessab and Schmeisser (2002) and asymmetric in-equality of Franjic (2009), as well as four point symmetric inequalites by Alomari (2012) and a variant with double internal nodes given by Liu and Park (2017).

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On a Modification of Romberg Quadrature

Canadian Mathematical Bulletin ◽

10.4153/cmb-1968-025-x ◽

1968 ◽

Vol 11 (2) ◽

pp. 217-224 ◽

Cited By ~ 1

Author(s):

R. Manohar ◽

C. Turnbull

Keyword(s):

Numerical Computation ◽

Error Term ◽

Numerical Results ◽

Numerical Quadrature ◽

Alternative Formulation

Meir and Sharma [1] have suggested a modification of Romberg quadrature using Newton-Cotes and, in particular, Simpson sums in place of trapezoidal sums. By comparing the error term with that obtained by Bulirsch [3] for trapezoidal sums, they concluded that the use of Simpson sums would lead to an improvement of the results. The procedure adopted by Meir and Sharma [1] permits them to obtain an expression for the error in the numerical quadrature. However, for the purpose of numerical computation, this procedure appears to be less suitable. In section 3, we give an alternative formulation which would enable us to carry out the computation, using Simpson sums, in the same wasy as is done in the case of Romberg quadrature with trapezoidal sums. Some numerical results are discussed in section 4.

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Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals

International Journal of Computer Mathematics ◽

10.1080/00207167708803139 ◽

1977 ◽

Vol 6 (3) ◽

pp. 219-227 ◽

Cited By ~ 32

Author(s):

C. G. Harris ◽

W. A. B. Evans

Keyword(s):

Numerical Quadrature ◽

Quadrature Formulae ◽

End Point

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Monotonicity of the error term in Gauss-Turán quadratures for analytic function

The ANZIAM Journal ◽

10.1017/s1446181100003229 ◽

2007 ◽

Vol 48 (4) ◽

pp. 567-581 ◽

Cited By ~ 3

Author(s):

Gradimir V. Milovanović ◽

Miodrag M. Spalević

Keyword(s):

Analytic Function ◽

Weight Function ◽

Error Estimates ◽

Complex Plane ◽

Error Term ◽

Quadrature Formulae

AbstractFor Gauss–Turán quadrature formulae with an even weight function on the interval [−1, 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than I, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some ℓ2-error estimates are considered.

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