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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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