Error of the Newton-Cotes and Gauss-Legendre quadrature formulas
Abstract: It was shown by P. J. Davis that the Newton-Cotes quadrature formula is convergent if the integrand is an analytic function that is regular in a sufficiently large region of the complex plane containing the interval of integration. In the present paper, a bound on the error of the Newton-Cotes quadrature formula for analytic functions is derived. Also the bounds on the Legendre polynomial and the Legendre function of the second kind are obtained. These bounds are employed to derive a bound on the error of the Gauss-Legendre quadrature formula for analytic functions.
-  Philip J. Davis and Philip Rabinowitz, Numerical integration, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1967. MR 0211604
-  Philip Davis, On a problem in the theory of mechanical quadratures, Pacific. J. Math. 5 (1955), 669–674. MR 0072258
-  Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR 0157156
-  E. T. Copson, Theory of Functions of a Complex Variable, Oxford Univ. Press, Oxfprd, 1935.
-  M. M. Chawla and M. K. Jain, Error estimates for Gauss quadrature formulas for analytic functions, Math. Comp. 22 (1968), 82–90. MR 0223093, https://doi.org/10.1090/S0025-5718-1968-0223093-3
-  M. M. Chawla, Asymptotic estimates for the error of the Gauss-Legendre quadrature formula, Comput. J. 11 (1968/1969), 339–340. MR 0237096, https://doi.org/10.1093/comjnl/11.3.339
-  Frank Stenger, Bounds on the error of Gauss-type quadratures, Numer. Math. 8 (1966), 150–160. MR 0196936, https://doi.org/10.1007/BF02163184
- P. J. Davis & P. Rabinowitz, Numerical Integration, Blaisdell, Waltham, Mass., 1967. MR 35 #2482. MR 0211604 (35:2482)
- P. J. Davis, "On a problem in the theory of mechanical quadratures," Pacific J. Math., v. 5, 1955, pp. 669-674. MR 17, 255. MR 0072258 (17:255f)
- P. J. Davis, Interpolation and Approximation, Blaisdell, Waltham, Mass., 1963. MR 28 #393. MR 0157156 (28:393)
- E. T. Copson, Theory of Functions of a Complex Variable, Oxford Univ. Press, Oxfprd, 1935.
- M. M. Chawla & M. K. Jain, "Error estimates for Gauss quadrature formulas for analytic functions." Math. Comp., v. 22, 1968, pp. 82-90. MR 36 #6142. MR 0223093 (36:6142)
- M. M. Chawla, "Asymptotic estimates for the error of the Gauss-Legendre quadrature formula," Comput. J., v. 11, 1968, pp. 339-340. MR 38 #5389. MR 0237096 (38:5389)
- F. Stenger, "Bounds on the error of Gauss-type quadratures," Numer. Math., v. 8, 1966, pp. 150-160. MR 33 #5120. MR 0196936 (33:5120)
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Keywords: Newton-Cotes quadrature formula, Cotes numbers, Legendre polynomial, Legendre function of the second kind, Gauss-Legendre quadrature formula, error bound, asymptotic bound, Davis' method
Article copyright: © Copyright 1970 American Mathematical Society