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Smooth interpolation to scattered data by bivariate piecewise polynomials of odd degree. (English) Zbl 0708.65012
An algorithm for constructing bivariate spline interpolants to scattered data is developed. Given scattered data points \(x^ 1,...,x^ n\) in \({\mathbb{R}}^ 2\) and a corresponding triangulation, the authors discuss the dimension of spaces of splines of degree k and smoothness m with respect to this triangulation, and the description of the smoothness properties with the aid of Bézier ordinates.
In the first step of the algorithm, a continuous spline p of degree k is constructed which interpolates given data at \(x^ 1,...,x^ n\) and satisfies certain differentiability conditions at these points. In a second step, the interpolant p is perturbed such that the resulting interpolating spline s is m-times continuously differentiable. This approach guarantees that s is as close as possible to p and that polynomials of degree m are reproduced. For a given smoothness m of the spline interpolants, the method keeps their degree k as low as possible.
A similar version was already given by T. A. Grandine [Comput. Aided Geom. Des. 4, 307-319 (1987; Zbl 0637.65008)] for the case of differentiable cubic splines. Finally, the authors describe several numerical examples for quintic splines of smoothness two.
Reviewer: G.Nürnberger

MSC:
65D07 Numerical computation using splines
65D05 Numerical interpolation
41A15 Spline approximation
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
Software:
nag; NAG; symrcm
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