On a heuristic algorithm for constructing optimal coefficients with optimization by the ℎ-function
https://doi.org/10.22405/2226-8383-2025-26-5-84-93
Abstract
In this paper, for 𝑠 ⩾ 3, we describe an algorithm for constructing sequences 𝑃𝐻(𝑠,⃗𝑎,𝑁)𝑖 —
𝑠-dimensional optimal coefficients ⃗𝑎 = (1, 𝑎, 𝑎2 (mod 𝑁), . . . , 𝑎𝑠−1 (mod 𝑁)) modulo N, such
that 𝑎𝑠 ≡ ±1 (mod 𝑁). We construct a sequence such that the error in numerically calculating
the integral of the boundary function of class 𝐸2 𝑠 ℎ(⃗𝑥) = 3𝑠Π︀𝑠 𝑖=1(1 − 2𝑥𝑖)2 over parallelepiped grids 𝑀(⃗𝑎,𝑁) on the cube [0, 1)𝑠 decreases with increasing 𝑁.
About the Authors
Yuri Alexandrovich Basalov
Tula State Lev Tolsoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Irina Nikolaevna Balaba
Tula State Lev Tolsoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Nikolay Nikolaevich Dobrovolsky
Lomonosov Moscow State University; Tula State Lev Tolsoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Nina Magomedrasulovna Isaeva
Tula State Lev Tolsoy Pedagogical University
Russian Federation
Russian Federation
candidate of biological sciences
Anna Dmitrievna Pankina
Tula State Lev Tolsoy Pedagogical University
Russian Federation
Russian Federation
student
References
1. Korobov, N.M. 1959, “Approximate computation of multiple integrals”, Doklady Akademii Nauk SSSR, 124, pp. 1207.
2. Korobov, N.M. 1959, “Computation of multiple integrals by the method of optimal coefficients”, Vestnik Moskovskogo Universiteta, 4, pp. 19–25.
3. Korobov, N.M. 1960, “Properties and calculation of optimal coefficients”, Doklady Akademii Nauk SSSR, 132(5), pp. 1009–1012.
4. Hlawka. E. 1962, “Zur angen¨aherten Berechnung mehrfacher Integrale”, Monatshefte f¨ur Mathematik, 66 (1962), 140–151.
5. Korobov, N.M. 1963, Number-Theoretic Methods in Approximate Analysis. Moscow: Fizmatgiz.
6. Sharigin, I.F. 1960, “On the application of number-theoretic methods of integration in the case of nonperiodic functions”, Doklady Akademii Nauk SSSR, 132(1), pp. 71–74.
7. Dobrovol’skii, N.N., Skobel’tsyn, S.A., Tolokonnikov, L.A. and Larin, N.V. 2022, “Application of number-theoretic grids in problems of sound diffraction on elastic bodies”, Chebyshevskii Sbornik, 23(5), pp. 206–226.
8. Korobov, N.M. 1967, “On some questions of the theory of Diophantine approximations”, Uspekhi Matematicheskikh Nauk, 22(3), pp. 83–118; Russian Mathematical Surveys, 22(3), pp. 80–118.
9. Nikol’skii, S.M. 1988, Quadrature Formulas, 4th edn, with additions by N.P. Korneichuk. Moscow: Nauka. (In Russian).
10. Korobov, N.M. 1994, “Quadrature formulas with combined nets”, Matematicheskie Zametki, 55(2), pp. 83–90; Mathematical Notes, 55(2), pp. 159–164.
11. Vinogradov, I.M. 1981, Elements of Number Theory, 9th rev. edn. Moscow: Nauka.
12. Korobov, N.M. 2004, Number-Theoretic Methods in Approximate Analysis, 2nd edn. Moscow: MCCME.
13. Bykovsky, V.A. 2004, “An algorithm for computing local minima of lattices”, Doklady Akademii Nauk, 399(5), pp. 585–589.
14. Dobrovol’skii N. N., Dobrovol’skii N. M., Basalov Yu. A., Rebrov E. D. 2023. “Fast Calculation of Parameters of Parallelepipedal Nets for Integration and Interpolation”, In: Alikhanov, A., Lyakhov, P., Samoylenko, I. (eds) Current Problems in Applied Mathematics and Computer Science and Systems. APAMCS 2022. Lecture Notes in Networks and Systems, vol 702. Springer, Cham. https://doi.org/10.1007/978-3-031-34127-4_16
15. Bahvalov, N.S. 1959, “On approximate computation of multiple integrals”, Vestnik Moskovskogo Universiteta, 4, pp. 3–18.
16. Dobrovol’skii, N. N., Dobrovolskii, N. M., Rebrova, I. Yu., Rebrov, E. D. 2023. “On Calculating the Hyperbolic Parameter of a Two-Dimensional Lattice of Linear Comparison Solutions”. In: Alikhanov, A., Lyakhov, P., Samoylenko, I. (eds) Current Problems in Applied Mathematics and Computer Science and Systems. APAMCS 2022. Lecture Notes in Networks and Systems, vol 702. Springer, Cham. https://doi.org/10.1007/978-3-031-34127-4_8
17. Sharigin, I.F. 1963, “Lower bounds for the error of quadrature formulas”, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 3(2), pp. 370–376; U.S.S.R. Computational Mathematics and Mathematical Physics, 3(2), pp. 489–497.
18. Belov, A.A. Tintul, M.A. 2023, “Multidimensional cubature formulas with super-polynomial convergence”, Doklady Rossiiskoi Akademii Nauk. Matematika, Informatika, Protsessy Upravleniya, 514(1), pp. 107–111; Doklady Mathematics, 108(3), pp. 514–518.
19. Zhubanysheva, A. Zh., Temirgaliev, N., Temirgalieva, Zh. N. 2009, “Application of divisor theory to the construction of tables of optimal coefficients for quadrature formulas”, Zh. Vychisl. Mat. Mat. Fiz., Vol. 49, No. 1, pp. 14–25.
Review
For citations:
Basalov Yu.A., Balaba I.N., Dobrovolsky N.N., Isaeva N.M., Pankina A.D. On a heuristic algorithm for constructing optimal coefficients with optimization by the ℎ-function. Chebyshevskii Sbornik. 2025;26(5):84-93. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-5-84-93
Views:
42
JATS XML
Email this article 