Hostname: page-component-7bb8b95d7b-dvmhs Total loading time: 0 Render date: 2024-09-18T16:44:22.676Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERAL THREE-POINT QUADRATURE FORMULAS OF EULER TYPE

Part of: Numerical approximation and computational geometry Inequalities

Published online by Cambridge University Press:  05 December 2011

IVA FRANJIĆ ,
JOSIP PEČARIĆ  and
IVAN PERIĆ
IVA FRANJIĆ*
Affiliation:
Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia (email: ifranjic@pbf.hr, iperic@pbf.hr)
JOSIP PEČARIĆ
Affiliation:
Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia (email: pecaric@element.hr)
IVAN PERIĆ
Affiliation:
Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia (email: ifranjic@pbf.hr, iperic@pbf.hr)
*
✉For correspondence; e-mail: ifranjic@pbf.hr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

General three-point quadrature formulas for the approximate evaluation of an integral of a function f over [0,1], through the values f(x), f(1/2), f(1−x), f′(0) and f′(1), are derived via the extended Euler formula. Such quadratures are sometimes called “corrected” or “quadratures with end corrections” and have a higher accuracy than the adjoint classical formulas, which only include the values f(x), f(1/2) and f(1−x) . The Gauss three-point, corrected Simpson, corrected dual Simpson, corrected Maclaurin and corrected Gauss two-point formulas are recaptured as special cases. Finally, sharp estimates of error are given for this type of quadrature formula.

Keywords

general three-point quadrature formulascorrected quadrature formulassharp estimates of errorBernoulli polynomialsextended Euler formula

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals 65D32: Quadrature and cubature formulas
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 3 , January 2011 , pp. 309 - 317
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Abramowitz, M. and Stegun (eds), I. A., Handbook of mathematical functions: with formulas, graphs and mathematical tables (Dover Publications, New York, 1992).Google Scholar
[2]Dedić, Lj., Matić, M. and Pečarić, J., “On generalizations of Ostrowski inequality via some Euler-type identities”, Math. Inequal. Appl. 3 (2000) 337–353.Google Scholar
[3]Franjić, I. and Pečarić, J., “Corrected Euler–Maclaurin’s formulae”, Rend. Circ. Mat. Palermo 54 (2005) 259–272; doi:10.1007/BF02874640.Google Scholar
[4]Franjić, I. and Pečarić, J., “On corrected dual Euler–Simpson formulae”, Soochow J. Math. 32 (2006) 575–587.Google Scholar
[5]Franjić, I., Perić, I. and Pečarić, J., “Quadrature formulae of Gauss type based on Euler identities”, Math. Comput. Modelling 45 (2007) 355–370; doi:10.1016/j.mcm.2006.05.009.Google Scholar
[6]Hildebrand, F. B., Introduction to numerical analysis (McGraw-Hill, New York, 1956).Google Scholar
[7]Krylov, V. I., Approximate calculation of integrals (Macmillan, New York–London, 1962).Google Scholar
[8]Lanczos, C., Applied analysis (Prentice-Hall, Englewood Cliffs, NJ, 1956).Google Scholar
[9]Pečarić, J. and Franjić, I., “Generalisation of a corrected Simpson’s formula”, ANZIAM J. 47 (2006) 367–385; doi:10.1017/S1446181100009895.CrossRefGoogle Scholar
[10]Pečarić, J., Perić, I. and Vukelić, A., “Sharp integral inequalities based on general Euler two-point formulae”, ANZIAM J. 46 (2005) 555–574; doi:10.1017/S1446181100009676.Google Scholar
[11]Ujević, N. and Roberts, A. J., “A corrected quadrature formula and applications”, ANZIAM J. 45(E) (2004) E41–E56.Google Scholar