×

\(d\)-fold Hermite-Gauss quadrature. (English) Zbl 0993.65033

Systems of orthogonal rational functions are constructed from systems of orthogonal polynomials and explicating the \((2^dn)\)-point \(d\)-fold Hermite-Gauss quadrature formula \[ \int_{-\infty}^\infty f(x)\exp\bigl(-[v^{[d](\gamma,\lambda)}(x)]^2\bigr) dx= \sum_{k=1}^{2^dn}f\bigl(h^{(\gamma,\lambda)}_{d,n,k}\bigr)H^{(\gamma,\lambda)}_{d,n,k}+E^{(\gamma,\lambda)}_{d,n}[f(x)] \] with parameters \(\gamma,\lambda>0\), where \(v^{[d](\gamma,\lambda)}(x)\) is the \(d\)-fold composition of \(v^{[d](\gamma,\lambda)}(x)=(1/\lambda)(x-\gamma/x)\) and where the abscissas \(h^{(\gamma,\lambda)}_{d,n,k}\) and the weight \(H^{(\gamma,\lambda)}_{d,n,k}\) are given recusively in terms of the abscissas and weights associated with the classical Hermite-Gauss quadrature. Error analysis, tables of numerical values for nodes, and examples and comparisons are included. These results extend those presented by P. E. Gaustafon and B. A. Hagler [J. Comput. Appl. Math. 105, No. 1-2, 317-326 (1999; Zbl 0949.65019)]; by B. A. Hagler [Ph. D. Thesis, University of Colorada (1997) and ibid. 104, No. 2, 163-171 (1999; Zbl 0949.41020)]; and by B. A. Hagler, W. B. Jones, and W. J. Thorn [Lect. Notes Pure Appl. Math. 199, 187-208 (1998; Zbl 0932.33005)].
Reviewer: Feng Qi (Henan)

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bultheel, A.; Diaz-Mendoza, C.; González-Vera, P.; Orive, R., Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part III: the unbounded case, J. Comput. Appl. Math., 87, 95-117 (1997) · Zbl 0891.41011
[2] Bultheel, A.; Diaz-Mendoza, C.; González-Vera, P.; Orive, R., Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part II: convergence, J. Comput. Appl. Math., 77, 53-76 (1997) · Zbl 0865.41017
[3] Bultheel, A.; González-Vera, P.; Hendriksen, E.; Njåstad, O., Orthogonal Rational Functions (1999), Cambridge University Press: Cambridge University Press New York · Zbl 1014.42017
[4] Bultheel, A.; González-Vera, P.; Orive, R., Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part I: algebraic aspects, J. Comput. Appl. Math., 65, 57-72 (1995) · Zbl 0847.41015
[5] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach New York · Zbl 0389.33008
[6] Cochran, L.; Clement Cooper, S., Orthogonal Laurent polynomials on the real line, (Cooper, S. C.; Thron, W. J., Continued Fractions and Orthogonal Functions: Theory and Applications, Proceedings, Leon, Norway, 1992. Continued Fractions and Orthogonal Functions: Theory and Applications, Proceedings, Leon, Norway, 1992, Lecture Notes in Pure and Applied Mathematics (1993), Marcel Dekker: Marcel Dekker New York), 47-100 · Zbl 0810.42013
[7] Cooper, S. C.; Gustafson, P., The strong Chebyshev distribution and orthogonal Laurent polynomials, J. Approx. Theory, 92, 3, 361-378 (1998) · Zbl 0930.42011
[8] Gautschi, W., A survey of Gauss-Christoffel quadrature formulae, (Butzer, P. L.; Feher, F., Christoffel Memorial Volume (1981), Birkhäuser: Birkhäuser Basel), 72-147 · Zbl 0479.65001
[9] Gustafson, P. E.; Hagler, B. A., Gaussian quadrature rules and numerical examples for strong extensions of mass distribution functions, J. Comput. Appl. Math., 105, 317-326 (1999) · Zbl 0949.65019
[10] B.A. Hagler, A transformation of orthogonal polynomial sequences into orthogonal Laurent polynomial sequences, Ph.D. Thesis, University of Colorado, 1997.; B.A. Hagler, A transformation of orthogonal polynomial sequences into orthogonal Laurent polynomial sequences, Ph.D. Thesis, University of Colorado, 1997.
[11] Hagler, B. A., Laurent-Hermite-Gauss quadrature, J. Comput. Appl. Math., 104, 163-171 (1999) · Zbl 0949.41020
[12] Hagler, B. A.; Jones, W. B.; Thron, W. J., Orthogonal Laurent polynomials of Jacobi, Hermite and Laguerre types, (Jones, W. B.; Sri Ranga, A., Orthogonal Functions, Moment Theory, and Continued Fractions: Theory and Applications. Orthogonal Functions, Moment Theory, and Continued Fractions: Theory and Applications, Lecture Notes in Pure and Applied Mathematics Series, Vol. 199 (1998), Marcel Dekker: Marcel Dekker New York), 187-208 · Zbl 0932.33005
[13] Hildebrand, F. B., Introduction to Numerical Analysis (1956), McGraw-Hill: McGraw-Hill New York · Zbl 0070.12401
[14] Jones, W. B.; Njåstad, Olav, Orthogonal Laurent polynomials and strong moment theory: a survey, J. Comput. Appl. Math., 105, 51-91 (1999) · Zbl 0943.30024
[15] Jones, W. B.; Thron, W. J., Orthogonal Laurent polynomials and Gaussian quadrature, (Gustafson, K. E.; Reinhardt, W. P., Quantum Mechanics in Mathematics, Chemistry, and Physics (1981), Plenum Press: Plenum Press New York), 449-455
[16] O. Njåstad, W.J. Thron, in: H. Waadeland, H. Wallin (Eds.), The theory of sequences of orthogonal \(L\)-polynomials, Det Kong. Norske Vid. Selsk. Skr. 1 1983, pp. 54-91.; O. Njåstad, W.J. Thron, in: H. Waadeland, H. Wallin (Eds.), The theory of sequences of orthogonal \(L\)-polynomials, Det Kong. Norske Vid. Selsk. Skr. 1 1983, pp. 54-91. · Zbl 0643.33011
[17] Sri Ranga, A., Symmetric orthogonal polynomials and the associated orthogonal \(L\)-polynomials, Proc. Amer. Math. Soc., 123, 10, 3135-3141 (1995) · Zbl 0861.33008
[18] Sri Ranga, A.; de Andrade, E. X.L.; Phillips, G. M., Associated symmetric quadrature rules, J. Comput. Appl. Math., 21, 175-183 (1996) · Zbl 0854.41026
[19] Salzer, H. E.; Zucker, R.; Capuano, R., Table of zeros and weight factors of the first twenty hermite polynomials, J. Research NBS, 48, 111-116 (1952)
[20] G. Szegö, Orthogonal Polynomials, AMS Colloquium Publications, Vol. 23, AMS, Providence, RI, 1975.; G. Szegö, Orthogonal Polynomials, AMS Colloquium Publications, Vol. 23, AMS, Providence, RI, 1975.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.