## 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