## Piecewise cubic curve fitting algorithm.(English)Zbl 0633.65012

The problem of data interpolation by differentiable piecewise cubic polynomials is investigated. Let a function f:[0,1]$$\to {\mathbb{R}}$$ and points $$0=x_ 0<x_ 1<...<x_ n=1$$ be given. An algorithm is developed for constructing a piecewise cubic polynomial $$s\in C^ 1[0,1]$$ with knots at $$x_ 1,...,x_{n-1}$$ and at appropriately chosen further points which satisfies the following conditions: $$s(x_ i)=f(x_ i)$$, $$i=0,...,n$$, and s is monotone, if $$f(x_{i+1})-f(x_ i)\geq 0$$, $$i=0,...,n-1$$. It is shown that for every monotone function $$f\in C^ 4[0,1]$$, $$\| f-s\|_{\infty}\leq 9\| f^{(4)}\|_{\infty}h^ 4,$$ where $$h=\max \{x_{i+1}-x_ i: i=0,...,n-1\}.$$ The author compares his method with some known algorithms and gives two numerical examples.
Reviewer: G.Nürnberger

### MSC:

 65D10 Numerical smoothing, curve fitting 65D07 Numerical computation using splines 41A15 Spline approximation
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