Chapter 1 Introduction
1.6 Shape Preserving FIF and FIS
When data arises from various natural and scientific phenomena, it may contains cer- tain geometric properties which may be expressed mathematically in terms of positiv- ity, monotonicity, convexity, etc. For example, positive data arises in monthly rainfall amounts, discharge of gas in certain chemical reactions etc. The negative graphical rep- resentation is meaningless for these physical quantities. Rate of dissemination of drug in blood [16], empirical option of pricing models in finance [69], approximation of couples and quasi couples in statistics [16] are some examples of monotonic data. Convexity plays vital role in the theory of nonlinear optimization. The problem of searching suffi- ciently smooth functions that preserve these shape properties hidden in the data set is generally referred to as shape preserving interpolation.
Definition 1.6.1. The univariate data {(xi, yi) : i = 1,2, . . . , N} is said to be positive data if yi >0 for i= 1,2, . . . , N.
Definition 1.6.2. The univariate data{(xi, yi) :i= 1,2, . . . , N}is said to be monoton- ically decreasing data if yi ≥ yi+1 for i = 1,2, . . . , N −1 and monotonically increasing data if yi ≤yi+1 for i= 1,2, . . . , N −1.
Definition 1.6.3. The univariate data {(xi, yi) : i = 1,2, . . . , N} is said to be convex data if ∆1 ≤ ∆2 ≤ . . . ≤ ∆i−1 ≤ ∆i ≤ ∆i+1 ≤ . . . ≤ ∆N−1 where ∆i = yi+1h−yi
i and
hi =xi+1−xi for 1≤i≤N −1.
The following result is used to show the interpolantion function is convex in Chapters 2 and 3:
Three Chords Lemma: Let f be a convex function on the domain I and x∗, y∗, z∗
∈I, where x∗ < y∗ < z∗. Then, f(y∗)−f(x∗)
y∗−x∗ ≤ f(z∗)−f(x∗)
z∗−x∗ ≤ f(z∗)−f(y∗) z∗−y∗ .
Definition 1.6.4. The bivariate data {(xi, yj, zi,j) :i= 1,2, . . . , M, j = 1,2, . . . , N} is said to be positive surface data if zi,j >0 for i= 1,2, . . . , M, j = 1,2, . . . , N.
Definition 1.6.5. The bivariate data {(xi, yj, zi,j) :i= 1,2, . . . , M, j = 1,2, . . . , N} is said to be monotonic surface data if monotonic in x-direction, i.e., zi,j ≤ zi+1,j (zi,j ≥ zi+1,j) for i= 1,2, . . . , M −1, j = 1,2, . . . , N and monotonic in y-direction, i.e., zi,j ≤ zi,j+1 (zi,j ≥zi,j+1) for i= 1,2, . . . , M, j = 1,2, . . . , N −1.
Definition 1.6.6. The bivariate data {(xi, yj, zi,j) :i= 1,2, . . . , M, j = 1,2, . . . , N} is said to be convex surface data if ∆1,j ≤ ∆2,j ≤ . . . ≤ ∆i−1,j ≤ ∆i,j ≤ ∆i+1,j ≤ . . . ≤
∆M−1,j, j = 1,2, . . . , N, and ∆1i,1 ≤∆1i,2 ≤. . .≤∆1i,j−1 ≤∆1i,j ≤∆1i,j+1 ≤. . .≤∆1i,N−1, i= 1,2, . . . , M, where ∆i,j = zi+1,jh−zi,j
i , hi =xi+1−xi, ∆1i,j = zi,j+1h∗−zi,j
j , h∗j =yj+1−yj. In order to preserve the shapes of the univariate data, various non-recursive tradi- tional interpolation methods and fractal interpolation methods are available. We first present a brief survey on the univariate shape preserving interpolation. Construction
of shape preserving interpolation was initiated by Schweikert [128] and he introduced tension splines through the solutions of suitable differentiable equations. Carl de Boor and Swartz [53] proved a number of theorems concerning the piecewise monotone in- terpolation of data by splines. Passow and Roulier [103] considered the possibility of a spline interpolant of pre-determined smoothness which is monotone or convex for a monotone and convex data respectively. Costantini [48] considered the problem of ex- istence of monotone or convex splines that having degree n and order of continuity k.
