An optimal quadrature formula exact to the exponential function by the phi function method
DOI:
https://doi.org/10.24193/subbmath.2024.3.11Keywords:
Hilbert space, phi-function method, optimal quadrature formula, error quadrature formulaAbstract
The numerical integration of definite integrals is essential in fundamental and applied sciences. The accuracy of approximate integral calculations is contingent upon the initial data and specific requirements, leading to the imposition of diverse conditions on the resultant computations. Classical methods for the numerical analysis of definite integrals are known, such as the quadrature formulas of Gregory, Newton-Cotes, Euler, Gauss, Markov, etc. Since the middle of the last century, the theory of constructing optimal formulas for numerical integration based on variational methods began to develop. It should be noted that there are optimal quadrature formulas in the sense of Nikolsky and Sard. In this paper, we study the problem of constructing an optimal quadrature formula in the sense of Sard. When constructing a quadrature formula, the method of ϕ-functions is used. The error of the formula is estimated from above using the integral of the square of the function ϕ from a specific Hilbert space. Next, such a ϕ function is selected, and the integral of the square in this interval takes the smallest value. The coefficients of the optimal quadrature formula are calculated using the resulting ϕ function. The optimal quadrature formula in this work is exact on the functions eσx and e−σx, where σ is a nonzero real parameter.
Mathematics Subject Classification (2010): 65D30, 65D32.
Received 08 November 2023; Accepted 02 July 2024.
References
Babaev, S.S., Hayotov, A.R., Optimal interpolation formulas in the space W_2^((m,m-1)), Calcolo, 56(2019), no. 23, 1066–1088. Boltaev, N.D., Hayotov, A.R., Khudayberdiev, M., Optimal quadrature formula for approximate calculation of Fourier coefficients in W_2^((1,0)) space, Problems of Computational and Applied Mathematics, Tashkent, 1(2015), no. 1, 71–77.
Boltaev, N.D., Hayotov, A.R., Milovanovi´c, G.V., Shadimetov, Kh.M., Optimal quadrature formulas for numerical evaluation of Fourier coefficients in W_2^((m,m-1)), J. Appl. Anal. Comput., 7(2017), no. 4, 1233–1266.
Boltaev, N.D., Hayotov, A.R., Shadimetov, Kh.M., Construction of optimal quadrature formula for numerical calculation of Fourier coefficients in Sobolev space L(1), Amer. J. Numer. Anal., 4(2016), 1–7.
Cătinaș, T., Coman, Gh., Optimal quadrature formulas based on the ϕ-function method, Stud. Univ. Babeș-Bolyai Math., 51(2006), no. 1, 49–64.
Coman, Gh., Formule de cuadratura˘ de tip Sard, Stud. Univ. Babeș-Bolyai Math.-Mech., 17(1972), no. 2, 73–77.
Coman, Gh., Monosplines and optimal quadrature formulae, Lp. Rend. Mat., 5(1972), no. 6, 567–577.
DeVore, R., Foucart, S., Petrova, G., Wojtaszczyk, P., Computing a quantity of interest from observational data, Constr. Approx., 49(2019), 461–508.
Ghizzett, A., Ossicini, A., Quadrature Formulae, Academie Verlag, Berlin, 1970.
Hayotov, A.R., Babaev, S.S., Optimal quadrature formulas for computing of Fourier integrals in W_2^((m,m-1)), space, AIP Conference Proceedings, 2365(2021), 020021.
Hayotov, A.R., Jeon, S., Lee, C.-O., On an optimal quadrature formula for approximation of Fourier integrals in the space L_2^((1)), J. Comput. Appl. Math., 372(2020), 112713.
Hayotov, A.R., Jeon, S., Shadimetov, Kh.M., Application of optimal quadrature formulas for reconstruction of CT images, J. Comput. Appl. Math., 388(2021), 113313.
Hayotov, A.R., Kuldoshev, H.M., An optimal quadrature formula with sigma parameter, Problems of Computational and Applied Mathematics, Tashkent, 48(2023), no. 2/1, 7–19.
Hayotov, A.R., Rasulov, R.G., The order of convergence of an optimal quadrature formula with derivative in the space W_2^((1,0)), Filomat, 34(2020), no. 11, 3835–3844.
Köhler, P., On the weights of Sard’s quadrature formulas, Calcolo, 25(1988), no. 3, 169–186.
Lanzara, F., On optimal quadrature formulae, J. Inequal. Appl., 5(2000), 201–225.
Meyers, L.F., Sard, A., Best approximate integration formulas, J. Math. and Phys., 29(1950), 118–123.
Nikolsky, S.M., On the issue of estimates of approximations by quadrature formulas (in Russian), Advances in Math. Sciences, 5(1950), no. 3, 165–177.
Nikolsky, S.M., Quadrature Formulas (in Russian), 4th ed., Nauka, Moscow, 1988.
Sard, A., Best approximate integration formulas, best approximate formulas, Amer. J. Math., 71(1949), 80–91.
Sard, A., Linear Approximation, 2nd ed., American Math. Society, Province, Rhode Island, 1963.
Schoenberg, I.J., On trigonometric spline interpolation, J. Math. Mech., 13(1964), 795–825.
Schoenberg, I.J., On monosplines of least deviation and best quadrature formulae, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2(1965), 144–170.
Schoenberg, I.J., On monosplines of least square deviation and best quadrature formulae II, SIAM J. of Numer. Anal., 3(1966), 321–328.
Schoenberg, I.J., Silliman, S.D., On semicardinal quadrature formulae, Math. Comp., 27(1973), 483–497.
Shadimetov, Kh.M., Hayotov, A.R., Optimal quadrature formulas in the sense of Sard in W_2^((m,m-1)) space, Calcolo, 51(2014), no. 2, 211–243.
Shadimetov, Kh.M., Hayotov, A.R., Optimal Approximation of Error Functionals of Quadrature and Interpolation Formulas in Spaces of Differentiable Functions (in Russian), Muhr Press, Tashkent, 2022.
Shadimetov, Kh.M., Hayotov, A.R., Akhmedov, D.M., Optimal quadrature formulas for Cauchy type singular integrals in Sobolev space, Appl. Math. Comput., 263(2015), 302–314.
Sobolev, S.L., Introduction to the Theory of Cubature Formulas (in Russian), Nauka, Moscow, 1974.
Sobolev, S.L., Coefficients of optimal quadrature formulas (in Russian), Doklady Akademii Nauk SSSR, 235(1977), no. 1, 34–37.
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