# Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 75-101. ISSN 2079-6641

Research Article

MSC 65D32

Euler-Maclaurin type optimal formulas for numerical integration in Sobolev space

A. R. Hayotov¹, F. A. Nuraliev¹, R. I. Parovik², Kh. M. Shadimetov¹

¹V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, M. Ulugbek str. 81, 100125, Tashkent, Uzbekistan
²Vitus Bering Kamchatka State University, Pogranichnaya str. 4, Petropavlovsk-Kamchatsky, 683032, Russia

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space  L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number Nm-3 and for any m ≥ 4 using S. L. Sobolev method which is based on the discrete analogue of the differential operator d2m/dx2m. In particular, for m = 4 and m = 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=6 new optimal quadrature formulas are obtained. At the end of this work some numerical results are presented.

Keywords: optimal quadrature formulas, the error functional, the extremal function S.L. Sobolev space, optimal coefficients

DOI: 10.26117/2079-6641-2020-32-3-75-101

Original article submitted: 01.09.2020

Revision submitted: 10.10.2020

For citation. Hayotov A. R., Nuraliev F. A., Parovik R. I., Shadimetov Kh. M. Euler-Maclaurin type optimal formulas for numerical integration in Sobolev space. Vestnik KRAUNC. Fiz.-mat. nauki. 2020, 32: 3, 75-101. DOI: 10.26117/2079-6641-2020-32-3-75-101

© Hayotov A. R., et al., 2020

Научная статья

УДК 519.644

Оптимальные формулы типа Эйлера-Маклорена для численного интегрирования в пространстве Соболева

А. Р. Хаётов¹, Ф. А. Нуралиев¹, Р. И. Паровик², Х.М. Шадиметов¹

¹Институт Математики имени В. И. Романовского Академии наук Узбекистана, г. Ташкент, ул. Мирзо Улугбека 85, 100170, Республика Узбекистан
²Камчатский государственный университет имени Витуса Беринга, г. Петропавловск-Камчатский, ул. Пограничная 4, 683032, Россия

В настоящей статье рассматривается задача построения оптимальных квадратурных формул в смысле Сарда в пространстве L2(m)(0,1). Здесь квадратурная сумма состоить из значений подынтегральной функции в узлах и значений первой и третьей производных подынтегральной функции на концах интервала интегрирования. Найдены коэффициенты оптимальных квадратурных формул и вычислена норма оптимального функционала погрешности для любого натурального числа Nm-3 и для любого m ≥ 4, используя метод С. Л. Соболева который основывается на дискретный аналог дифференциального оператора d2m/dx2m. В частности, при m = 4 и m = 5 получен оптимальность классической формулы Эйлера-Маклорена. Начиная с m = 6 получены новые оптимальные квадратурные формулы. В конце работы приведаны некоторые численные результаты.

Ключевые слова: оптимальные квадратурные формулы, функционал погрешности, экстремальная функция, пространство Соболева, оптимальные коэффициенты.

DOI: 10.26117/2079-6641-2020-32-3-75-101

Поступила в редакцию: 01.09.2020

В окончательном варианте: 10.10.2020

Для цитирования. Hayotov A. R., Nuraliev F. A., Parovik R. I., Shadimetov Kh. M. Euler-Maclaurin type optimal formulas for numerical integration in Sobolev space // Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 75-101. DOI: 10.26117/2079-6641-2020-32-3-75-101

References

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6. Boltaev N. D., Hayotov A. R., Shadimetov Kh. M., “Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L2(m) ”, Numerical Algorithms, 74 (2017), 307-336.
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32. Shadimetov Kh. M., “Construction of weight optimal quadrature formulas in the space L2(m)(0,N)”, Siberian J. Comput. Math., 5:3 (2002), 275–293 (in Russian).
33. Shadimetov Kh. M., Hayotov A. R., “Optimal quadrature formulas with positive coefficients in L2(m)(0,1) space”, J. Comput. Appl. Math., 235 (2011), 1114–1128.
34. Shadimetov Kh. M., Hayotov A. R., “Optimal quadrature formulas in the sense of Sard in W2(m,m-1) space”, Calcolo, 51 (2014), 211-243.
35. Shadimetov Kh. M., Hayotov A. R., Azamov S. S., “Optimal quadrature formula in K2(P2) space”, Applied Numerical Mathematics, 62 (2012), 1893-1909.
36. Shadimetov Kh. M., Hayotov A. R., Nuraliev F. A., “On an optimal quadrature formula in Sobolev space L2(m)(0,1)”, Journal of Computational and Applied Mathematics, 243 (2013), 91-112.
37. Sobolev S. L., Introduction to the Theory of Cubature Formulas, Nauka, Moscow, 1974 (in Russian).
38. Sobolev S. L., “The coefficients of optimal quadrature formulas”, Selected Works of S.L. Sobolev, Springer, 2006, 561–566.
39. Sobolev S. L., Vaskevich V. L., The Theory of Cubature Formulas, Kluwer Academic Publishers Group, Dordrecht, 1997.
40. Zagirova F.Ya., On construction of optimal quadrature formulas with equal spaced nodes, Preprint No. 25, Institute of Mathematics SD of AS of USSR, Novosibirsk, 1982 (in Russian), 28 pp.
42. Zhensikbaev A. A., “Monosplines of minimal norm and the best quadrature formulas”, Uspekhi Matem. Nauk, 36 (1981), 107–159 (in Russian).

