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 $\rho (x)= \alpha +\beta (x-c)+\gamma/(x-g)$ 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 $f^{(4)}(x) f^{\prime\prime}(x)>0$ 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 $C^2$ 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.
