Optimization of quadrature formulas with derivatives.

Shadimetov Xolmatvay Maxkambayevich; Nuraliyev  Farxod Abduganievich

676

In this paper in the space L( )2m (0,1) the problem of construction of optimal quadrature formulas is considered. Here the
quadrature sum consists on values of integrand at nodes and values of first, third, fifth and seventh derivatives of
integrand at the end points of the integration interval. In the work the coefficients of the optimal quadrature formula
are found and the norm of the error functional of the formula is calculated.

Журнал номиHisoblash va amaliy matematika muammolari
Нашр номи№1(1) 2015
Кўришлар сони 676

Internet ҳавола

DOI

UzSCI тизимида яратилган сана 13-11-2019

Ўқишлар сони 647

Нашр санаси 05-11-2015

Мақола тилиIngliz

Саҳифалар сони61-70

Калит сўзлар

optimal quadrature formulas

English

In this paper in the space L( )2m (0,1) the problem of construction of optimal quadrature formulas is considered. Here the
quadrature sum consists on values of integrand at nodes and values of first, third, fifth and seventh derivatives of
integrand at the end points of the integration interval. In the work the coefficients of the optimal quadrature formula
are found and the norm of the error functional of the formula is calculated.

Калит сўзлар

optimal quadrature formulas

№ Муаллифнинг исми Лавозими Ташкилот номи

1 Shadimetov X.M. professor O'zbekiston Milliy universitet

2 Nuraliyev F.A. Katta ilmiy xodim O'zR Fanlar akademiyasi V.I. Romanovskiy nomidagi Matematika instituti

№ Ҳавола номи

1 Babuška I. Optimal quadrature formulas // Dokladi Akad. Nauk SSSR. – 1963. – № 149. Pp. 227–229.

2 Blaga P., Coman Gh. Some problems on optimal quadrature // Stud. Univ. Babe-Bolyai Math. – 2007. – Vol. 52. – № 4. – Pp. 21-44.

3 Bojanov B. Optimal quadrature formulas // Uspekhi Mat. Nauk. – 2005. – Vol. 60. – № 6 (366). Pp. 33-52.

4 Catina T., Coman Gh. Optimal quadrature formulas based on the -function method // Stud. Univ. Babe-Bolyai Math. – 2006. – Vol. 51. – № 1. – Pp. 49-64.

5 Chakhkiev M.A. Linear differential operators with real spectrum, and optimal quadrature formulas // Izv. Akad. Nauk SSSR. Ser. Mat. – 1984. – Vol. 48. – № 5. – Pp. 1078-1108.

6 Coman Gh. Quadrature formulas of Sard type // Studia Univ. Babe-Bolyai Ser. Math.-Mech. – 1972. – Vol. 17. – № 2. Pp. 73-77.

7 Coman Gh. Monosplines and optimal quadrature formulae // Rend. Mat. – 1972. – Vol. 6. – № 5. – Pp. 567-577.

8 Gelfond A.O. Calculus of Finite Differences. – Moscow: Nauka, 1967.

9 Ghizzetti A., Ossicini A. Quadrature Formulae. – Berlin: Akademie Verlag, 1970.

10 Hamming R.W. Numerical methods for Scientists and Engineers. – USA: McGraw Bill Book Company, 1962.

11 Hayotov A.R., Milovanovi G.V., Shadimetov Kh.M. On an optimal quadrature formula in the sense of Sard // Numerical Algorithms. – 2011. – Vol. 57. № 4. – Pp. 487-510.

12 Hayotov A.R., Nuraliev F.A., Shadimetov Kh.M. Optimal Quadrature Formulas with Derivative in the Space // American Journal of Numerical Analysis. – 2014. – Vol. 2. – № 4. – Pp. 115-127.

13 Köhler P. On the weights of Sard’s quadrature formulas // Calcolo. – 1988. – № 25. – Pp.169-186.

14 Lanzara F. On optimal quadrature formulae // J. Ineq. Appl. – 2000. – № 5. – Pp. 201-225.

15 Maljukov A.A., Orlov I.I. Construction of coefficients of the best quadrature formula for the class with equally spaced nodes // Optimization methods and operations research, applied mathematics. – 1976. – № 191. – Pp. 174- 177.

16 Meyers L.F., Sard A. Best approximate integration formulas // J. Math. Physics. – 1950. – № 29. – Pp. 118-123.

17 Nikol’skii S.M. To question about estimation of approximation by quadrature formulas // Uspekhi Matem. Nauk. – 1950. – № 5:2. – Pp. 165-177.

18 Nikol’skii S.M. Quadrature Formulas. – Moscow: Nauka, 1988.

19 Sard A. Best approximate integration formulas // Amer. J. Math. – 1949. – № 71. – Pp. 80-91.

20 Sard A. Linear approximation. – AMS, 1963.

21 Schoenberg I.J. On monosplines of least deviation and best quadrature formulae // J. Soc. Indust. Appl. Math. Ser. – 1965. – Pp. 144-170.

