Quadrature rules for $C^0$ and $C^1$ splines, a recipe

Abstract

Closed formulae for all Gaussian or optimal, 1-parameter quadrature rules in a compact interval [a, b] with non uniform, asymmetric subintervals, arbitrary number of nodes per subinterval for the spline classes $S_{2N,0}$ and $S_{2N+1,1}$, i.e. even and odd degree are presented. Also rules for the 2 missing spline classes $S_{2N-1,0}$ and $S_{2N,1}$ (the so called 1/2-rules), i.e. odd and even degree are presented. These quadrature rules are explicit in the sense, that they compute the nodes and their weights in the first/last boundary subinterval and, via a recursion the other nodes/weights, parsing from the first/last subinterval to the middle of the interval. These closed formulae are based on the semi-classical Jacobi type orthogonal polynomials.

29 pages, 6 figures