Optimal Local Parameters For Hermite Type Cubic Splines

The local derivatives of Hermite type cubic splines are optimally determined on each subinterval of a given mesh through a set of five classical interpolation procedures in order to minimize a certain objective function. We construct an iterative method that selects on each subinterval, the adequate algorithm for providing the local derivatives which minimize the objective function such that the values of the local derivatives computed at the previous subinterval are preserved in order to keep the smoothness of the spline. The admissible set of classical interpolation procedures consists of the natural cubic spline, the Catmull-Rom cubic spline, the Akima's cubic spline, and the cubic splines with minimal derivative oscillation and those with minimal deviation by the data polygon. A numerical experiment is presented in order to illustrate the performances of the algorithm.