Abstract
A survey is given on contour integration methods for estimating the remainder of quadrature rules involving analytic functions. In addition to historical remarks, recent results are summarized concerning the remainder of Gauss-type formulae.
This work was supported in part by the National Science Foundation under grant DMS-9023403.
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References
Akrivis, G., Fehlerabschätzungen für Gauss-Quadraturformeln, Numer. Math. 44 (1984), 261–278.
Akrivis, G., Die Fehlernorm spezieller Gauss-Quadraturformeln, in Constructive Methods for the Practical Treatment of Integral Equations (G. Hämmerlin and K.-H. Hoffmann, eds.), pp. 13–19. ISNM 73, Birkhäuser, Basel, 1985.
Akrivis, G., The error norm of certain Gaussian quadrature formulae, Math. Comp. 45 (1985), 513–519.
Akrivis, G. and Burgstaller, A., Fehlerabschätzungen für nichtsymmetrische Gauss-Quadraturformeln, Numer. Math. 47 (1985), 535–543.
Barnhill, R.E., Complex quadratures with remainders of minimum norm, Numer. Math. 7 (1965), 384–390.
Barnhill, R.E., Optimal quadratures in L 2(EP). I, II, SIAM J. Numer. Anal. 4 (1967), 390–397; ibid. 534-541.
Barnhill, R.E., Asymptotic properties of minimum norm and optimal quadratures, Numer. Math. 12 (1968), 384–393.
Barnhill, R.E. and Wixom, J.A., Quadratures with remainders of minimum norm. I, II, Math. Comp. 21 (1967), 66–75; ibid. 382-387.
Chawla, M.M. and Kaul, V., Optimal rules for numerical integration round the unit circle, BIT 13 (1973), 145–152.
Chawla, M.M. and Raina, B.L., Optimal quadratures for analytic functions, BIT 12 (1972), 489–502.
Chen, T.H.C., An upper bound for the Gauss-Legendre quadrature error for analytic functions, BIT 22 (1982), 530–532.
Davis, P.J., Errors of numerical approximation for analytic functions, J. Rational Mech. Anal. 2 (1953), 303–313.
Davis, P.J. and Rabinowitz, P., On the estimation of quadrature errors for analytic functions, Math. Tables Aids Comput. 8 (1954), 193–203.
Davis, P.J. and Rabinowitz, P., Methods of Numerical Integration, 2nd ed., Academic Press, Orlando, 1984.
Eckhardt, U., Einige Eigenschaften Wilfscher Quadraturformeln, Numer. Math. 12 (1968), 1–7.
Fock, V., On the remainder term of certain quadrature formulae (Russian), Bull. Acad. Sci. Leningrad (7) (1932), 419–448.
Gautschi, W., A survey of Gauss-Christoffel quadrature formulae, in E.B. Christoffel (P.L. Butzer and F. Fehér, eds.), pp. 72–147. Birkhäuser, Basel, 1981.
Gautschi, W., Minimal solutions of three-term recurrence relations and orthogonal polynomials, Math. Comp. 36 (1981), 547–554.
Gautschi, W., On Padé approximants associated with Hamburger series, Calcolo 20 (1983), 111–127.
Gautschi, W., On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures, Rocky Mountain J. Math., 1991, to appear.
Gautschi, W. and Li, S., The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, J. Comput. Appl. Math. 33.(1990), 315–329.
Gautschi, W. and Varga, R.S., Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983), 1170–1186.
Gautschi, W., Tychopoulos, E. and Varga, R.S., A note on the contour integral representation of the remainder term for a Gauss-Chebyshev quadrature rule, SIAM J. Numer. Anal. 27 (1990), 219–224.
Hämmerlin, G., Über ableitungsfreie Schranken für Quadraturfehler, Numer. Math. 5 (1963), 226–233.
Hämmerlin, G., Über ableitungsfreie Schranken für Quadraturfehler, II. Ergänzungen und Möglichkeiten zur Verbesserung, Numer. Math. 7 (1965), 232–237.
Hämmerlin, G., Zur Abschätzung von Quadraturfehlern für analytische Funktionen, Numer. Math. 8 (1966), 334–344.
Heine, E., Anwendungen der Kugelfunctionen und der verwandten Functionen, 2nd ed., Reimer, Berlin, 1881. [I. Theil: Mechanische Quadratur, 1-31.]
Henrici, P., Applied and Computational Complex Analysis, vol. 3, Wiley, New York, 1986.
Hermite, C., Sur la formule d’interpolation de Lagrange, J. Reine Angew. Math. 84 (1878), 70–79. [Oeuvres III, 432-443.]
Hunter, D.B., Some properties of orthogonal polynomials, Math. Comp. 29 (1975), 559–565.
Hunter, D.B. and Okecha, G.E., A modified Gaussian quadrature rule for integrals involving poles of any order, BIT 26 (1986), 233–240.
Knauff, W. and Kress, R., Optimale Approximation mit Nebenbedingungen an lineare Funktionale auf periodischen Funktionen, Numer. Math. 25 (1975/76), 149–152.
Paulik, A., On the optimal approximation of bounded linear functionals in Hilbert spaces of analytic functions, BIT 16 (1976), 298–307.
Raina, B.L. and Kaul, N., A class of optimal quadrature formulae, IMA J. Numer. Anal. 3 (1983), 119–125.
Raina, B.L. and Kaul, N., Estimating errors of certain Gauss quadrature formulae, Calcolo 22 (1985), 229–240.
Rivlin, T.J., Gauss quadrature for analytic functions, in Orthogonal Polynomials and their Applications (M. Alfaro et al., eds.), pp. 178–192. Lecture Notes Math. 1329, Springer, Berlin, 1988.
Smith, H.V., Global error bounds for Gauss-Gegenbauer quadrature, BIT 21 (1981), 481–490.
Smith, H.V., Global error bounds for the Clenshaw-Curtis quadrature formula, BIT 22 (1982), 395–398.
Takahasi, H. and Mori, M., Error estimation in the numerical integration of analytic functions, Rep. Comput. Centre Univ. Tokyo 3 (1970), 41–108.
Takahasi, H. and Mori, M., Estimation of errors in the numerical quadrature of analytic functions, Applicable Anal. 1 (1971), 201–229.
Wimp, J., Computation with Recurrence Relations, Pitman, Boston, 1984.
Yanagiwara, H., A new method of numerical integration of Gaussian type (Japanese), Bull. Fukuoka Gakugei Univ. III 6 (1956), 19–25.
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Gautschi, W. (1992). Remainder Estimates for Analytic Functions. In: Espelid, T.O., Genz, A. (eds) Numerical Integration. NATO ASI Series, vol 357. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2646-5_11
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DOI: https://doi.org/10.1007/978-94-011-2646-5_11
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