Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2752910.22363/2658-4670-2021-29-3-251-259Research ArticleAsymptotically accurate error estimates of exponential convergence for the trapezoidal ruleBelovAleksandr A.<p>Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Researcher of Faculty of Physics, M.V. Lomonosov Moscow State University</p>aa.belov@physics.msu.ruhttps://orcid.org/0000-0002-0918-9263KhokhlachevValentin S.<p>Master’s degree student of Faculty of Physics</p>valentin.mycroft@yandex.ruhttps://orcid.org/0000-0002-6590-5914M.V. Lomonosov Moscow State UniversityPeoples’ Friendship University of Russia (RUDN University)3009202129325125930092021Copyright © 2021, Belov A.A., Khokhlachev V.S.2021<p style="text-align: justify;">In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only 10 steps.</p>trapezoidal ruleexponential convergenceerror estimateasymptotically sharp estimatesформула трапецийэкспоненциальная сходимостьоценки точностиасимптотически точные оценки[L. N. Trefethen and J. A. C. Weideman, “The exponentially convergent trapezoidal rule,” SIAM Review, vol. 56, no. 3, pp. 385-458, 2014. DOI: 10.1137/130932132.][J. Mohsin and L. N. Trefethen, “A trapezoidal rule error bound unifying the Euler-Maclaurin formula and geometric convergence for periodic functions,” in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 470, 2014, p. 20130571. DOI: 10.1098/rspa.2013.0571.][J. A. C. 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