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On quadratic spline interpolation. (English) Zbl 0631.41009

P\({}_ i(x_ i,y_ i)\) \((i=0,...,n)\) are the data points with the additional requirement that the knots \(\{x_ i\}\) satisfy \(\Delta_ n=\{a=x_ 0<...<x_ n=b\}.\) S(x) is said to be a quadratic spline over the partition \(\Delta_ n\) of [a,b] if \(s(x)\in C^ 1[a,b]\) and s(x) restricted to the subinterval \([x_ i,x_{i+1}]\) is a quadratic polynomial for \(0\leq i\leq n-1\). If in addition this function also satisfies \(s(x_ i)=y_ i\) \((i=0,...,n)\), then s(x) is said to be an \(S(\Delta_ n,z)\)-interpolate of y(x). The author proves the existence and uniqueness of a quadratic spline interpolate s(x) with equally spaced knots \(x=a+ih\), \(h=(b-a)n\) \((i=0,...,n)\) and satisfying \(s(x_ i)=y_ i\), \(s'(x_ 0)\equiv s_ 0'=y_ 0'\). Employing s(x), the author gives formulas for the approximation of y’ and y” at selected knots by orders up to \(O(h^ 4)\) over a uniform partition of the interval.
Reviewer: Sun Yongsheng

MSC:

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

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