A quadrature formula with positive coefficients is constructed with the use of three nodes a, b, and c=(a+b)/2 and rational interpolants of the form ρ(x)=α+β(x−c)+γ/(x−g) with a pole g determined by nodes outside the integration interval [a,b]. The error of the constructed formula is smaller than the error of the corresponding Simpson quadrature formula if the integrand f(x) has a continuous derivative f(4)(x) on the interval [a,b] and the inequality f(4)(x)f′′(x)>0 is satisfied.
Keywords: rational interpolant, quadrature formula, Simpson formula
Received February 20, 2021
Revised May 17, 2021
Accepted June 15, 2021
A.-R.K. Ramazanov, Dr. Phys.-Math., Prof., Dagestan State University, the Republic of Dagestan, Makhachkala, 367002 Russia; Dagestan Scientific Center RAN, the Republic of Dagestan, Makhachkala, 367025 Russia, e-mail: ar-ramazanov@rambler.ru
V.G. Magomedova, Cand. Sci. (Phys.-Math.), Dagestan State University, the Republic of Dagestan, Makhachkala, 367002 Russia, e-mail: vazipat@rambler.ru
REFERENCES
1. Ahlberg J., Nilson E., Walsh J. The theory of splines and their applications. ТН: Acad. Press, 1967, 284 p. ISBN: 9781483222950 . Translated to Russian under the title Teoriya splainov i ee prilozheniya, Moscow: Mir Publ., 1972, 316 p.
2. Stechkin S.B., Subbotin Yu.N. Splainy v vychislitel’noi matematike [Splines in computational mathematics]. Moscow: Nauka Publ., 1976, 248 p.
3. Zav’yalov Yu.S., Kvasov B.I., Miroshnichenko V.L. Metody splain-funktsii [Methods of spline-functions]. Moscow: Nauka Publ., 1980, 352 p.
4. Berdyshev V.I., Subbotin Yu.N. Chislennye metody priblizheniya funktsii [Numerical methods for approximation of functions]. Sverdlovsk: Sredne-Ural’skoe Kn. Izd-vo, 1979, 120 p.
5. Arushanyan I.O. Primenenie metoda kvadratur dlya chislennogo resheniya integral’nykh uravnenii vtorogo roda [Application of the quadrature method for numerical solutions of integral equations of the second order]. Moscow: Izd-vo TsPI Mekh.-Mat. MGU, 2018, 61 p.
6. Edeo A., Gofeb G., Tefera T. Shape preserving C2 rational cubic spline interpolation. American Scientific Research Journal for Engineering, Technology and Sciences, 2015, vol. 12, no. 1. pp. 110–122.
7. Magomedova V.G., Ramazanov A.-R.K. Approximate solution of differential equations with the help of rational spline functions. Comput. Math. and Math. Phys., 2019, vol. 59, no. 4, pp. 542–549. doi: 10.1134/S0965542519040110
8. Nikolskiy S.M. Kvadraturnie formuli [Quadrature formulae]. Moscow: Nauka Publ., 1988, 255 p. ISBN: 5-02-013786-3 .
9. Bakhvalov N.S., Zhidkov N.P., Kobel’kov G.M. Chislennye metody [Numerical methods]. Moscow: Laboratoriya Bazovykh Znanii Publ., 2002, 632 p. ISBN: 5-93208-043-4 .
10. Krylov V.I., Bobkov V.V., Monastyrnyi P.I. Vychislitel’nye metody [Computational methods]. Vol. 1. Moscow: Nauka Publ., 1976, 304 p.
11. Ramazanov A.-R.K., Magomedova V.G. Splines for three-point rational interpolants with autonomous poles. Dagestan. Elektron. Mat. Izv., 2017, no. 7, pp. 16–28 (in Russian). doi: 10.31029/demr.7.2
12. Householder A.S. Principles of numerical analysis. NY; Toronto; London: McGRAW-HILL, 1953, 274 p. Translated to Russian under the title Osnovy chislennogo analiza, Moscow: Izd-vo Inostr. Lit-ry, 1956, 320 p.
Cite this article as: A.-R.K.Ramazanov, V.G.Magomedova. Comparison of the remainders of the Simpson quadrature formula and the quadrature formula for three-point rational interpolants, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 4, pp. 102–110.
