A Computational Method for Subdivision Depth of Ternary Schemes

• Main Authors: Faheem Khan, Ghulam Mustafa, Aamir Shahzad, Dumitru Baleanu, Maysaa M. Al-Qurashi
• Format: Article
• Language: English
• Published: MDPI AG 2020-05-01
• Series: Mathematics
• Subjects: subdivision schemes; convolution; error bounds; subdivision depth; subdivision level
• Online Access: https://www.mdpi.com/2227-7390/8/5/817

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at <i>k</i>-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.