22 Schoenberg I.J. On monosplines of least square deviation and best quadrature formulae // SIAM J. – 1966. – Vol. 3. – № 2. – Pp. 321-328.

23 Schoenberg I.J., Silliman S.D. On semicardinal quadrature formulae // Math. Comp. – 1974. – Vol. 126. – Pp. 483-497.

24 Shadimetov Kh.M. Optimal formulas of approximate integration for differentiable functions: candidate dissertation. – Novosibirsk, 1983. – 140 p.

25 Shadimetov Kh.M. Optimal quadrature formulas // Dokl. Akad. Nauk UzSSR. – 1983. – № 3. – Pp. 5-8.

26 Shadimetov Kh.M. The discrete analogue of the differential operator and its construction // Questions of Computations and Applied Mathematics. – Tashkent, 1985. – Pp. 22-35.

27 Shadimetov Kh.M. Optimal Lattice Quadrature and Cubature Formulas // Doklady Mathematics. – 2001. – Vol. 63. – № 1. Pp. 92-94.

28 Shadimetov Kh.M. Construction of weight optimal quadrature formulas in the space // Siberian J. Comput. Math. – 2002. – Vol. 5. – № 3. – Pp. 275-293.

29 Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas with positive coefficients in space // J. Comput. Appl. Math. – 2011. – № 235. – Pp.1114-1128.

30 Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas in the sense of Sard in space // Calcolo. – 2014. – № 51. – Pp. 211-243.

31 Shadimetov Kh.M., Hayotov A.R., Azamov S.S. Optimal quadrature formula in space // Applied Numerical Mathematics. – 2012. – № 62. – Pp. 1893-1909.

32 Shadimetov Kh.M., Hayotov A.R., Nuraliev F.A. On an optimal quadrature formula in Sobolev space // Journal of Computational and Applied Mathematics. – 2013. – № 243. – Pp. 91-112.

33 obolev S.L. Introduction to the Theory of Cubature Formulas. – Moscow: Nauka, 1974.

34 Sobolev S.L. The coefficients of optimal quadrature formulas // Selected Works of S.L. Sobolev. – 2006. – Pp. 561-566.

35 Sobolev S.L., Vaskevich V.L. The Theory of Cubature Formulas. – Dordrecht: Kluwer Academic Publishers Group, 1997.

36 Zagirova F.Ya. On construction of optimal quadrature formulas with equal spaced nodes. – Novosibirsk, 1982. – 28 p.

37 Zhamalov Z.Zh., Shadimetov Kh.M. About optimal quadrature formulas // Dokl. Akademii Nauk UzSSR. – 1980. – № 7. – Pp. 3-5.

38 Zhensikbaev A.A. Monosplines of minimal norm and the best quadrature formulas // Uspekhi Matem. Nauk. – 1981. – № 36. – Pp. 107-159.

Кутилмоқда

№	Муаллифнинг исми	Лавозими	Ташкилот номи
1	Shadimetov X.M.	professor	O'zbekiston Milliy universitet
2	Nuraliyev F.A.	Katta ilmiy xodim	O'zR Fanlar akademiyasi V.I. Romanovskiy nomidagi Matematika instituti

