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Evaluation of tension splines

Department of Mathematics, University of Osijek
Publication Date
  • Tension Spline
  • B-Spline
  • Evaluation
  • Computer Science
  • Mathematics


Tension spline of order $k$ is a function that, for a given partition mbox{$x_0<x_1<ldots<x_n$}, on each interval mbox{$[x_i,x_{i+1}]$} satisfies differential equation mbox{$(D^k-rho_{i}^2 D^{k-2})u=0$}, where $rho_i$'s are prescribed nonnegative real numbers. Most articles deal with tension splines of order four, applied to the problem of convex (or monotone) interpolation or to the two-point boundary value problem for ordinary differential equations. Higher order tension splines are described in several papers, but no application is given. Possible reason for this is a lack of an appropriate algorithm for their evaluation. Here we present an explicit algorithm for evaluation of tension splines of arbitrary order. We especially consider stable and accurate computation of hyperbolic-like functions used in our algorithm.

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