Abstract
In this paper, the problem of designing optimal smoothing spline curves for a given set of discrete data is studied. For designing such curves, the normalized uniform B-splines are employed as the basis functions. First we derive a concise expression for the optimal solution of smoothing spline curves, yielding straightforward computational procedure. The so-called leave-out-one method is also employed for determining the smoothing parameter. Then, we develop an algorithm for computing the extrema, i.e. local minima and maxima, for the optimal curve. Since the curve is a piece-wise polynomial and is represented by the control points, the algorithm for detecting and computing the extrema is simple and easy numerically. In particular, it is shown that the algorithm yields extrema exactly under certain conditions. The validities of the proposed methods are examined by numerical experiments.