№	Ҳавола номи
1	Babuška I. Optimal quadrature formulas // Dokladi Akad. Nauk SSSR. – 1963. – № 149. Pp. 227–229.
2	Blaga P., Coman Gh. Some problems on optimal quadrature // Stud. Univ. Babe-Bolyai Math. – 2007. – Vol. 52. – № 4. – Pp. 21-44.
3	Bojanov B. Optimal quadrature formulas // Uspekhi Mat. Nauk. – 2005. – Vol. 60. – № 6 (366). Pp. 33-52.
4	Catina T., Coman Gh. Optimal quadrature formulas based on the -function method // Stud. Univ. Babe-Bolyai Math. – 2006. – Vol. 51. – № 1. – Pp. 49-64.
5	Chakhkiev M.A. Linear differential operators with real spectrum, and optimal quadrature formulas // Izv. Akad. Nauk SSSR. Ser. Mat. – 1984. – Vol. 48. – № 5. – Pp. 1078-1108.
6	Coman Gh. Quadrature formulas of Sard type // Studia Univ. Babe-Bolyai Ser. Math.-Mech. – 1972. – Vol. 17. – № 2. Pp. 73-77.
7	Coman Gh. Monosplines and optimal quadrature formulae // Rend. Mat. – 1972. – Vol. 6. – № 5. – Pp. 567-577.
8	Gelfond A.O. Calculus of Finite Differences. – Moscow: Nauka, 1967.
9	Ghizzetti A., Ossicini A. Quadrature Formulae. – Berlin: Akademie Verlag, 1970.
10	Hamming R.W. Numerical methods for Scientists and Engineers. – USA: McGraw Bill Book Company, 1962.
11	Hayotov A.R., Milovanovi G.V., Shadimetov Kh.M. On an optimal quadrature formula in the sense of Sard // Numerical Algorithms. – 2011. – Vol. 57. № 4. – Pp. 487-510.
12	Hayotov A.R., Nuraliev F.A., Shadimetov Kh.M. Optimal Quadrature Formulas with Derivative in the Space // American Journal of Numerical Analysis. – 2014. – Vol. 2. – № 4. – Pp. 115-127.
13	Köhler P. On the weights of Sard’s quadrature formulas // Calcolo. – 1988. – № 25. – Pp.169-186.
14	Lanzara F. On optimal quadrature formulae // J. Ineq. Appl. – 2000. – № 5. – Pp. 201-225.
15	Maljukov A.A., Orlov I.I. Construction of coefficients of the best quadrature formula for the class with equally spaced nodes // Optimization methods and operations research, applied mathematics. – 1976. – № 191. – Pp. 174- 177.
16	Meyers L.F., Sard A. Best approximate integration formulas // J. Math. Physics. – 1950. – № 29. – Pp. 118-123.
17	Nikol’skii S.M. To question about estimation of approximation by quadrature formulas // Uspekhi Matem. Nauk. – 1950. – № 5:2. – Pp. 165-177.
18	Nikol’skii S.M. Quadrature Formulas. – Moscow: Nauka, 1988.
19	Sard A. Best approximate integration formulas // Amer. J. Math. – 1949. – № 71. – Pp. 80-91.
20	Sard A. Linear approximation. – AMS, 1963.
21	Schoenberg I.J. On monosplines of least deviation and best quadrature formulae // J. Soc. Indust. Appl. Math. Ser. – 1965. – Pp. 144-170.
22	Schoenberg I.J. On monosplines of least square deviation and best quadrature formulae // SIAM J. – 1966. – Vol. 3. – № 2. – Pp. 321-328.
23	Schoenberg I.J., Silliman S.D. On semicardinal quadrature formulae // Math. Comp. – 1974. – Vol. 126. – Pp. 483-497.
24	Shadimetov Kh.M. Optimal formulas of approximate integration for differentiable functions: candidate dissertation. – Novosibirsk, 1983. – 140 p.
25	Shadimetov Kh.M. Optimal quadrature formulas // Dokl. Akad. Nauk UzSSR. – 1983. – № 3. – Pp. 5-8.
26	Shadimetov Kh.M. The discrete analogue of the differential operator and its construction // Questions of Computations and Applied Mathematics. – Tashkent, 1985. – Pp. 22-35.
27	Shadimetov Kh.M. Optimal Lattice Quadrature and Cubature Formulas // Doklady Mathematics. – 2001. – Vol. 63. – № 1. Pp. 92-94.
28	Shadimetov Kh.M. Construction of weight optimal quadrature formulas in the space // Siberian J. Comput. Math. – 2002. – Vol. 5. – № 3. – Pp. 275-293.
29	Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas with positive coefficients in space // J. Comput. Appl. Math. – 2011. – № 235. – Pp.1114-1128.
30	Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas in the sense of Sard in space // Calcolo. – 2014. – № 51. – Pp. 211-243.
31	Shadimetov Kh.M., Hayotov A.R., Azamov S.S. Optimal quadrature formula in space // Applied Numerical Mathematics. – 2012. – № 62. – Pp. 1893-1909.
32	Shadimetov Kh.M., Hayotov A.R., Nuraliev F.A. On an optimal quadrature formula in Sobolev space // Journal of Computational and Applied Mathematics. – 2013. – № 243. – Pp. 91-112.
33	obolev S.L. Introduction to the Theory of Cubature Formulas. – Moscow: Nauka, 1974.
34	Sobolev S.L. The coefficients of optimal quadrature formulas // Selected Works of S.L. Sobolev. – 2006. – Pp. 561-566.
35	Sobolev S.L., Vaskevich V.L. The Theory of Cubature Formulas. – Dordrecht: Kluwer Academic Publishers Group, 1997.
36	Zagirova F.Ya. On construction of optimal quadrature formulas with equal spaced nodes. – Novosibirsk, 1982. – 28 p.
37	Zhamalov Z.Zh., Shadimetov Kh.M. About optimal quadrature formulas // Dokl. Akademii Nauk UzSSR. – 1980. – № 7. – Pp. 3-5.
38	Zhensikbaev A.A. Monosplines of minimal norm and the best quadrature formulas // Uspekhi Matem. Nauk. – 1981. – № 36. – Pp. 107-159.