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DOI: 10.46698/v590959661536u On Representation of Certain Integrals Using the Values of a Function and its Derivatives
Shustov, V. V.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 2.
Abstract:
The problem of integrating a function on the basis of its approximation by twopoint Hermite interpolation polynomials is considered. Quadrature formulas are obtained for the general case, when the orders of the derivatives given at the endpoints of the segment can be not equal to each other. The formula for the remainder term is presented and the error of numerical integration is estimated. Examples of integrating functions with data on error and its estimation are given. A twopoint approximation of the integrals is compared with a method based on the EulerMaclaurin formula. Comparison of the twopoint integration method with the approach based on the use of the EulerMaclaurin formula showed that for sufficiently smooth functions the accuracy of twopoint integration is significantly higher than by the EulerMaclaurin formula. An example of an integral is given for which its approximations obtained using the EulerMaclaurin formula diverge, and those obtained by the formula twopoint integration converge quickly enough. We also note that, in contrast to the EulerMaclaurin formula, the twopoint integration formula is also applicable in the case when the maximum orders of the derivatives at the ends of the integration interval may not be equal to each other, which is important in practical applications.
Keywords: quadrature of functions, twopoint Hermite interpolation polynomial, quadrature formulas using derivatives, estimation of the integration error, EulerMaclaurin formula, convergence of approximations.
Language: Russian
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For citation:
Shustov, V. V. On Representation of Certain Integrals Using the Values of a Function and its Derivatives, Vladikavkaz Math. J., 2020, vol. 22, no. 2, pp.8397 (in Russian).
DOI 10.46698/v590959661536u← Contents of issue 
 

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