On estimates for the weights in Gaussian quadrature in the ultraspherical case
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- by Klaus-Jürgen Förster and Knut Petras PDF
- Math. Comp. 55 (1990), 243-264 Request permission
Abstract:
In this paper the Christoffel numbers $a_{v,n}^{(\lambda )G}$ for ultraspherical weight functions ${w_\lambda }$, ${w_\lambda }(x) = {(1 - {x^2})^{\lambda - 1/2}}$, are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by $\theta _{v,n}^{(\lambda )}$ the trigonometric representation of the Gaussian nodes, we obtain for $\lambda \in [0,1]$ the inequalities \[ \begin {array}{*{20}{c}} {\frac {\pi }{{n + \lambda }}{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}}} \right \}} \\ { \leq a_{v,n}^{(\lambda )G} \leq \frac {\pi }{{n + \lambda }}\;{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}} \\ \end {array} \] and similar results for $\lambda \notin (0,1)$. Furthermore, assuming that $\theta _{v,n}^{(\lambda )}$ remains in a fixed closed interval, lying in the interior of $(0,\pi )$ as $n \to \infty$, we show that, for every fixed $\lambda > - 1/2$, \[ a_{v,n}^{(\lambda )G} = \frac {\pi }{{n + \lambda }}\;{\sin ^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}} - \frac {{\lambda (1 - \lambda )\;[3(\lambda + 1)(\lambda - 2) + 4{{\sin }^2}\theta _{v,n}^{(\lambda )}]}}{{8{{(n + \lambda )}^4}{{\sin }^4}\theta _{v,n}^{(\lambda )}}}} \right \} + O({n^{ - 7}}).\]References
-
M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions, 7th printing, Dover, New York, 1970.
- Shafique Ahmed, Martin E. Muldoon, and Renato Spigler, Inequalities and numerical bounds for zeros of ultraspherical polynomials, SIAM J. Math. Anal. 17 (1986), no. 4, 1000–1007. MR 846402, DOI 10.1137/0517070
- Giampietro Allasia, Proprietà statistiche delle formule di quadratura, Rend. Sem. Mat. Univ. e Politec. Torino 35 (1976/77), 339–348 (1978) (Italian, with English summary). MR 0483317
- S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 17–26 (Russian, with French summary). MR 0002699
- Helmut Brass, Quadraturverfahren, Studia Mathematica: Skript, vol. 3, Vandenhoeck & Ruprecht, Göttingen, 1977 (German). MR 0443305 —, Zur Quadraturtheorie konvexer Funktionen, Numerical Integration (G. Hämmerlin, ed.), ISNM 57, Birkhäuser, Basel, 1982, pp. 34-47.
- C. Eugene Buell, The zeros of Jacobi and related polynomials, Duke Math. J. 2 (1936), no. 2, 304–316. MR 1545925, DOI 10.1215/S0012-7094-36-00225-9 P. L. Chebyshev, Sur les quadratures, J. Math. Pures Appl. (2) 19 (1874), 19-34.
- P. J. Davis and P. Rabinowitz, Some geometrical theorems for abscissas and weights of Gauss type, J. Math. Anal. Appl. 2 (1961), 428–437. MR 128613, DOI 10.1016/0022-247X(61)90021-X —, Methods of numerical integration, 2nd ed., Academic Press, London, 1984.
- Á. Elbert, Some inequalities concerning Bessel functions of first kind, Studia Sci. Math. Hungar. 6 (1971), 277–285. MR 310308
- F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of gamma functions, Pacific J. Math. 1 (1951), 133–142. MR 43948, DOI 10.2140/pjm.1951.1.133 K.-J. Förster, A comparison theorem for linear functionals and its applications in quadrature, Numerical Integration (G. Hämmerlin, ed.), ISNM 57, Birkhäuser, Basel, 1982, pp. 66-76. K.-J. Förster and K. Petras, On the zeros of ultraspherical polynomials and Bessel functions (submitted).
- L. Gatteschi and G. Vinardi, The degree of accuracy of quadrature formulas of Čebyšev type, Calcolo 15 (1978), no. 1, 59–85 (Italian, with English summary). MR 524809, DOI 10.1007/BF02576046 L. Gatteschi, Una nuova rappresentazione asintotica dei polinomi ultrasferici, Calcolo 16 (1979), 447-458.
