A Newton-type method for constrained least-squares data-fitting with easy-to-control rational curves

While the mathematics of constrained least-squares data-fitting is neat and clear, implementing a rapid and fully automatic fitter that is able to generate a fair curve approximating the shape described by an ordered sequence of distinct data subject to certain interpolation requirements, is far more difficult. The novel idea presented in this paper allows us to solve this problem with efficient performance by exploiting a class of very flexible and easy-to-control piecewise rational Hermite interpolants that make it possible to identify the desired solution with only a few computations. The key step of the fitting procedure is represented by a fast Newton-type algorithm which enables us to automatically compute the weights required by each rational piece to model the shape that best fits the given data. Numerical examples illustrating the effectiveness and efficiency of the new method are presented.