Error of the Newton-Cotes and Gauss-Legendre quadrature formulas

Kambo, N. S.

doi:10.1090/S0025-5718-1970-0275671-2

Error of the Newton-Cotes and Gauss-Legendre quadrature formulas
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by N. S. Kambo PDF

Math. Comp. 24 (1970), 261-269 Request permission

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.

References

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 72258
Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1963. MR 0157156

Theory of Functions of a Complex Variable

M. M. Chawla and M. K. Jain, Error estimates for Gauss quadrature formulas for analytic functions, Math. Comp. 22 (1968), 82–90. MR 223093, DOI 10.1090/S0025-5718-1968-0223093-3
M. M. Chawla, Asymptotic estimates for the error of the Gauss-Legendre quadrature formula, Comput. J. 11 (1968/69), 339–340. MR 237096, DOI 10.1093/comjnl/11.3.339
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