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In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. These families of schemes are constructed by using dynamic iterative re-weighed least squares method. These schemes are suitable for fitting noisy data with outliers. The codes are designed in a Python environment to numerically fit the given data points. Although these schemes are non-interpolatory, but have the ability to preserve the shape of the initial polygon in case of non-noisy initial data. The numerical examples illustrate that the schemes constructed by non-linear polynomials give better performance than the schemes that are constructed by linear polynomials (Mustafa et al., 2015). Moreover, the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data. Furthermore, families of non-linear bivariate subdivision schemes are also presented that are based on linear and non-linear bivariate polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalized Refine-Smooth algorithm to design a family of a -point b -ary approximating subdivision schemes with bell-shaped mask, where a ≥ 3 and b ≥ 2. We use the combination of corner cutting b -ary subdivision scheme and weighted average of ( b + 1 ) -points to construct the proposed family. We demonstrate that the proposed family has smaller complexity and support width and higher continuity than the existing Refine-Smooth subdivision schemes. We also study the shape preserving properties of the proposed family. In addition, it is observed that the proposed family is suitable for fitting the locally noisy, oscillatory, and irregular data. [ABSTRACT FROM AUTHOR]