Orthogonal polynomials and generalized Gauss-Rys quadrature formulae
DOI:
https://doi.org/10.48129/kjs.v49i1.10665Keywords:
Quadrature rule, orthogonal polynomials, recurrence relation, nodes, weightsAbstract
Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type with respect to the even weight function
$\omega^{\lambda}(t;x)=\exp(-x t^2)(1-t^2)^{\lambda-1/2}$ on $(-1,1)$, with parameters $\lambda>-1/2$ and $x>0$, are considered.
For $\lambda=1/2$ these quadrature rules reduce to the so-called
Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis, Rys, King (1976 and 1983), Sagar (1992), Schwenke (2014), Shizgal (2015), King (2016), Milovanovi\'c (2018), etc. In this generalized case
the method of modified moments is used, as well as a transformation of quadratures on $(-1, 1)$ with $N$ nodes to ones on $(0,1)$ with only $(N+1)/2$
nodes. Such an approach provides a stable and very efficient numerical construction.
References
harvarditem{Asanov textit{et al.}}{1917}{ASANOV} Asanov, A., Matanova, K. B., Asanov, R. A.: (2017) A class of linear and nonlinear Fredholm integral equations of the third kind. textit{Kuwait J. Sci.} {bf 44} (1), 17--28.
harvarditem{Cvetkovi'c and Milovanovi'c}{2004}{ASCGVM} Cvetkovi'c, A. S., Milovanovi'c, G. V. (2004) The Mathematica package ``OrthogonalPolynomials''. textit{Facta Univ. Ser. Math. Inform.} {bf 19}, 17--36.
harvarditem{Dupuis textit{et al.}}{1976}
{dupuis} Dupuis, M., Rys, J., King, H.F. (1976) Evaluation of molecular integrals over Gaussian basis functions. textit{J. Chem. Phys.} {bf 65}, 111--116.
harvarditem{Gautschi}{1982}
{GautschiSIAM} Gautschi, W. (1982) On generating orthogonal polynomials. textit{SIAM J. Sci.
Statist. Comput.} {bf 3}, 282--317.
harvarditem{Gautschi}{2004}
{GautschiOxford} Gautschi, W. (2004)
textit{Orthogonal Polynomials: Computation and Approximation}. Clarendon Press, Oxford.
harvarditem{Gautschi}{2005}{Gautschi05}
Gautschi, W. (2005) Orthogonal polynomials (in Matlab). textit{J. Comput. Appl. Math.} {bf178}, 215--234.
harvarditem{Gautschi}{2016}
{Gautschi2016} Gautschi, W. (2016)
textit{Orthogonal Polynomials in Matlab: Exercises and Solutions, Software-Enviroments-Tools}. SIAM, Philadelphia, PA.
harvarditem{Gautschi}{2018}{Gautschi18}
Gautschi, W. (2018) textit{A Software Repository for Orthogonal Polynomials. Software, Environments and Tools}. Vol. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
harvarditem{Golub and Welsch}{1969}{Golub} Golub, G., Welsch, J. H. (1969) Calculation of Gauss quadrature rules. textit{Math. Comp.} {bf 23}, 221--230.
harvarditem{King}{2016}{King} King, H.F. (2016) Strategies for evaluation of Rys roots and weights. textit{J. Phys. Chem. A} {bf 120}, 9348--9351.
harvarditem{Masjad-Jamei and Milovanovi'c}{2017}{Masjad} Masjad-Jamei, M., Milovanovi'c, G. V. (2017) Construction of Gaussian formulas for even weight function. textit{Appl. Anal. Discrete Math.} {bf 11}, 177--198.
harvarditem{Mastroianni and Milovanovi'c}{2008}{GMGVM} Mastroianni, G., Milovanovi'c, V. G. (2008) textit{Interpolation Processes -- Basic
Theory and Applications}. Springer Monographs in Mathematics, Springer -- Verlag, Berlin --Heindelberg.