- Luigi Gatteschi, New inequalities for the zeros of Jacobi polynomials, SIAM J. Math. Anal. 18 (1987), no. 6, 1549–1562. MR 911648, DOI 10.1137/0518111
- Luigi Gatteschi, Uniform approximation of Christoffel numbers for Jacobi weight, Numerical integration, III (Oberwolfach, 1987) Internat. Schriftenreihe Numer. Math., vol. 85, Birkhäuser, Basel, 1988, pp. 49–59. MR 1021522
- Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060
- A. Ghizzetti and A. Ossicini, Quadrature formulae, Academic Press, New York, 1970. MR 0269116, DOI 10.1007/978-3-0348-5836-6 G. Hämmerlin (ed.), Numerical integration, ISNM 57, Birkhäuser Verlag, Basel, 1982.
- David K. Kahaner, On equal and almost equal weight quadrature formulas, SIAM J. Numer. Anal. 6 (1969), 551–556. MR 286279, DOI 10.1137/0706049
- Darryl Katz, Optimal quadrature points for approximating integrals when function values are observed with error, Math. Mag. 57 (1984), no. 5, 284–290. MR 765644, DOI 10.2307/2689602
- Vladimir Ivanovich Krylov, Approximate calculation of integrals, The Macmillan Company, New York-London, 1962, 1962. Translated by Arthur H. Stroud. MR 0144464
- M. Kütz, On the asymptotic behaviour of some Cotes numbers, Z. Angew. Math. Mech. 66 (1986), no. 8, 373–375. MR 864494, DOI 10.1002/zamm.19860660815
- H. N. Laden, Fundamental polynomials of Lagrange interpolation and coefficients of mechanical quadrature, Duke Math. J. 10 (1943), 145–151. MR 7814, DOI 10.1215/S0012-7094-43-01013-0
- Giovanni Monegato, Some new inequalities related to certain ultra-spherical polynomials, SIAM J. Math. Anal. 11 (1980), no. 4, 663–667. MR 579558, DOI 10.1137/0511061
- Alessandro Ossicini, Sulle costanti di Christoffel della formula di quadratura di Gauss-Jacobi, Ist. Lombardo Accad. Sci. Lett. Rend. A 101 (1967), 169–180 (Italian). MR 241872
- Alexander M. Ostrowski, On trends and problems in numerical approximation, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Publication of the Mathematics Research Center, U.S. Army, the University of Wisconsin, no. 1, University of Wisconsin Press, Madison, Wis., 1959, pp. 3–10. Edited by R. E. Langer. MR 0100956
- Petras Knut, Asymptotic behaviour of Peanokernels of fixed order, Numerical integration, III (Oberwolfach, 1987) Internat. Schriftenreihe Numer. Math., vol. 85, Birkhäuser, Basel, 1988, pp. 186–198. MR 1021534 J. A. Shohat and C. Winston, On mechanical quadratures, Rend. Circ. Mat. Palermo 58 (1934), 153-160.
- A. H. Stroud, Estimating quadrature errors for functions with low continuity, SIAM J. Numer. Anal. 3 (1966), 420–424. MR 215528, DOI 10.1137/0703036
- A. H. Stroud and Don Secrest, Gaussian quadrature formulas, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0202312 G. Szegö, Asymptotische Entwicklungen der Jacobischen Polynome, Schriften der Königsberger Gelehrten Gesellschaft 10 (1933), 35-112. [Collected papers, vol. 2, pp. 401-477.]
- Gabriel Szegö, Inequalities for the zeros of Legendre polynomials and related functions, Trans. Amer. Math. Soc. 39 (1936), no. 1, 1–17. MR 1501831, DOI 10.1090/S0002-9947-1936-1501831-2 —, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1939 (revised edition 1985).
- Francesco G. Tricomi, Sugli zeri dei polinomi sferici ed ultrasferici, Ann. Mat. Pura Appl. (4) 31 (1950), 93–97 (Italian). MR 42555, DOI 10.1007/BF02428258
- E. L. Whitney, Estimates of weights in Gauss-type quadrature, Math. Comp. 19 (1965), 277–286. MR 202313, DOI 10.1090/S0025-5718-1965-0202313-2
- C. Winston, On mechanical quadratures formulae involving the classical orthogonal polynomials, Ann. of Math. (2) 35 (1934), no. 3, 658–677. MR 1503185, DOI 10.2307/1968756
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 55 (1990), 243-264
- MSC: Primary 65D32; Secondary 41A55
- DOI: https://doi.org/10.1090/S0025-5718-1990-1023758-1
- MathSciNet review: 1023758