Discrete and Continuous Models and Applied Computational ScienceDiscrete 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)3510910.22363/2658-4670-2023-31-2-128-138XAUSJAArticlesСтатьиResearch ArticleQuadratures with super power convergenceКвадратуры со сверхстепенной сходимостьюhttps://orcid.org/0000-0002-0918-926357191950560Q-5064-2016BelovAleksandr A.БеловА. А.

Candidate of Physical and Mathematical Sciences, Researcher of Faculty of Physics, M. V. Lomonosov Moscow State University; Assistant Professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia

aa.belov@physics.msu.ruhttps://orcid.org/0000-0002-5466-1221TintulMaxim A.ТинтулМ. А.

Master’s Degree Student of Faculty of Physics

maksim.tintul@mail.ruhttps://orcid.org/0000-0002-6590-5914KhokhlachevValentin S.ХохлачевВ. С.

Master’s Degree Student of Faculty of Physics

valentin.mycroft@yandex.ruM. V. Lomonosov Moscow State UniversityМосковский государственный университет им. М. В. ЛомоносоваRUDN UniversityРоссийский университет дружбы народов30062023312VOL 31, NO2 (2023)ТОМ 31, №2 (2023)12813829062023Copyright ©; 2023, Belov A.A., Tintul M.A., Khokhlachev V.S.Copyright ©; 2023, Белов А.А., Тинтул М.А., Хохлачев В.С.2023Belov A.A., Tintul M.A., Khokhlachev V.S.Белов А.А., Тинтул М.А., Хохлачев В.С.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/35109

The calculation of quadratures arises in many physical and technical applications. The replacement of integration variables is proposed, which dramatically increases the accuracy of the formula of averages. For infinitely smooth integrand functions, the convergence law becomes super power. It is significantly faster than the power law and is close to exponential one. For integrals with bounded smoothness, power convergence is realized with the maximum achievable order of accuracy.

Вычисление квадратур возникает во многих физических и технических приложениях. В статье предложена замена переменных интегрирования, кардинально повышающая точность формулы средних. Для бесконечно гладких подынтегральных функций закон сходимости становится сверхстепенным. Он существенно быстрее степенного и близок к экспоненциальному. Для подынтегральных функций с ограниченной гладкостью реализуется степенная сходимость с максимально достижимым порядком точности.

trapezoid ruleexponential convergenceerror estimateasymptotically sharp estimatesформула трапецийэкспоненциальная сходимостьоценки точностиасимптотически точные оценкиN. N. Kalitkin and E. A. Alshina, Numerical Methods. Vol. 1: Numerical Analysis [Chislennye Metody. T. 1: Chislennyi analiz]. Moscow: Akademiya, 2013, in Russian.N. N. Kalitkin, A. B. Alshin, E. A. Alshina, and V. B. Rogov, Computations with Quasi-Uniform Grids [Vychisleniya na kvaziravnomernykh setkakh]. Moscow: Fizmatlit, 2005, in Russian.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.N. N. Kalitkin and S. A. Kolganov, “Quadrature formulas with exponential convergence and calculation of the Fermi-Dirac integrals,” Doklady Mathematics, vol. 95, no. 2, pp. 157-160, 2017. DOI: 10.1134/S1064562417020156.N. N. Kalitkin and S. A. Kolganov, “Computing the Fermi-Dirac functions by exponentially convergent quadratures,” Mathematical Models and Computer Simulations, vol. 10, no. 4, pp. 472-482, 2018. DOI: 10.1134/S2070048218040063.T. Sag and G. Szekeres, “Numerical evaluation of high-dimensional integrals,” Math. Comp., vol. 18, pp. 245-253, 1964. DOI: 10.1090/S0025-5718-1964-0165689-X.A. Sidi, “Numerical evaluation of high-dimensional integrals,” International Series Numer. Math., vol. 112, pp. 359-373, 1993. DOI: 10.1007/978-3-0348-6338-4_27.M. Iri, S. Moriguti, and Y. Takasawa, “On a certain quadrature formula,” International Series Numer. Math., vol. 17, pp. 3-20, 1987. DOI: 10.1016/0377-0427(87)90034-3.M. Mori, “An IMT-Type Double Exponential Formula for Numerical Integration,” Publ. Res. Inst. Math. Sci. Kyoto Univ., vol. 14, no. 3, pp. 713-729, 1978. DOI: 10.2977/prims/1195188835.A. A. Belov, N. N. Kalitkin, and V. S. Khokhlachev, “Improved error estimates for an exponentially convergent quadratures [Uluchshennyye otsenki pogreshnosti dlya eksponentsial’no skhodyashchikhsya kvadratur],” Preprints of IPM im. M. V. Keldysh, no. 75, 2020, in Russian. DOI: 10.20948/prepr-2020-75.V. S. Khokhlachev, A. A. Belov, and N. N. Kalitkin, “Improvement of error estimates for exponentially convergent quadratures [Uluchsheniye otsenok pogreshnosti dlya eksponentsial’no skhodyashchikhsya kvadratur],” Izv. RAN. Ser. fiz., vol. 85, no. 2, pp. 282-288, 2021, in Russian. DOI: 10.31857/S0367676521010166.