Abscissas and weights for Lobatto quadrature of high order

In recent years, Gaussian quadrature has become the standard method for numerical integration in many computer installations [1, 2]. In general, Gaussian rules are most economical since an n-point rule is exact for polynomials up to degree 2n 1 and no rule can do better. However, for particular classes of functions and for particular applications, other rules may be more efficient. Thus there are occasions when we prefer a closed rule, i.e. one which includes among its abscissas the two end points of the integration interval. This is the case when the integrand vanishes at the two end points as it does in Longman's method for evaluating integrals of oscillating functions [3]. If we want to check a quadrature over a given interval by doing two additional quadratures, each over half the interval, then a closed rule will save at least two evaluations of the integrand. Let us normalize our integration interval to (-1, 1) and consider the closed symmetric n-point integration rule with n odd, n = 2m + 1,