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Making the difference interpolation method for splines more stable. (English) Zbl 0725.65013

Let s be a spline of order m associated to a grid without multiple knots. The purpose of this paper is to compute the values \((\Delta^ k_ hs(t))^{m-2}_{k=0}\) where, for any function f: \({\mathbb{R}}\to {\mathbb{R}}\), the forward differences \(\Delta^ k_ hf(t)\) are defined by \(\Delta^ 0_ hf(t)=f(t)\) and \(\Delta^ k_ hf(t)=\Delta_ h^{k-1}f(t+h)- \Delta_ h^{k-1}f(t)\) for any positive integer k, any \(t\in {\mathbb{R}}\), and any \(h\in {\mathbb{R}}\), \(h>0\). It is shown that the values \((\Delta^ k_ hs(t))^{m-2}_{k=0}\) can be computed by solving a linear system of equations of the form \[ \sum^{m-2- k}_{j=0}\gamma^{(\lambda)}_{k,j}\Delta_ h^{k+j}s(t)=\Delta^ k_{\lambda h}s(t)-\sum^{\lambda k- m+1}_{i=0}\theta_{k,i}^{(\lambda,m-1)}\Delta_ h^{m-1}s(t+i\cdot h) \] for \(k=0,...,m-2\), where \(\lambda\) is any integer which is greater than or equal to 2 and the quantities \(\Delta_ h^{m-1}s(t+i\cdot h)\) can be obtained with the aid of a well-known formula. For the computation of the coefficients \(\gamma^{(\lambda)}_{k,j}\) and \(\theta_{k,i}^{(\lambda,m-1)}\) an efficient and numerically stable method is derived.

MSC:

65D07 Numerical computation using splines
65D05 Numerical interpolation
41A15 Spline approximation
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References:

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