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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