Weighted quadrature formulas  for semi-infinite range integrals

Gradimir V. Milovanović

doi:10.33993/jnaat441-1063

Authors

Gradimir V. Milovanović Serbian Academy of Sciences and Arts http://orcid.org/0000-0002-3255-8127

DOI:

https://doi.org/10.33993/jnaat441-1063

Keywords:

Gaussian quadrature rules, nodes, Christoffel numbers, non-exponentially decreasing integrands.

Abstract views: 432

Abstract

Weighted quadrature formulas on the half line \((a,+\infty)\), \(a>0\), for non-exponentially decreasing integrands are developed. Such \(n\)-point quadrature rules are exact for all functions of the form \(x\mapsto x^{-2}P(x^{-1})\), where \(P\) is an arbitrary algebraic polynomial of degree at most \(2n-1\). In particular, quadrature formulas with respect to the weight function \(x\mapsto w(x)=x^\beta\log^m x\) (\(0\le \beta<1\), \(m\in \mathbb{N}_0\)) are considered and several numerical examples are included.

Downloads

Download data is not yet available.

References

A.S. Cvetkovic and G.V. Milovanovic, The Mathematica Package “Orthogonal Polynomials”, Facta Univ. Ser. Math. Inform.,19, pp. 17-36, 2004.

G.A. Evans, Some new thoughts on Gauss-Laguerre quadrature, Int. J. Comput. Math. 82, pp. 721-730, 2005. http://doi.org/10.1080/00207160512331323399 DOI: https://doi.org/10.1080/00207160512331323399

W. Gautschi, On generating orthogonal polynomials , SIAM J. Sci. Statist. Comput., 3, pp. 289–317, 1982. http:// doi.org/10.1137/0903018 DOI: https://doi.org/10.1137/0903018

W. Gautschi, Algorithm 726: ORTHPOL – A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Trans. Math. Software, 20, pp. 21-62, 1994. http://doi.org/10.1145/174603.174605 DOI: https://doi.org/10.1145/174603.174605

W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Clarendon Press, Oxford, 2004. DOI: https://doi.org/10.1093/oso/9780198506720.001.0001

W. Gautschi, Gauss quadrature routines for two classes of logarithmic weight functions, Numer. Algor. 55, pp. 265-277, 2010. http://doi.org/10.1007/s11075-010-9366-0 DOI: https://doi.org/10.1007/s11075-010-9366-0

G. Golub and J.H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23, pp. 221–230, 1969. http://doi.org/10.2307/2004418 DOI: https://doi.org/10.1090/S0025-5718-69-99647-1

A. Markov, Sur la méthode de Gauss pour le calcul approché des intégrales , Math. Ann., 25, pp. 427 432, 1885. http://doi.org/10.1007/BF01443287 DOI: https://doi.org/10.1007/BF01443287

G. Mastroianni and G. Monegat, Truncated quadrature rules over (0, ?) and Nyström-type methods , SIAM J. Numer. Anal. 41, pp. 1870-1892, 2003. http://doi.org/10.1137/S0036142901391475 DOI: https://doi.org/10.1137/S0036142901391475

G. Mastroianni and G.V. Milovanovic, Interpolation Processes – Basic Theory and Applications, Springer-Verlag, Berlin – Heidelberg – New York, 2008.

G.V. Milovanovic, Construction and applications of Gaussian quadratures with non-classical and exotic weight function, Stud. Univ. Babes-Bolyai Math., 60, pp. 211-233, 2015.

G.V. Milovanovic, Generalized Gaussian quadratures for integrals with logarithmic singularity, FILOMAT (to appear). https://www.jstor.org/stable/24898684

G.V. Milovanovic and A.S. Cvetkovic, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica, 26, pp. 169-184, 2012.

G.V. Milovanovic, T.S. Igic and D. Turnic, Generalized quadrature rules of Gaussian type for numerical evaluation of singular integrals, J. Comput. Appl. Math. 278, pp. 306-25, 2015. http://doi.org/10.1016/j.cam.2014.10.009 DOI: https://doi.org/10.1016/j.cam.2014.10.009

C. Posse, Sur les quadratures, Nouv. Ann. Math. (2) 14, pp. 49-62, 1875.

T.J. Stieltjes, Quelques recherches sur la théorie des quadratures dites mécaniques, Ann. Sci. Ec. Norm. Paris, Sér. 2, 1, pp. 409-426, 1884 DOI: https://doi.org/10.24033/asens.245

J.V. Uspensky, On the convergence of quadrature formulas related to an infinite interval, Trans. Amer. Math. Soc., 30, pp. 542-559, 1928. http://doi.org/10.1090/S0002-9947-1928-1501444-8 DOI: https://doi.org/10.1090/S0002-9947-1928-1501444-8

Zhenhua Xu and G.V. Milovanovic, Efficient method for the computation of oscillatory Bessel transform and Bessel Hilbert transform (submitted). https://doi.org/10.1016/j.cam.2016.05.031 DOI: https://doi.org/10.1016/j.cam.2016.05.031