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Monotonicity of the error term in Gauss-Turán quadratures for analytic function

Published online by Cambridge University Press:  17 February 2009

Gradimir V. Milovanović and
Miodrag M. Spalević
Gradimir V. Milovanović
Affiliation:
Department of Mathematics, University of Niš, Faculty of Electronic Engineering, P. O. Box 73, 18000 Niš Serbia; e-mail: grade@ni.ac.yu.
Miodrag M. Spalević
Affiliation:
Department of Mathematics and Informatics, University of Kragujevac, Faculty of Science, P. O. Box 60, 34000 Kragujevac, Serbia; e-mail: spale@kg.ac.yu.
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Abstract

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For Gauss–Turán quadrature formulae with an even weight function on the interval [−1, 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than I, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some ℓ2-error estimates are considered.

Keywords

Gauss–Turán quadratureerror termmonotonicityweight functionanalytic functions

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 4 , April 2007, pp. 567 - 581
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bernstein, S., “Sur les polynomes orthogonaux relatifs à un segment fini”, J. Math. Pures Appl. 9 (1930) 127–177.Google Scholar
[2]Brass, H., “Monotonie bei den Quadraturverfahren von Gauss and Newton—Cotes”, Numer. Math. 30 (1978) 349–354.CrossRefGoogle Scholar
[3]Gautschi, W. and Milovanović, G. V., “S-orthogonality and construction of Gauss–Turán type quadrature formulae”, J. Comput. Appl. Math. 86 (1997) 205–218.CrossRefGoogle Scholar
[4]Gautschi, W. and Varga, R. S., “Error bounds for Gaussian quadrature of analytic functions”, SIAM J. Numer. Anal. 20 (1983) 1170–1186.CrossRefGoogle Scholar
[5]Ghizzetti, A. and Ossicini, A., Quadrature Formulae (Akademie Verlag, Berlin, (1970)CrossRefGoogle Scholar
[6]Gori, L. and Micchelli, C. A., “On weight functions which admit explicit Gauss–Turán quadrature formulas”, Math. Comp. 69 (1996) 269–282.Google Scholar
[7]Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products (Academic Press, New York, (1980)Google Scholar
[8]Hunter, D. B., “Some error expansions for Gaussian quadrature”, BIT 35 (1995) 64–82.CrossRefGoogle Scholar
[9]Milovanović, G. V., “Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation”, J. Comput. Appl. Math. 127 (2001) 267–286.CrossRefGoogle Scholar
[10]Milovanović, G. V. and Spalević, M. M., “Quadrature formulae connected to σ-orthogonal polynomial”, J. Comput. Appl. Math. 140 (2002) 619–637.CrossRefGoogle Scholar
[11]Milovanović, G. V. and Spalević, M. M., “Error bounds for Gauss–Turán quadrature formulae of analytic functions”, Math. Comp. 72 (2003) 855–187.CrossRefGoogle Scholar
[12]Milovanović, G. V. and Spalević, M. M., “Error analysis in some Gauss–Turán–Radau and Gauss–Turán–Lobatto quadratures for analytic functions”, J. Comput. Appl. Math. 164—165 (2004) 569–586.CrossRefGoogle Scholar
[13]Milovanović, G. V. and Spalević, M. M., “Bounds of the error of Gauss–Turán–type quadratures”, J. Comput. Appl. Math. 178 (2005) 333–346.CrossRefGoogle Scholar
[14]Milovanović, G. V. and Spalević, M. M., “An error expansion for Gauss–Turán quadratures and L1-estimates of the remainder term”, BIT 45 (2005) 117–136.CrossRefGoogle Scholar
[15]Milovanović, G. V. and Spalević, M. M., “Quadrature rules with multiple nodes for evaluating integrals with strong singularities”, J. Comput. Appl. Math. 189 (2006) 689–702.CrossRefGoogle Scholar
[16]Milovanović, G. V., Spalević, M. M. and Cvetković, A. S., “Calculation of Gaussian type quadratures with multiple nodes”, Math. Comput. Modelling 39 (2004) 325–347.CrossRefGoogle Scholar
[17]Newman, D. J., “Monotonicity of quadrature approximations”, Proc. Amer. Math. Soc. 42 (1974) 251–257.CrossRefGoogle Scholar
[18]Ossicini, A. and Rosati, F., “Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli”, Boll. Unione Mat. Ital. 11 (1975) 224–237.Google Scholar
[19]Stenger, F., “Bounds on the error of Gauss-type quadratures”, Numer. Math. 8 (1966) 150–160.CrossRefGoogle Scholar
[20]Wilf, H., “Exactness conditions in numerical quadrature”, Numer. Math. 6 (1964) 315–319.CrossRefGoogle Scholar