A (Formula presented.)-continuous rational cubic fractal interpolation function was introduced and its monotonicity aspect was investigated in [Adv. Difference Eq. (30) 2014]. Using this univariate interpolant and a blending technique, in this article, we develop a monotonic rational fractal interpolation surface (FIS) for given monotonic surface data arranged on the rectangular grid. The analytical properties like convergence and stability of the rational cubic FIS are studied. Under some suitable hypotheses on the original function, the convergence of the rational cubic FIS is studied by calculating an upper bound for the uniform error of the surface interpolation. The stability results are studied when there is a small perturbation in the corresponding scaling factors. We also provide numerical examples to corroborate our theoretical results. © Springer India 2015.

Arya Kumar Bedabrata Chand

Chand A.K.B

Vijender N.

Blending

Fractals

interpolation

ANALYTICAL PROPERTIES

BLENDING FUNCTION

CONVERGENCE AND STABILITY

FRACTAL INTERPOLATION FUNCTIONS

FRACTAL INTERPOLATION SURFACE

Monotonicity

Small perturbations

SURFACE INTERPOLATION

Rational functions

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A Monotonic Rational Fractal Interpolation Surface and Its Analytical Properties

Springer Proceedings in Mathematics and Statistics

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of C1- rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form (Formula presented.) where pi(x) and qi(x) are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in C2 is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results. © 2013 Springer-Verlag Italia.

Calcolo

Shape preservation of scientific data through rational fractal splines

This paper is concerned with interpolation subject to a strip condition on the first order derivative using a class of rational cubic Fractal Interpolation Functions (FIFs). This facilitates the FIF to generate monotonic curves for a given set of monotonic data. The proposed monotonicity preserving rational FIF subsumes and supplements a classical monotonic rational cubic spline. In models leading to the monotonicity preserving interpolation problem wherein the first order derivative of the constructed interpolant is to be nondifferentiable in a finite or dense subset of the interpolation interval, the developed fractal scheme is well-suited in contrast to its classical nonrecursive counterpart. It is shown that the present fractal interpolation scheme has O(h4) accuracy, provided the original function belongs to C4(I) and the parameters involved in the FIF are appropriately chosen. © 2014 Elsevier Inc. All rights reserved.

Applied Mathematics and Computation

A fractal procedure for monotonicity preserving interpolation

Fractal interpolation function defined through suitable iterated function system provides a method to perturb a function f∈C(I) so as to yield a class of functions fα∈C(I), where α is a free parameter, called scale vector. For suitable values of scale vector α, the fractal functions fα simultaneously interpolate and approximate f. Further, the iterated function system can be selected suitably so that the corresponding fractal function fα shares the quality of smoothness or non-smoothness of f. The objective of the present paper is to choose elements of the iterated function system appropriately in order that fα preserves fundamental shape properties, namely positivity, monotonicity, and convexity in addition to the regularity of f in the given interval. In particular, the scale factors (elements of the scale vector) must be restricted to satisfy two inequalities that provide numerical lower and upper bounds for the multipliers. As a consequence of this process, fractal versions of some elementary theorems in shape preserving interpolation/approximation are obtained. For instance, positive approximation (that is to say, using a positive function) is extended to the fractal case if the factors verify certain inequalities. © 2014 Elsevier Inc.

Fulltext

Journal of Mathematical Analysis and Applications

Fractal perturbation preserving fundamental shapes: Bounds on the scale factors

In this paper we consider the (inverse) problem of determining the iterated function system (IFS) which produces a shaped fractal interpolant. We develop a new type of rational IFS by using functions of the form Ei Fi , where Ei are cubics and Fi are preassigned quadratics having 3-shape parameters. The fixed point of the developed rational cubic IFS is in C1, but its derivative varies from a piecewise differentiable function to a continuous nowhere differentiable function. An upper bound of the uniform error between the fixed point of a rational IFS and an original function φ ∈ C4 is deduced for the convergence results. The automatic generations of the scaling factors and shape parameters in the rational IFS are formulated so that its fixed point preserves the positive/monotonic features of prescribed data. The presence of scaling factors provides additional freedom to the shape of the fractal interpolant over its classical counterpart in the modeling of discrete data. © 2014 Chand et al.; licensee Springer.

Advances in Difference Equations

Rational iterated function system for positive/monotonic shape preservation

Advances in Numerical Analysis

Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions

A monotonic rational fractal interpolation surface and its analytical properties

Journal	Data powered by TypesetSpringer Proceedings in Mathematics and Statistics
Publisher	Data powered by TypesetSpringer New York LLC
ISSN	21941009
Open Access	No