Uniform Cubic Spline Interpolation with Some Additional Techniques for Functions with Strong Gradients

Spline interpolation is commonly applied in many applications because several desirable interpolation properties are evident and related to the smoothness and accuracy. It may distort somewhat because of the Gibbs phenomenon though it is less noticeable if there is a gap or if the change from one slope to another is steep. Moreover, the issues of data consistency are needed in some applications, but these peculiarities are not checked automatically by interpolation. Thus, it becomes imperative to apply a few more additional technologies to facilitate the uniformity. Orden and repetitiveness of C^2 are not ensured simultaneously, and thus, the last interpolation is not a spline in actuality. The analysis of the given problem is carried out in the context of developing a sufficient condition for generating a homogeneous cubic spline based on the cubic Hermite interpolator and the identification of two construction methods based on nonlinear formulas. As such, these methods are aimed at reducing the regions where accuracy of order is not maximum or where interpolation is less than C^2 order. We compare the order of accuracy and the intrapolar pattern of the extended and the original method and perform many numbers experiment to check with theory.