Asymptotic error estimates for the Gauss quadrature formula

Introduction. Classical error estimates for the Gauss quadrature formula using derivatives can be used, but they are not of great practical value since the derivatives are not usually available. Davis and Rabinowitz [1] give a more convenient method for obtaining an upper bound for the error in the quadrature of analytic functions. McNamee [2] has discussed complex-variable methods for obtaining upper bounds for the errors of the Gaussian quadratures applied to analytic functions, and Barrett [3] has discussed their convergence. The error of the n-point Gauss quadrature depends on n and also on the function to be integrated. The object of the present paper is to obtain asymptotic estimates for the error of the Gauss quadrature formula, for large n, according to the nature of the integrand f(x). The analysis also brings out the effect of the nature of f(z) on the rate of convergence of the Gauss quadrature formula.