Gregory type quadrature based on quadratic nodal spline interpolation

Using a method based on quadratic nodal spline interpolation, we define a q uadrature rule with respect to arbitrary nodes, and which in the case of un iformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which is an important example of a trapezoidal rule with end point corrections. The resulting weights are explicitly calculated, and Pea no kernel techniques are then employed to establish error bounds in which t he associated error constants are shown to grow at most linearly with respe ct to the mesh ratio parameter. Specializing these error estimates to the c ase of uniform nodes, we deduce non-optimal order error constants for the L acroix rule, which are significantly smaller than those calculated by crude r methods in previous work, and which are shown here to compare favourably with the corresponding error constants for the Simpson rule.