Shape preserving C² rational quartic interpolation spline with two parameters

Abstract

Constructing shape-preserving interpolation spline has been a hot topic in industrial design and scientific data visualization during the past 30 years. In the existing shape preserving $C^2$ interpolation spline methods, however, some methods can be only used to preserve the monotonic data set, while others can be only used to preserve the convex data set, and often for $C^2$ continuity, it is requested to solve a linear system of consistency equations for the derivative values at the knots. In this paper, a new explicit representation of a $C^2$ rational quartic interpolation spline with two local tension parameters is developed. A convergence analysis establishes an error bound and shows that the order of approximation is $O\left(h^2\right)$ accuracy. Sufficient conditions for the proposed interpolation spline to preserve the shape of positive, monotonic, and convex set of data are derived.

Keywords:

• rational quartic interpolation spline
• convergence analysis
• approximation order
• shape-preserving interpolation

2000 AMS Subject Classifications::

• 65D05
• 65D07
• 65D17

Acknowledgments

We thank the anonymous reviewers for their valuable remarks for improvements. The research is supported by the National Natural Science Foundation of China (No. 10871208, No. 11271376) and Graduate Students Scientific Research Innovation Project of Hunan Province (No. CX2012B111).