C1 sign, monotonicity and convexity preserving Hermite polynomial splines of variable degree

Abstract

This paper proposes an algorithm for constructing interpolatory Hermite polynomial splines of variable degree, which preserve the sign, the monotonicity and the convexity of the data. The polynomial segments are represented as Bézier curves. The degree of each segment plays the role of the tension parameter of the spline. We discuss extensively the monotonicity and convexity criteria, detailing a strict and a weak form of monotonicity preservation, as well as their implications on the Hermite interpolation. We also propose a global method for estimating the nodal derivatives of the spline. We evaluate the results of this method in an extensive set of examples, comparing them with a number of local derivative estimation methods from the pertinent literature. The algorithm is numerically stable, simple to implement, and it is of linear complexity, since the related mathematical conditions can be expressed as linear inequalities with respect to the control points of the spline.