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

TL;DR

This paper derives explicit Gaussian quadrature rules for specific classes of $C^0$ and $C^1$ spline functions over non-uniform intervals, using semi-classical Jacobi orthogonal polynomials, including formulas for all degrees and missing classes.

Contribution

It provides closed-form, recursive quadrature rules for all degrees and spline classes, including previously missing cases, based on semi-classical Jacobi polynomials.

Findings

Explicit formulas for quadrature nodes and weights for all spline classes.

Recursive computation method from boundary to center of interval.

Includes rules for missing spline classes $S_{2N-1, 0}$ and $S_{2N, 1}$.

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.