References (GOST)

1. Akhmedov D. M., Hayotov A. R., Shadimetov Kh. M. Optimal quadrature formulas with derivatives for Cauchy type singular integrals // Applied Mathematics and Computation. vol. 317. 2018. pp. 150-159.
2. Babuˇska I. Optimal quadrature formulas // Dokladi Akad. Nauk SSSR. vol. 149. 1963. pp. 227–229 (in Russian).
3. Blaga P., Coman Gh. Some problems on optimal quadrature // Stud. Univ. Babe¸s-Bolyai Math. 2007. vol. 52. no. 4. pp. 21–44.
4. Bojanov B. Optimal quadrature formulas // Uspekhi Mat. Nauk. 2005. vol. 60. no. 6(366). pp. 33–52 (in Russian).
5. Boltaev N. D., Hayotov A. R., Milovanovi´c G.V., Shadimetov Kh. M. Optimal quadrature formulas for numerical evaluation of Fourier coefficients in W2(m,m-1) // Journal of Applied Analysis and Computation. 2017. vol. 7. no. 4. pp. 1233-1266.
6. Boltaev N. D., Hayotov A. R., Shadimetov Kh. M. Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space L2(m) // Numerical Algorithms. 2017. vol. 74. pp. 307-336.
7. Catina¸s T., Coman Gh. Optimal quadrature formulas based on the f-function method // Stud. Univ. Babe¸s-Bolyai Math. 2006. vol. 51. no. 1. pp. 49–64.
8. Chakhkiev M. A. Linear differential operators with real spectrum, and optimal quadrature formulas // Izv. Akad. Nauk SSSR Ser. Mat. 1984. vol. 48. no. 5. pp. 1078–1108 (in Russian).
9. Coman Gh. Quadrature formulas of Sard type // Studia Univ. Babe¸s-Bolyai Ser. Math.-Mech. 1972. vol. 17. no. 2. pp. 73–77.
10. Coman Gh. Monosplines and optimal quadrature formulae in Lp // Rend. Mat. 1972. vol. 6. no. 5. pp. 567–577.
11. Gelfond A. O. Calculus of Finite Differences. Moscow: Nauka, 1967 (in Russian).
13. Hamming R.W. Numerical methods for Scientists and Engineers. USA: McGraw Bill Book Company, Inc., 1962.
14. Hayotov A. R., Jeon S., Lee C.-O. On an optimal quadrature formula for approximation of Fourier integrals in the space L2(1) // Journal of Computational and Applied Mathematics. 2020. vol. 372. 112713.
15. Hayotov A. R., Milovanovi´c G.V., Shadimetov Kh. M. On an optimal quadrature formula in the sense of Sard // Numerical Algorithms. 2011. vol. 57. no. 4. pp. 487-510.
16. Hayotov A. R., Nuraliev F. A., Shadimetov Kh. M. Optimal Quadrature Formulas with Derivative in the Space L2(m)(0,1) // American Journal of Numerical Analysis. 2014. vol. 2. no. 4. pp. 115-127.
17. K¨ohler P. On the weights of Sard’s quadrature formulas // Calcolo. 1988. vol. 25. 1988. pp. 169–186.
18. Lanzara F. On optimal quadrature formulae // J. Ineq. Appl. 2000. vol. 5. pp. 201–225.
19. Maljukov A. A., Orlov I. I. Construction of coefficients of the best quadrature formula for the class WL2(2)(M;ON)  with equally spaced nodes // Optimization methods and operations research, applied mathematics. 1976. vol. 191. pp. 174–177 (in Russian).
20. Meyers L. F., Sard A. Best approximate integration formulas // J. Math. Physics. 1950. vol. 29. pp. 118–123.
21. Nikol’skii S. M. To question about estimation of approximation by quadrature formulas // Uspekhi Matem. Nauk. 1950. vol. 5. no. 2(36). pp. 165–177 (in Russian).
22. Nikol’skii S. M. Quadrature Formulas. Moscow: Nauka, 1988 (in Russian).
23. Sard A. Best approximate integration formulas; best approximation formulas // Amer. J. Math. 1949. vol. 71. pp. 80–91.
24. Sard A. Linear approximation. AMS, 1963.
25. Schoenberg I.J. On monosplines of least deviation and best quadrature formulae // J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 1965. vol. 2. pp. 144–170.
26. Schoenberg I. J. On monosplines of least square deviation and best quadrature formulae II // SIAM J. Numer. Anal. 1966. vol. 3. no. 2. pp. 321–328.
27. Schoenberg I. J. , Silliman S. D. On semicardinal quadrature formulae // Math. Comp. 1974. vol. 126. pp. 483–497.
28. Shadimetov Kh. M. Optimal formulas of approximate integration for differentiable functions. Candidate dissertation. Novosibirsk, 1983. 140 c.
29. Shadimetov Kh. M. Optimal quadrature formulas in  L2m(Ω) and L2m(R1) // Dokl. Akad. Nauk UzSSR. 1983. no. 3 . pp. 5–8 (in Russian).
30. Shadimetov Kh. M. The discrete analogue of the differential operator d2m/dx2m and its construction // Questions of Computations and Applied Mathematics. 1985. pp. 22-35.
31. Shadimetov Kh. M. Optimal Lattice Quadrature and Cubature Formulas // Doklady Mathematics. 2001. vol. 63. no. 1. pp. 92-94.
32. Shadimetov Kh. M. Construction of weight optimal quadrature formulas in the space L2(m)(0,N) // Siberian J. Comput. Math. 2002. vol. 5. no. 3. pp. 275–293 (in Russian).
33. Shadimetov Kh. M., Hayotov A. R. Optimal quadrature formulas with positive coefficients in L2(m)(0,1) space // J. Comput. Appl. Math. 2011. vol. 235. pp. 1114–1128.
34. Shadimetov Kh. M., Hayotov A. R. Optimal quadrature formulas in the sense of Sard in W2(m,m-1) space // Calcolo. 2014. vol. 51. pp. 211-243.
35. Shadimetov Kh. M., Hayotov A. R., Azamov S. S. Optimal quadrature formula in K2(P2) space // Applied Numerical Mathematics. 2012. vol. 62. pp. 1893-1909.
36. Shadimetov Kh. M., Hayotov A. R., Nuraliev F. A. On an optimal quadrature formula in Sobolev space L2(m)(0,1) Journal of Computational and Applied Mathematics. 2013. vol. 243. pp. 91-112.
37. Sobolev S. L. Introduction to the Theory of Cubature Formulas. Moscow: Nauka, 1974 (in Russian).
38. Sobolev S. L. The coefficients of optimal quadrature formulas. Selected Works of S.L. Sobolev: Springer, 2006. pp. 561–566.
39. Sobolev S. L., Vaskevich V. L. The Theory of Cubature Formulas. Dordrecht: Kluwer Academic Publishers Group, 1997.
40. Zagirova F.Ya. On construction of optimal quadrature formulas with equal spaced nodes. Preprint No. 25. Novosibirsk: Institute of Mathematics SD of AS of USSR, 1982. 28 c (in Russian).
41. Zhamalov Z. Zh., Shadimetov Kh. M. About optimal quadrature formulas // Dokl. Akademii Nauk UzSSR. 1980. vol. 7. pp. 3–5 (in Russian).
42. Zhensikbaev A. A. Monosplines of minimal norm and the best quadrature formulas // Uspekhi Matem. Nauk. 1981. vol. 36. pp. 107–159 (in Russian).

