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arXiv · 1908.06199

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

Helmut Ruhland
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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.

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Helmut Ruhland. 2019-08-16. Quadrature rules for $C^0$ and $C^1$ splines, a recipe. https://arxiv.org/abs/1908.06199

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