Spline Fitting

Abstract

Given are the m points (xi,yi), i=1,2,…,m. Spline functions are introduced, and it is noticed that the interpolation task in the case of natural splines has a unique solution. The interpolating natural cubic spline is constructed. For the construction of smoothing splines, different optimization problems are formulated. A selected problem is looked at in detail. The construction of the solution is carried out in two steps. In the first step the unknown Di=s(xi) are calculated via a linear system of equations. The second step is the construction of the interpolating natural cubic spline with respect to these (xi,Di), i=1,2,…,m. Every optimization problem contains a smoothing parameter. A method of estimation of the smoothing parameter from the given data is motivated briefly.