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

08/16/2019

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by   Helmut Ruhland, et al.

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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.