Construction of Three Quadrature Formulas of Eighth Order and Their Application for Approximating Series

Affiliation(s)

Faculty of Applied Mathematics, AGH University of Science and Technology, Cracow, Poland.

Faculty of Applied Mathematics, AGH University of Science and Technology, Cracow, Poland.

ABSTRACT

In this paper, three types of three-parameters families of quadrature formulas for the Riemann’s integral on an interval of the real line are carefully studied. This research is a continuation of the results in the [1]-[3]. All these quadrature formulas are not based on the integration of an interpolant as so as the Gregory rule, a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order (see [4]). In some natural restrictions on the parameters we construct the only one quadrature formula of the eight order which belongs to the first, second and third family. For functions whose 8th derivative is either always positive or always negative, we use these quadrature formulas to get good two-sided bound on . Additionally, we apply these quadratures to obtain the approximate sum of slowly convergent series , where .

In this paper, three types of three-parameters families of quadrature formulas for the Riemann’s integral on an interval of the real line are carefully studied. This research is a continuation of the results in the [1]-[3]. All these quadrature formulas are not based on the integration of an interpolant as so as the Gregory rule, a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order (see [4]). In some natural restrictions on the parameters we construct the only one quadrature formula of the eight order which belongs to the first, second and third family. For functions whose 8th derivative is either always positive or always negative, we use these quadrature formulas to get good two-sided bound on . Additionally, we apply these quadratures to obtain the approximate sum of slowly convergent series , where .

Cite this paper

Bożek, B. , Solak, W. and Szydełko, Z. (2015) Construction of Three Quadrature Formulas of Eighth Order and Their Application for Approximating Series.*Applied Mathematics*, **6**, 1031-1046. doi: 10.4236/am.2015.66095.

Bożek, B. , Solak, W. and Szydełko, Z. (2015) Construction of Three Quadrature Formulas of Eighth Order and Their Application for Approximating Series.

References

[1] Bozek, B., Solak, W. and Szydelko, Z. (2008) A Note on a Family of Quadrature Formulas and Some Applications. Opuscula Mathematica, 28, 109-121.

[2] Bozek, B., Solak, W. and Szydelko, Z. (2009) A Note of Some Quadrature Rules with Gregory End Corrections. Opuscula Mathematica, 29, 117-129.

[3] Bozek, B., Solak, W. and Szydelko, Z. (2012) On Some Quadrature Rules with Laplace End Corrections. Central European Journal of Mathematics, 10, 1172-1184.

http://dx.doi.org/10.2478/s11533-012-0034-6

[4] de Villiers, J.M. (1993) A Nodal Spline Interpolant for the Gregory Rule of Even Order. Numerische Mathematik, 66, 123-137. http://dx.doi.org/10.1007/BF01385690

[5] Kincaid, D. and Cheney, W. (2002) Numerical Analysis, Mathematics of Scientific Computing. 3rd Edition, The University of Texas at Austin, Brooks/Cole-Thomson Learning.

[1] Bozek, B., Solak, W. and Szydelko, Z. (2008) A Note on a Family of Quadrature Formulas and Some Applications. Opuscula Mathematica, 28, 109-121.

[2] Bozek, B., Solak, W. and Szydelko, Z. (2009) A Note of Some Quadrature Rules with Gregory End Corrections. Opuscula Mathematica, 29, 117-129.

[3] Bozek, B., Solak, W. and Szydelko, Z. (2012) On Some Quadrature Rules with Laplace End Corrections. Central European Journal of Mathematics, 10, 1172-1184.

http://dx.doi.org/10.2478/s11533-012-0034-6

[4] de Villiers, J.M. (1993) A Nodal Spline Interpolant for the Gregory Rule of Even Order. Numerische Mathematik, 66, 123-137. http://dx.doi.org/10.1007/BF01385690

[5] Kincaid, D. and Cheney, W. (2002) Numerical Analysis, Mathematics of Scientific Computing. 3rd Edition, The University of Texas at Austin, Brooks/Cole-Thomson Learning.