C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation

In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

A New GM(1,1) Model Based on Cubic Monotonicity-Preserving Interpolation Spline

In the classical GM(1,1) model, an accumulated generating operation is made on the original non-negative sequence to obtain a monotone increasing 1-AGO sequence, and the forecasting model is established based on the 1-AGO sequence. A great number of scholars have improved the accuracy of grey model prediction through better developed background value and the equation for the time response. In this work, we reconstruct the background value based on a new developed monotonicity-preserving piecewise cubic interpolations spline, and thereby establish a new GM(1,1) model. Numerical examples show that the new GM(1,1) model has better prediction quality of data than the original GM(1,1) model and improves the precision of prediction in practice.