harvarditem{Mastroianni and Milovanovi'c}{2009}{GMGVMNLAA} Mastroianni, G., Milovanovi'c, V. G. (2009) Well-conditioned matrices for numerical treatment of Fredholm integral equations of the second kind. textit{Numer. Linear Algebra Appl.} {bf16}, 995--1011.
harvarditem{Milovanovi'c}{2014}{gvmCh11} Milovanovi'c, G. V. (2014) Chapter 11: Orthogonal polynomials on the real line. In: textit{Walter Gautschi: Selected Works with Commentaries, Volume 2} (C. Brezinski, A. Sameh, eds.), pp. 3--16, Birkh"auser, Basel.
harvarditem{Milovanovi'c}{2015}{GVMNAAT15} Milovanovi'c, G. V. (2015) Construction and applications of Gaussian quadratures with nonclassical and exotic weight functions. textit{Stud. Univ. Babec{s}-Bolyai Math.} {bf60}, 211--233.
harvarditem{Milovanovi'c}{2018}{GVMBulletin} Milovanovi'c, G. V. (2018) An efficient computation of parameters in the Rys quadrature formula. textit{Bull. Cl. Sci. Math. Nat. Sci. Math.} {bf 43}, 39--64.
harvarditem{Milovanovi'c}{2019}{GVMIJPAM} Milovanovi'c, G. V. (2019) A note on extraction of orthogonal polynomials from generating function for reciprocal of odd numbers. textit{Indian J. Pure Appl. Math.} {bf 50}, 15--22.
harvarditem{Milovanovi'c and Cvetkovi'c}{2012}{GVMASC} Milovanovi'c, G. V., Cvetkovi'c, A. S. (2012) Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type. textit{Math. Balkanica} {bf 26}, 169--184.
harvarditem{Milovanovi'c and Rathie}{2019}{Mil-Rat19} Milovanovi'c, G. V., Rathie, A. K. (2019) On a quadratic transformation due to Exton and its generalization. textit{Hacet. J. Math. Stat.} {bf48}(6), 1706--1711.
harvarditem{Milovanovi'c textit{et al.}}{2018}{MilParRat1} Milovanovi'c, G. V., Parmar, R. K., Rathie, A. K. (2018) A study of generalized summation theorems for the series ${}_2F_1$ with an applications to Laplace transforms of convolution type integrals involving Kummer's functions ${}_1F_1$. textit{Appl. Anal. Discrete Math.} {bf12}, 257--272.
harvarditem{Ostrovska and Turan}{2017}{SofTur} Ostrovska, S., Turan, M.: On the powers of the Ku-mmer distribution. textit{Kuwait J. Sci.} {bf 44} (2), 1--8.
harvarditem{Prudnikov textit{et al.}}{1986}{Prudnikov} Prudnikov, A. P., Brychkov, Yu. A., Marichev, O. I. (1986) textit{Integrals and Series,
Vol. 2 Special Functions}, %Translated from the Russian by N. M. Queen.
Gordon & Breach Science Publisher, New York.
harvarditem{Rys textit{et al.}}{1983}{rys} Rys, J., Dupuis, M., King, H. F. (1983) Computation of electron repulsion integrals using Rys quadrature method. textit{J. Comput. Chem.} {bf 4}, 154--157.
harvarditem{Sagar and Smith}{1992}{sagar} Sagar, R. P., Smith, V. H. (1992) On the calculation of Rys polynomials and quadratures. textit{Int. J. Quant. Chem.} {bf 42}(4), 827--836.
harvarditem{Schwenke}{2014}{scwenke} Schwenke, R. P. (2014) On the computation of high order Rys quadrature weights and nodes. textit{Comput. Phys. Comm.} {bf 185}, 762--763.
harvarditem{Shizgal}{2015}{Shizgal} Shizgal, B. D. (2015) A novel Rys quadrature algorithm for use in the calculation of electron repuslion integrals. textit{Comput. Theor. Chem.} {bf 1074},
--184.
