Abstract
The Fejér and Clenshaw–Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like \(O(\varrho^{-2N})\), where \(\varrho\) is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw–Curtis rules is almost indistinguishable from that of the more celebrated Gauss–Legendre quadrature rule. For larger N, however, the error decreases at the rate \(O(\varrho^{-N})\), i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.
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Ablowitz M.J. and Fokas A.S. (2003). Complex Variables: Introduction and Applications, 2nd edn. Cambridge University Press, Cambridge
Abramowitz M. and Stegun I.A. (1970). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publications Inc., New York
Brass, H.: Quadraturverfahren. Vandenhoeck & Ruprecht, Göttingen. Studia Mathematica, Skript 3 (1977)
Chawla M.M. (1968). Error estimates for the Clenshaw–Curtis quadrature. Math. Comp. 22: 651–656
Davis P.J. (1975). Interpolation and Approximation. Dover Publications Inc., New York
Davis P.J. and Rabinowitz P. (1984). Methods of Numerical Integration, 2nd edn. Academic Press Inc., Orlando, FL
Elliott, D., Johnston, B.M., Johnston, P.R.: Clenshaw–Curtis and Gauss–Legendre quadrature for boundary element integrals (2006, submitted)
Erdélyi A., Magnus W., Oberhettinger F. and Tricomi F.G. (1953). Higher Transcendental Functions. vol. I. McGraw-Hill Book Company Inc., New York
Favati P., Lotti G. and Romani F. (1993). Bounds on the error of Fejér and Clenshaw–Curtis type quadrature for analytic functions. Appl. Math. Lett. 6: 3–8
Ferreira, C., López, J.L.: Asymptotic expansions of the Hurwitz–Lerch zeta function. J. Math. Anal. Appl. 298 (2004)
Gautschi W. (1981). A survey of Gauss–Christoffel quadrature formulae. In: Christoffel, E.B. (eds) (Aachen/ Monschau, 1979), pp 72–147. Birkhäuser, Basel
Krylov V.I. (1962). Approximate Calculation of Integrals. The Macmillan Co., New York
Mason J.C. and Handscomb D.C. (2003). Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton, FL
Notaris S.E. (2006). Integral formulas for Chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions. Math. Comp. 75: 1217–1231
Petras K. (1995). Gaussian integration of Chebyshev polynomials and analytic functions. Numer. Algorithms 10: 187–202
Petras K. (1998). Gaussian versus optimal integration of analytic functions. Constr. Approx. 14: 231–245
Petras K. (2000). On the Smolyak cubature error for analytic functions. Adv. Comput. Math. 12: 71–93
Scherer R. and Schira T. (2000). Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems. Numer. Math. 84: 497–518
Szegő, G.: Orthogonal polynomials, 4th edn. American Mathematical Society, Providence, R.I. (1975). American Mathematical Society, Colloquium Publications, vol. XXIII
Takahasi H. and Mori M. (1971). Estimation of errors in the numerical quadrature of analytic functions. Appl. Anal. 1: 201–229
Trefethen, L.N.: Is Gauss quadrature better than Clenshaw–Curtis? SIAM Review (2007, to appear)
Weideman J.A.C. and Laurie D.P. (2000). Quadrature rules based on partial fraction expansions. Numer. Algorithms 24: 159–178
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This work was supported by the Royal Society of the UK and the National Research Foundation of South Africa under the South Africa–UK Science Network Scheme. The first author also acknowledges grant FA2005032300018 of the NRF.
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Weideman, J.A.C., Trefethen, L.N. The kink phenomenon in Fejér and Clenshaw–Curtis quadrature. Numer. Math. 107, 707–727 (2007). https://doi.org/10.1007/s00211-007-0101-2
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DOI: https://doi.org/10.1007/s00211-007-0101-2