Хаётов Абдулло Рахмонович – доктор физико-математических наук, заведующий лаборатории «Вычислительная математика» Института математики им. В. И. Романовского Академии наук Узбекистана, г. Ташкент, Республика Узбекистан.

Hayotov Abdullo Rakhmonovich – D. Sci. (Phys & Math), Head of Laboratory of Computational Mathematics, V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan.

Нуралиев Фарход Абдуганиевич – кандидат физико-математических наук, старший научный сотрудник лаборатории «Вычислительная математика» Института математики им. В. И. Романовского Академии наук Узбекистана, г. Ташкент, Республика Узбекистан.

Nuraliev Farhod Abduganievich
– Ph. D. (Phys. & Math.), Senior scientific researcher of Laboratory of Computational Mathematics, V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan.

Паровик Роман Иванович – доктор физико-математических наук, доцент, декан физико-математического факультета Камчатского государственного университета имени Витуса Беринга, ведущий научный сотрудник лаборатории моделирования физических процессов Института космофизических исследований и
распространения радиоволн ДВО РАН, Камчатский край, Паратунка, Россия, ORCID 0000-0002-1576-1860.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Dean of the Fac. of Phys. & Math., Vitus Bering Kamchatka State University, Leading Researcher of the Phys. Proc. Modeling Lab. of the Institute of Cosmophysical Research and Propagation of Radio Waves, FEB RAS, Russia, ORCID 0000-0002-1576-1860.

Шадиметов Холмат Махкамбаевич – доктор физико-математических наук, профессор, главный научный сотрудник лаборатории «Вычислительная математика» Института математики им. В. И. Романовского Академии наук Узбекистана, г. Ташкент, Республика Узбекистан.

Shadiemtov Kholmat Makhkambaevich – D. Sci. (Phys. & Math.), professor, principal scientific researcher of Laboratory of Computational Mathematics, V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan.