Rational iterated function system for positive/monotonic shape preservation
 AKB Chand^{1}Email author,
 N Vijender^{1} and
 RP Agarwal^{2, 3}
https://doi.org/10.1186/16871847201430
© Chand et al.; licensee Springer. 2014
Received: 2 July 2013
Accepted: 16 December 2013
Published: 27 January 2014
Abstract
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 $\frac{{E}_{i}}{{F}_{i}}$, where ${E}_{i}$ are cubics and ${F}_{i}$ are preassigned quadratics having 3shape parameters. The fixed point of the developed rational cubic IFS is in ${\mathcal{C}}^{1}$, 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 $\mathrm{\Phi}\in {\mathcal{C}}^{4}$ 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.
Keywords
1 Introduction
Setting a novel platform for the approximation of natural objects such as trees, clouds, feathers, leaves, flowers, landscapes, glaciers, galaxies, and torrents of water, Mandelbrot [1] introduced the term fractal in the literature. Since fractals capture the nonlinear structures of various objects effectively, the fractal geometry has been successfully used in different problems in applied sciences and engineering [1–4]. The iterated function system (IFS) was introduced by Hutchinson [5] for the construction of various types of fractal sets, and popularized by Barnsley [6]. An IFS is a dynamical system consisting of a finite collection of continuous maps. Based on the IFS theory, Barnsley [7] constructed a class of functions that are known as FIFs. The graph of a FIF is the fixed point of an IFS. Also a FIF is the fixed point of the ReadBajraktarević operator on a suitable function space. Common features between a FIF and a piecewise polynomial interpolation are that they are geometrical in nature, and they can be computed rapidly, but the main difference is the fractal character, i.e., a FIF satisfies a functional relation related to the selfsimilarity on smaller scales. In the direction of smooth fractal curves, Barnsley and Harington [8] initiated the construction of a restricted class of differentiable FIF or ${\mathcal{C}}^{k}$FIF that interpolates the prescribed data if the values of ${\mathrm{\Phi}}^{(p)}$, $p=1,2,\dots ,k$, at the initial end point of the interval are given, where Φ is the original function. This method is based on the recursive nature of an algorithm, and specifying the boundary conditions similar to the classical splines was found to be quite difficult to handle in this construction. The fractal splines with general boundary conditions have been studied recently [9–13] by restricting their IFSs parameters suitably.
The motivation of this work is the research on different types of splines by several authors; see, for instance, Schmidt and Heß [14], Fritsch and Carlson [15], Schumaker [16], and Brodlie and Butt [17], and references therein. The uniqueness of spline representation for a given data set turns out to be a disadvantage for shape modification problems. The use of rational functions with the shape parameters was introduced by Späth [18] to preserve different geometric properties attached to a given set of data. Rational interpolants are often used in data visualization problems due to their excellent asymptotic properties, capability to model complicated smooth structures, better interpolation properties, and excellent extrapolating powers. Gregory and Delbourgo [19] introduced the rational cubic spline with one family of shape parameters, and this work inspired a large amount of research in shapepreserving rational spline interpolations, see [20, 21] and references therein.
In this paper, we introduce the rational cubic IFS with 3shape parameters in each subinterval of the interpolation domain such that its fixed point generalizes the corresponding classical rational cubic spline functions [20]. The developed rational cubic spline FIF is bounded, and is unique by fixed point theory for a given set of scaling factors and shape parameters. Because of the recursive nature of FIF, the necessary conditions for monotonicity on the derivative values at knots alone may not ensure the monotonicity of a rational cubic fractal interpolant for a given monotonic data. Based on the appropriate condition on the rational IFS parameters: (i) the scaling factors that depend only on given data, and (ii) the shape parameters that depend on both the interpolation data and scaling factors, we construct the shapepreserving rational cubic FIFs for a prescribed positive and/or monotonic data. By varying the scaling factors (within the shapepreserving interval) and shape parameters (according to the conditions derived in our theory), we can make the fixed point of a rational cubic IFS more pleasant and suitable for aesthetic requirements in a modeling problem. The proposed method is suitable for the shapepreserving interpolation problems where a data set originates from an unknown function $\mathrm{\Phi}\in {\mathcal{C}}^{1}$ and its derivative ${\mathrm{\Phi}}^{\prime}$ is a continuous nowhere differentiable function, for instance, the motion of single inverted pendulum in nonlinear control theory [22].
Comparison of the proposed rational cubic FIF over some existing schemes:

When all the scaling factors are zero, the proposed rational cubic FIF reduces to the classical rational cubic interpolant [20], see Remark 1, Section 3.

To generate shapepreserving interpolants, our construction does not need additional knots in contrast to methods due to Schumaker [16] and Brodlie and Butt [17], which require additional knots for the shapepreserving interpolants.

The classical interpolants [15, 23] are suitable only for monotonicity interpolation whereas the proposed rational cubic FIF is suitable for both monotonicity and positivity interpolation. Moreover, the rational quadratic interpolant [23] is a special case of our rational cubic FIF for the particular choice of the scaling factors and shape parameters, see Remark 2, Section 3.

For given monotonic data, the monotonic curve generated by the rational quadratic interpolant [23] is unique for fixed shape parameters, whereas for the same monotonic data an infinite number of monotonic curves will be obtained using our rational FIF by suitable modifications in the associated scaling factors. Thus, when the shape parameters are incapable to change the shape of an interpolant in given intervals, then the scaling factors can be used to alter the shape of the interpolant in our method.

Where monotonicity is concerned, our construction does not need an additional condition on derivatives at knots except for the necessary conditions. But the construction of Fritsch and Carlson [15] needs some restrictions on derivatives at knots apart from the necessary conditions for the same problem.

The derivatives of the shapepreserving interpolants [15–17, 23] are piecewise smooth, whereas the derivative of our rational cubic FIF may be piecewise smooth to a nondifferentiable function according to the choice of the scaling factors. Owing to this special feature, the proposed method is preferable over the classical shapepreserving interpolants when the approximation is taken for data originating with an unknown function $\mathrm{\Phi}\in {\mathcal{C}}^{1}$ having a shape with fractality in ${\mathrm{\Phi}}^{\prime}$.
This paper is organized as follows. In Section 2, the general constructions of fractal interpolants and ${\mathcal{C}}^{r}$rational cubic FIFs based on IFSs are summarized. Section 3 is devoted to the construction of a suitable rational IFS so that its fixed point is the desired interpolant that can be used for shape preservation. Then we deduce an upper bound of the uniform error bound between the original function and the rational cubic FIF. The fixed point of this rational IFS does not follow any shape constraints. The restrictions on the rational IFS parameters are deduced for a positivity shape in Section 4, and the results are illustrated with suitably chosen examples. In Section 5, the monotonicity problem is considered through the developed rational cubic IFS.
2 IFS for fractal functions
The existence of a spline FIF based on a polynomial IFS is given in [8]. We have extended this result to the rational IFS with 3shape parameters in the following.
then the fixed point of the rational IFS ${\mathcal{I}}^{\ast}$ is the graph of the ${\mathcal{C}}^{r}$rational FIF.
When all $2r+2$ arbitrary constants in ${q}_{i}(x)$ are determined from equation (8), then $\psi (x)$ exists. By using similar arguments as in [7], it can be shown that IFS ${\mathcal{I}}^{\ast}$ has a unique fixed point, and that it is the graph of the rational FIF $\psi \in {\mathcal{C}}^{r}[{x}_{1},{x}_{n}]$. □
3 Rational cubic IFS
The construction of the desired rational cubic IFS is given in Section 3.1 such that its fixed point is used for shape preservation in the sequel. The error analysis of the fixed point of rational cubic IFS with an original function is studied in Section 3.2 for convergence results.
3.1 Construction
The fixed point ψ of the above rational cubic IFS is unique for every fixed set of scaling factors and shape parameters. Thus by taking different sets of scaling and shape parameters, we can generate an infinite number of fixed points for the above rational cubic IFS. In most applications, the derivatives ${d}_{i}$ ($i=1,2,\dots ,n$) are not given, and hence they must be calculated either from the given data or by using numerical approximation methods [24].
described in the literature [20] with $\rho =\frac{x{x}_{i}}{{x}_{i+1}{x}_{i}}$, $x\in [{x}_{i},{x}_{i+1}]$.
Now from equation (12), we conclude that for the above choice of ${\alpha}_{i}$, ${\beta}_{i}$, and ${\gamma}_{i}$, our rational cubic FIF ψ reduces to a monotonicity preserving rational quadratic FIF [25] constructed by our group. Also it is easy to verify that, if ${\xi}_{i}=0$, ${\alpha}_{i}={\beta}_{i}=1$ and ${\gamma}_{i}=\frac{{d}_{i}+{d}_{i+1}}{{\mathrm{\u25b3}}_{i}}$, $i=1,2,\dots ,n1$, then the rational cubic FIF reduces to the rational quadratic function as in [23].
3.2 Error analysis of fixed point of rational cubic IFS
where ${E}_{i}^{\u2020}(\mathrm{\Phi})={h}_{i}^{3}{\parallel {\mathrm{\Phi}}^{(4)}\parallel}_{\mathrm{\infty}}A(\mathrm{\Phi})+16{\zeta}_{i}{h}_{i}^{2}{\parallel {\mathrm{\Phi}}^{(3)}\parallel}_{\mathrm{\infty}}+24{\zeta}_{i}{h}_{i}^{2}{\parallel {\mathrm{\Phi}}^{(2)}\parallel}_{\mathrm{\infty}}$, ${A}_{i}(\mathrm{\Phi})={\parallel {\mathrm{\Phi}}^{(1)}\parallel}_{\mathrm{\infty}}+\frac{{\zeta}_{i}}{2}$, ${\zeta}_{i}=max\{{\mathrm{\Phi}}^{\prime}({x}_{i}){d}_{i},{\mathrm{\Phi}}^{\prime}({x}_{i+1}){d}_{i+1}\}$, $h={max}_{1\le i\le n1}{h}_{i}$, ${\nu}_{i}={min}_{{x}_{i}\le x\le {x}_{i+1}}{\mathrm{\Phi}}^{\prime}(x)$, ${\omega}_{i}=max\{{\alpha}_{i},{\beta}_{i}\}$, ${\tau}_{i}=min\{{\alpha}_{i},{\beta}_{i}\}$, ${H}_{1}(h)={max}_{1\le i\le n1}\{3({f}_{i}+{f}_{i+1})+{h}_{i}({d}_{i}+{d}_{i+1})\}$, ${H}_{2}(h)={max}_{1\le i\le n1}\{3{a}_{i}({f}_{1}+{f}_{n})+{h}_{i}({d}_{1}+{d}_{n})\}$.
Now using equation (20) in equation (19), we get $S(x)\le 3{f}_{i}+3{f}_{i+1}+{h}_{i}({d}_{i}+{d}_{i+1})$.
From equation (24), it is evident that for ${\xi}_{i}=0$, $i=1,2,\dots ,n1$, the fixed point of rational cubic IFS (11) coincides with the corresponding classical rational cubic interpolant.
Therefore, using equations (24)(25) together with the inequality ${\parallel \mathrm{\Phi}\psi \parallel}_{\mathrm{\infty}}\le {\parallel fS\parallel}_{\mathrm{\infty}}+{\parallel S\psi \parallel}_{\mathrm{\infty}}$, we get the bound for ${\parallel \mathrm{\Phi}\psi \parallel}_{\mathrm{\infty}}$, and it completes the proof of theorem. □
Corollary 1 (Convergence results)
 (i)
Since ${a}_{\mathrm{\infty}}=\frac{h}{{x}_{n}{x}_{1}}$, we conclude from equation (13) that the fixed point of rational cubic IFS equation (11) converges uniformly to the original function Φ as $h\to 0$.
 (ii)
Again from the error estimation (13), $O({h}^{p})$ ($p=2,3,4$) convergence can be obtained if the derivative values are available such that ${\zeta}_{i}=O({h}_{i}^{p1})$ ($p=2,3,4$), and the scaling factors are chosen as ${\xi}_{i}\le \kappa {a}_{i}^{p1}$ ($p=2,3,4$) for $i=1,2,\dots ,n1$.
4 Positivity preserving rational cubic FIF
The ${\mathcal{C}}^{1}$rational cubic fractal interpolation function developed in Section 3 has deficiencies as far as the positivity preserving issue is concerned. Because of the recursive nature of FIFs, we assume all the scaling factors are nonnegative so that it is easy to derive the sufficient conditions for a positive fixed point of the rational cubic IFS (11). It requires one to assign appropriate restrictions on the scaling factors ${\xi}_{i}$ and shape parameters ${\alpha}_{i}$, ${\beta}_{i}$ and ${\gamma}_{i}$, for $i=1,2,\dots ,n1$, so that the positivity feature of a given set of positive data is preserved in the fixed point of the rational cubic IFS (11). In Section 4.1, the suitable restrictions are developed on the scaling factors and shape parameters for a positivity preserving ${\mathcal{C}}^{1}$rational cubic spline FIF. The importance of suitable restrictions on the rational IFS parameters is illustrated in Section 4.2.
4.1 Restrictions on IFS parameters for positivity
 (i)the scaling factors ${\xi}_{i}$, $i=1,2,\dots ,n1$, are selected as${\xi}_{i}\in \{\begin{array}{ll}[0,min\{\frac{{f}_{i}}{{a}_{i}{f}_{1}},\frac{{f}_{i+1}}{{a}_{i}{f}_{n}}\}]& \mathit{\text{if}}min\{\frac{{f}_{i}}{{a}_{i}{f}_{1}},\frac{{f}_{i+1}}{{a}_{i}{f}_{n}}\}\kappa ,\\ [0,\kappa ]& \mathit{\text{if}}min\{\frac{{f}_{i}}{{a}_{i}{f}_{1}},\frac{{f}_{i+1}}{{a}_{i}{f}_{n}}\}\ge \kappa ,\end{array}$(26)
 (ii)with ${\gamma}_{i}>0$, the shape parameters ${\alpha}_{i}$, ${\beta}_{i}$, $i=1,2,\dots ,n1$, are chosen as$\{\begin{array}{ll}{\alpha}_{i}>0& \mathit{\text{if}}{\lambda}_{i}\ge 0,\\ {\alpha}_{i}\in (0,\frac{{\gamma}_{i}{s}_{i}^{\ast}}{{\lambda}_{i}})& \mathit{\text{if}}{\lambda}_{i}0,\\ {\beta}_{i}0& \mathit{\text{if}}{\mu}_{i}\ge 0,\\ {\beta}_{i}\in (0,\frac{{\gamma}_{i}{p}_{i}^{\ast}}{{\mu}_{i}})& \mathit{\text{if}}{\mu}_{i}0,\end{array}$(27)
where ${p}_{i}^{\ast}={f}_{i+1}{\xi}_{i}{f}_{n}{a}_{i}$, ${s}_{i}^{\ast}={f}_{i}{\xi}_{i}{f}_{1}{a}_{i}$, ${\lambda}_{i}={f}_{i}+{d}_{i}{h}_{i}{\xi}_{i}({h}_{i}{d}_{1}+{f}_{1}{a}_{i})$, ${\mu}_{i}={f}_{i+1}{d}_{i+1}{h}_{i}{\xi}_{i}({f}_{n}{a}_{i}{h}_{i}{d}_{n})$, then for fixed ${\xi}_{i}$, ${\alpha}_{i}$, ${\beta}_{i}$, ${\gamma}_{i}$ ($i=1,2,\dots ,n1$), the unique fixed point ψ of the rational IFS (11) is positive.
By substituting $\theta =\frac{s}{s+1}$ in equation (28), ${E}_{i}(\theta )>0$ for all $\theta \in [0,1]$ is equivalent to ${\mathrm{\Omega}}_{i}(s)={p}_{i}^{\ast}{s}^{3}+{q}_{i}^{\ast}{s}^{2}+{r}_{i}^{\ast}s+{s}_{i}^{\ast}>0$ for all $s\ge 0$, where ${p}_{i}^{\ast}={p}_{i}+{q}_{i}+{r}_{i}+{s}_{i}$, ${q}_{i}^{\ast}={q}_{i}+2{r}_{i}+3{s}_{i}$, ${r}_{i}^{\ast}={r}_{i}+3{s}_{i}$, ${s}_{i}^{\ast}={s}_{i}$.
If ${\lambda}_{i}\ge 0$, ${r}_{i}^{\ast}={\alpha}_{i}{\lambda}_{i}+{\gamma}_{i}{s}_{i}^{\ast}>0$ is true from equation (29), and in this case ${\alpha}_{i}>0$ can be chosen arbitrarily. Otherwise, ${\lambda}_{i}<0$, we have ${r}_{i}^{\ast}={\alpha}_{i}{\lambda}_{i}+{\gamma}_{i}{s}_{i}^{\ast}>0\iff {\alpha}_{i}<\frac{{\gamma}_{i}{s}_{i}^{\ast}}{{\lambda}_{i}}$. Similarly ${q}_{i}^{\ast}={\beta}_{i}{\mu}_{i}+{\gamma}_{i}{p}_{i}^{\ast}>0$ is true when (i) ${\mu}_{i}\ge 0$, ${\beta}_{i}>0$ arbitrary (ii) ${\mu}_{i}<0$, ${\beta}_{i}<\frac{{\gamma}_{i}{p}_{i}^{\ast}}{{\mu}_{i}}$. Another set of restrictions on ${\xi}_{i}$, ${\alpha}_{i}$, ${\beta}_{i}$, and ${\gamma}_{i}$ can be derived if $({p}_{i}^{\ast},{q}_{i}^{\ast},{r}_{i}^{\ast},{s}_{i}^{\ast})\in {R}_{2}$. But we have not considered it here due to the complexity involved in the calculations. The above discussions yield equation (27).
Therefore, ${E}_{i}(\theta )\ge 0$ whenever equations (29) and (27) are true. Now it is easy to see that the fixed point of the rational cubic IFS (11) is positive if the scaling factors and shape parameters involved in the IFS (11) satisfy equations (26) and (27), respectively. □
4.2 Examples and discussion
Rational IFS parameters for positive fractal interpolants
Figure  Scaling factors  Shape parameters 

1(a)  ${\xi}_{1}=0.123$, ${\xi}_{2}=0.107$, ${\xi}_{3}=0.179$, ${\xi}_{4}=0.005$, ${\xi}_{5}=0.005$, ${\xi}_{6}=0.124$  ${\beta}_{i}={\gamma}_{i}=1$, i = 1(1)6, ${\alpha}_{i}=1$, i = 1,2,3,6, ${\alpha}_{4}=0.445$, ${\alpha}_{5}=0.512$ 
1(b)  ${\xi}_{1}=\mathbf{0.01}$, ${\xi}_{2}=0.107$, ${\xi}_{3}=0.179$, ${\xi}_{4}=0.005$, ${\xi}_{5}=0.005$, ${\xi}_{6}=0.124$  ${\beta}_{i}={\gamma}_{i}=1$, i = 1(1)6, ${\alpha}_{i}=1$, i = 1,2,3,6, ${\alpha}_{4}=0.445$, ${\alpha}_{5}=0.512$ 
1(c)  ${\xi}_{1}=0.123$, ${\xi}_{2}=0.107$, ${\xi}_{3}=\mathbf{0.01}$, ${\xi}_{4}=0.005$, ${\xi}_{5}=0.005$, ${\xi}_{6}=0.124$  ${\beta}_{i}={\gamma}_{i}=1$, i = 1(1)6, ${\alpha}_{i}=1$, i = 1,2,3,6, ${\alpha}_{4}=0.445$, ${\alpha}_{5}=0.512$ 
1(d)  ${\xi}_{1}=0.123$, ${\xi}_{2}=0.107$, ${\xi}_{3}=0.179$, ${\xi}_{5}=0.005$, ${\xi}_{6}=0.124$  ${\beta}_{i}={\gamma}_{i}=1$, i = 1(1)6, ${\xi}_{4}=0.005$, ${\alpha}_{i}=1$, i = 1,2,3, ${\alpha}_{4}=0.445$, ${\alpha}_{5}=0.512$, ${\alpha}_{6}={\mathbf{10}}^{\mathbf{9}}$ 
1(e)  ${\xi}_{1}=0.123$, ${\xi}_{2}=0.107$, ${\xi}_{3}=\mathbf{0.01}$, ${\xi}_{4}=0.005$, ${\xi}_{5}=0.005$, ${\xi}_{6}=0.124$  ${\beta}_{i}={\gamma}_{i}=1$, i∈{1,2,4,5,6}, ${\gamma}_{3}=1$, ${\beta}_{3}={\mathbf{10}}^{\mathbf{3}}$, ${\alpha}_{i}=1$, i = 1,2,3,6, ${\alpha}_{4}=0.445$, ${\alpha}_{5}=0.512$ 
1(f)  ${\xi}_{1}={\xi}_{2}={\xi}_{3}={\xi}_{4}={\xi}_{5}={\xi}_{6}={\xi}_{7}=\mathbf{0}$  ${\beta}_{i}={\gamma}_{i}=1$, i = 1(1)6, ${\alpha}_{i}=1$, i = 1,2,3,6, ${\alpha}_{4}=0.445$, ${\alpha}_{5}=0.512$ 
Uniform errors between Φ and rational fractal interpolants, and their derivatives
5 Monotonicity preserving rational cubic FIF
The fixed point ψ of a rational cubic IFS may not preserve the monotonic feature of a given set of monotonic data. For an automatic generation of rational IFS parameters, we restrict them in Section 5.1, and the results are implemented in Section 5.2 through suitable examples.
5.1 Restrictions on IFS parameters for monotonicity
then for a fixed ξ, ${\alpha}_{i}$, ${\beta}_{i}$, ${\gamma}_{i}$ ($i=1,2,\dots ,n1$), the unique fixed point ψ of the rational cubic IFS (11) is monotonic in nature.
Due to the recursive nature of rational fractal function (33), the necessary conditions (30) are not sufficient to ensure the monotonicity of fixed point ψ of the rational cubic IFS (11). We impose additional restrictions on the scaling factors ${\xi}_{i}$, and shape parameters ${\alpha}_{i}$, ${\beta}_{i}$, and ${\gamma}_{i}$, $i=1,2,\dots ,n1$, so that these conditions together with the necessary conditions (30) yield the monotonic feature of the fixed point ψ of the IFS (11).
Case I: Monotonically increasing data
i.e., ${\xi}_{i}<\frac{{\mathrm{\Delta}}_{i}({x}_{n}{x}_{1})}{{f}_{n}{f}_{1}}$. We search for sufficient conditions that make ${A}_{2,i}\ge 0$. For this purpose, we make each term in ${A}_{2,i}$ nonnegative. The selection of ${\xi}_{i}$ with respect to equations (34) and (36) gives ${d}_{i}{\xi}_{i}{d}_{1}\ge 0$ and ${\mathrm{\u25b3}}_{i}{\xi}_{i}\frac{{f}_{n}{f}_{1}}{{x}_{n}{x}_{1}}>0$, respectively. Now it remains to make ${\alpha}_{i}({\gamma}_{i}+{\beta}_{i})\ge 0$ and ${\alpha}_{i}{\beta}_{i}\ge 0$. In these two inequalities, the product of the shape parameters is involved. Therefore these inequalities are true if we restrict the shape parameters ${\alpha}_{i}$, ${\beta}_{i}$, and ${\gamma}_{i}$, $i=1,2,\dots ,n1$, respectively, as in equation (32).
Justification for equation (32)
Let $sgn({\alpha}_{i})=sgn({\beta}_{i})$ be negative, then from equations (32) and (34)(36), we can conclude that $sgn({\gamma}_{i})$ is negative. Therefore, ${\alpha}_{i}({\gamma}_{i}+{\beta}_{i})\ge 0$ and ${\alpha}_{i}{\beta}_{i}\ge 0$. Similarly, it can be shown that $sgn({\alpha}_{i})=sgn({\beta}_{i})$ being positive gives similar results.
The above discussion led to the following procedure to make ${A}_{2,i}\ge 0$: first choose the scaling factors with respect to equations (34)(36), then select the shape parameters according to equation (32).
From the final expression of ${A}_{3,i}$, it is easy to verify that equations (32) and (34)(36) are sufficient for ${A}_{3,i}\ge 0$. Hence we have proved that the fixed point ψ of the rational cubic IFS (11) is monotonically increasing over $[{x}_{1},{x}_{n}]$, if the scaling factors and shape parameters are chosen according to equation (31) and equation (32), respectively. In the case of ${\mathrm{\u25b3}}_{i}=0$, the fixed point of the rational cubic IFS (11) is a constant throughout that subinterval with the value ${f}_{i}$, and ${\xi}_{i}=0$.
Case II: Monotonically decreasing data
Suppose $\{({x}_{i},{f}_{i}),i=1,2,\dots ,n\}$ is a given monotonically decreasing data set. It is easy to see that the sufficient conditions for monotonicity of equation (10) on $[{x}_{i},{x}_{i+1}]$ are ${A}_{j,i}\le 0$, $j=1,2,\dots ,5$. As explained in Case I, it is easy to verify that selections of the scaling factors and shape parameters according to equation (31) and (32), respectively, are sufficient for ${A}_{j,i}\le 0$, $j=1,2,\dots ,5$.
