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Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.42 no.2 Bogotá July/Dec. 2008

 

An optimal 3-point quadrature formula of closed type and error bounds

Una fórmula de cuadratura óptima de 3 puntos de tipo cerrado y error de frontera

NENAD UJEVIC1, LUCIJA MIJIC2

1University of Split, Split, Croatia. Email: ujevic@pmfst.hr
2University of Split, Split, Croatia. Email: lucmij@pmfst.hr


Abstract

An optimal 3-point quadrature formula of closed type is derived. The obtained optimal quadrature formula has better estimations of error than the well-known Simpson's formula. A few error inequalities for this formula are established.

Key words: Optimal quadrature formula, error inequalities, Ostrowski-like inequalities.


2000 Mathematics Subject Classification: 26D10, 41A55.

Resumen

Se establece una fórmula de cuadratura óptima de 3 puntos de tipo cerrado. Dicha fórmula mejora la estimación de error de la bien conocida fórmula de Simpson. Se establecen algunas desigualdades de error para esta fórmula.

Palabras clave: Fórmula de cuadratura óptima, desigualdades de error, desigualdades de tipo de Ostrowski.


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References

[1] Atkinson, K. & Han, W., Theoretical Numerical Analysis: A Functional Analysis Framework, Springer-Verlag, New York-Berlin-Heidelberg, 2001.         [ Links ]

[2] Cerone, P., `Three points rules in numerical integration´, Nonlinear Anal. Theory Methods Appl. 47, 4 (2001), 2341-2352.         [ Links ]

[3] Cruz-Uribe, D. & Neugebauer, C., `Sharp error bounds for the trapezoidal rule and Simpson's rule´, J. Inequal. Pure Appl. Math. 3, 4 (2002), 1-22.         [ Links ]

[4] Dragomir, S., Agarwal, R. & Cerone, P., `On Simpson's inequality and applications´, J. Inequal. Appl. 5, (2000), 533-579.         [ Links ]

[5] Dragomir, S., Cerone, P. & Roumeliotis, J., `A new generalization of Ostrowski's integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means´, Appl. Math. Lett. 13, (2000), 19-25.         [ Links ]

[6] Dragomir, S., Pecaric, J. & Wang, S., `The unified treatment of trapezoid, Simpson and Ostrowski type inequalities for monotonic mappings and applications´, Math. Comput. Modelling 31, (2000), 61-70.         [ Links ]

[7] Mitrinovic, D., Pecaric, J. & Fink, A., Classical and New Inequalities in Analysis, Kluwer Acad. Publ., Dordrecht-Boston-London, 1993.         [ Links ]

[8] Pearce, C., Pecaric, J., Ujevic, N. & Varosanec, S., `Generalizations of some inequalities of Ostrowski-Grüss type´, Math. Inequal. Appl. 3, 1 (2000), 25-34.         [ Links ]

[9] Ujevic, N., `Inequalities of Ostrowski-Grüss type and applications´, Appl. Math. 29, 4 (2002), 465-479.         [ Links ]

[10] Ujevic, N., `An optimal quadrature formula of open type´, Yokohama Math. J. 50, (2003), 59-70.         [ Links ]

[11] Ujevic, N., `Error inequalities for a quadrature formula and applications´, Comput. Math. Appl. 48, 10-11 (2004a), 1531-1540.         [ Links ]

[12] Ujevic, N., `Two sharp Ostrowski-like inequalities and applications´, Meth. Appl. Analysis 10, 3 (2004b), 477-486.         [ Links ]

[13] Volkov, E., Numerical Methods, Mir Publishers, Moscow, 1986.         [ Links ]

(Recibido en marzo de 2008. Aceptado en julio de 2008)

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv42n2a07,
    AUTHOR  = {Ujevic, Nenad and Mijic, Lucija},
    TITLE   = {{An optimal 3-point quadrature formula of closed type and error bounds}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2008},
    volume  = {42},
    number  = {2},
    pages   = {209-220}
}

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