Therefore from the arguments in Case I and Case II, we conclude that if the scaling factors and shape parameters are chosen according to (31) and (32), respectively, then the fixed point ψ of the rational cubic IFS (11) is monotone for given monotonic data. □
Remark 3 Convergence results in Corollary 1 are valid for the shapepreserving rational cubic FIFs.
5.2 Examples and discussion
Rational IFS parameters for monotonic fractal interpolants
Figure  Scaling factors  Shape parameters 

3(a)  ${\xi}_{6}=0.034$, ${\xi}_{7}=0.44$, ${\xi}_{8}=0.789$, ${\xi}_{9}=0.578$, ${\xi}_{10}=0.999$  ${\alpha}_{i}={\beta}_{i}=1$, i = 6(1)10, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=503.3320$, ${\gamma}_{8}=1.5805$, ${\gamma}_{9}=20.5370$, ${\gamma}_{10}=2.4994$ 
3(b)  ${\xi}_{6}=0.034$, ${\xi}_{7}=\mathbf{0.1}$, ${\xi}_{8}=0.789$, ${\xi}_{9}=0.578$, ${\xi}_{10}=0.999$  ${\alpha}_{i}={\beta}_{i}=1$, i = 6(1)10, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=\mathbf{14.3808}$, ${\gamma}_{8}=1.5805$, ${\gamma}_{9}=20.5370$, ${\gamma}_{10}=2.4994$ 
3(c)  ${\xi}_{6}=0.034$, ${\xi}_{7}=0.44$, ${\xi}_{8}=0.789$, ${\xi}_{9}=\mathbf{0.1}$, ${\xi}_{10}=0.999$  ${\alpha}_{i}={\beta}_{i}=1$, i = 6(1)10, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=503.3320$, ${\gamma}_{8}=1.5805$, ${\gamma}_{9}=\mathbf{9.6296}$, ${\gamma}_{10}=2.4994$ 
3(d)  ${\xi}_{6}=0.034$, ${\xi}_{7}=\mathbf{0.1}$, ${\xi}_{8}=\mathbf{0.1}$, ${\xi}_{9}=\mathbf{0.1}$, ${\xi}_{10}=\mathbf{0.1}$  ${\alpha}_{i}={\beta}_{i}=1$, i = 6(1)10, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=\mathbf{14.3808}$, ${\gamma}_{8}=\mathbf{1.4227}$, ${\gamma}_{9}=\mathbf{20.537}$, ${\gamma}_{10}=\mathbf{2.0408}$ 
3(e)  ${\xi}_{6}=0.034$, ${\xi}_{7}=0.44$, ${\xi}_{8}=0.789$, ${\xi}_{9}=0.578$, ${\xi}_{10}=0.999$  ${\alpha}_{i}={\beta}_{i}=1$, i∈{6,7,8,10}, ${\alpha}_{9}={\mathbf{10}}^{\mathbf{4}}$, ${\beta}_{9}=\mathbf{10}$, ${\alpha}_{10}=1$, ${\beta}_{10}=1$, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=503.3320$, ${\gamma}_{8}=1.5805$, ${\gamma}_{9}=\mathbf{1.1857}$, ${\gamma}_{10}=2.4994$ 
3(f)  ${\xi}_{6}=0.034$, ${\xi}_{7}=\mathbf{0.1}$, ${\xi}_{8}=0.789$, ${\xi}_{9}=0.578$, ${\xi}_{10}=0.999$  ${\alpha}_{i}={\beta}_{i}=1$, i∈{6,8,9,10}, ${\alpha}_{7}={\mathbf{10}}^{\mathbf{4}}$, ${\beta}_{7}=\mathbf{10}$, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=\mathbf{6.3276}$, ${\gamma}_{8}=1.5805$, ${\gamma}_{9}=20.5370$, ${\gamma}_{10}=2.4994$ 
3(g)  ${\xi}_{6}=0.034$, ${\xi}_{7}=0.44$, ${\xi}_{8}=0.789$, ${\xi}_{9}=\mathbf{0.1}$, ${\xi}_{10}=0.999$  ${\alpha}_{i}={\beta}_{i}=1$, i∈{6,7,8,10}, ${\alpha}_{9}={\mathbf{10}}^{\mathbf{4}}$, ${\beta}_{9}=1$, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=503.3320$, ${\gamma}_{8}=1.5805$, ${\gamma}_{9}=\mathbf{5.556}$, ${\gamma}_{10}=2.4994$ 
3(h)  ${\xi}_{6}=0.034$, ${\xi}_{7}=0.44$, ${\xi}_{8}=0.789$, ${\xi}_{9}=\mathbf{0.1}$, ${\xi}_{10}=0.999$  ${\alpha}_{i}={\beta}_{i}=1$, i∈{6,7,8,10}, ${\alpha}_{9}=1$, ${\beta}_{9}={\mathbf{10}}^{\mathbf{4}}$, ${\gamma}_{6}=4.2927$, ${\gamma}_{7}=503.3320$, ${\gamma}_{8}=1.5805$, ${\gamma}_{9}=\mathbf{4.0796}$, ${\gamma}_{10}=2.4994$ 
3(i)  ${\xi}_{i}=\mathbf{0}$, i = 6(1)10  ${\alpha}_{i}={\beta}_{i}=1$, i∈{6,7,8,10}, ${\alpha}_{9}={10}^{4}$, ${\beta}_{9}=1$, ${\gamma}_{6}=\mathbf{0.0003}$, ${\gamma}_{7}=\mathbf{0.0011}$, ${\gamma}_{8}=\mathbf{0.0001}$, ${\gamma}_{9}=\mathbf{5.0004}$, ${\gamma}_{10}=\mathbf{0.0002}$ 
Uniform errors between Φ and rational fractal interpolants, and their derivatives
Monotonic RCFIF  Uniform distance with monotonic RCFIF in Figure 3(a)  Derivative of monotonic RCFIF  Uniform distance with derivative in Figure 3(a) 

Figure 3(b)  0.7871  Figure 3(b)  18.736 
Figure 3(c)  2.3678  Figure 3(c)  27.0239 
Figure 3(d)  2.3783  Figure 3(d)  44.6254 
Figure 3(e)  0.8223  Figure 3(e)  3.6775 
Figure 3(f)  2.0038  Figure 3(f)  18.6884 
Figure 3(g)  4.6438  Figure 3(g)  29.531 
Figure 3(h)  1.5456  Figure 3(h)  23.4984 
From the examples in Sections 45, it is observed that proper interactive adjustments of the scaling factors and shape parameters give us a wide variety of positivity and/or monotonicity preserving fixed points of our rational cubic IFS (11) that can be used in various scientific and engineering problems for aesthetic modifications. In order to get an optimal choice of the fixed point of our rational cubic IFS (11), one can employ a genetic algorithm interactively until the desired accuracy is obtained with the original function.
6 Conclusion
A new type of rational cubic IFS with 3shape parameters is introduced in this work such that its fixed point can be used for shaped data. The developed FIF in this paper includes the corresponding classical rational cubic interpolant as a special case. An upper bound of uniform error between the rational cubic FIF ψ and an original function Φ in ${\mathcal{C}}^{4}[{x}_{1},{x}_{n}]$ is estimated, and consequently we have found that ψ converges uniformly to Φ as $h\to 0$. When the accurate derivatives of $O({h}_{i}^{p})$, $i=1,2,\dots ,n$ are available, and the scaling factors are chosen as ${\xi}_{i}<\kappa {a}_{i}^{p1}$, $i=1,2,\dots ,n1$, it is possible to get $O({h}^{p})$ ($p=2,3,4$) convergence for the rational cubic FIF. Automatic data dependent restrictions are derived on the scaling factors and shape parameters of rational cubic IFS so that its fixed point preserves the positivity or monotonicity features of a given set of data. The effects of a change in the scaling factors and shape parameters on the local control of the shape of rational cubic FIF are demonstrated through various examples. Our rational cubic FIFs are more flexible and more suitable for shape related problems in computer graphics, CAD/CAM, CAGD, medical imaging, finance, and engineering applications, and apply equally well to data with or without derivatives. In particular, the proposed method will be an ideal tool in shapepreserving interpolation problems where the data set originates from a positive and/or monotonic function $\mathrm{\Phi}\in {\mathcal{C}}^{1}$, but its derivative ${\mathrm{\Phi}}^{\prime}$ is a continuous and nowhere differentiable function.
Declarations
Acknowledgements
AKBC is thankful to the Department of Science and Technology, Govt. of India for the SERC DST Project No. SR/S4/MS: 694/10. The authors are grateful to the anonymous referees for the valuable comments and suggestions, which improved the presentation of the paper.
Authors’ Affiliations
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