Also, the interpolating splines are obtained by using Bernstein polynomials. Fritsch and Carlson [61] proposed an algorithm which constructs aC1- monotone piecewise cu- bic interpolant to a monotonic data. Fritsch and Butland [60] described a method for producing piecewise cubic interpolant to a monotonic data. Also, the method is local and extremely simple to implement. Schumaker [127] designed an algorithm to con- struct C1-quadratic splines in such a way that monotonicity or convexity of the data is preserved. Lamberti [86] described a global method for construction of a C2-shape preserving interpolating function based on parametric cubic curve. He used step length as tension parameters and he selected tension parameters suitably so that the interpola- tion function preserves the positivity, monotonicity and convexity of the data. Also, we can see various types of shape preserving methods were developed by different authors in [49, 78, 79, 122].
The aforementioned shape preserving methods were developed using polynomials or tension splines. Alternative to the polynomial or tensional spline interpolation is rational spline interpolation and it was introduced by Sp¨ath [131]. Gregory and Delbourgo [54, 63] constructed rational quadratic splines which does not involve shape parameters to preserve the shapes of the monotonic data. By introducing shape parameter on each interval, Gregory and Delbourgo [55] have developed the rational cubic spline. Shape of the interpolation curves can be controlled by these shape parameters and shapes of the data can be preserved by selecting suitable choices of parameters. Recently, huge number of shape preserving rational interpolants were developed, for example [1, 123]. Using fractal methodology, variety of rational FIFs with shape parameters were developed by Chand and co-authors, to preserve the shapes of the univariate data [34, 142–146].
In order to preserve the shapes of the bivariate data, variety of numerical schemes have been developed. For example, Carlson and Fritsch [27] constructed a monotone C1-piecewise bicubic spline on a rectangular grid and the interpolant is determined by the first partial derivatives and mixed partial derivatives at the mesh points. Beatson and Ziegler [15] proposed an algorithm for C1-quadratic spline surface to preserve the monotonicity of the data that lies on the rectangular grid. Renka [119] developed an algorithm for construction ofC1-convex surface that interpolates a convex data. Zhang et al. [150] constructed bivariate rational interpolation surface based on function values to preserve the convexity. Kouibia and Pasadas [85] presented an approximation problem of parametric curves and surfaces from a Lagrange or Hermite data set. Also, one can see in references [28, 50, 68], a variety of numerical schemes have been developed to preserve the shapes of the bivariate data. Also, using fractal techniques, Chand et al. [32, 33, 35, 37, 38, 141] constructed different kinds of FISs in the domain of shape preserving.
In this thesis,C1-FIFs andC1-FISs are constructed using the IFS that involves ratio- nal functions. To construct FIFs or FISs, derivative values or partial derivative values at the knot points are needed. Many situations derivative values or partial derivative values may not be supplied and only data points are available. In that situations esti- mation of the derivative values or partial derivative values are necessary. To compute derivative values, the arithmetic mean method [34] is commonly used and they are given by the following: At the interior point xi, i= 2,3, . . . , N −1, set
di =
0, if ∆i = 0 or ∆i−1 = 0,
hi∆i−1+hi−1∆i
hi+hi−1 , otherwise.
At the end points x1 and xN, set
d1 =
0, if ∆1 = 0 orsgn(d∗1)6=sgn(∆1), d∗1 = ∆1+ (∆1h−∆2)h1
1+h2 , otherwise, dN =
0, if ∆N−1 = 0 orsgn(d∗N)6=sgn(∆N−1),
d∗N = ∆N−1+(∆N−1h −∆N−2)hN−1
N−1+hN−2 , otherwise.
To compute the partial derivatives, we used the arithmetic mean method [68] for the bivariate data as follows: For each fixed j = 1,2, . . . , N, we compute the x-direction partial derivatives as
z1,jx = ∆1,j + (∆1,j−∆2,j) h1
h1+h2, zxi,j = ∆i,j+ ∆i−1,j
2 , i= 2,3, . . . , M −1, zM,jx = ∆M−1,j+ (∆M−1,j−∆M−2,j) hM−1
hM−1+hM−2
.
For each fixedi= 1,2, . . . , M, we compute the y-direction partial derivatives as zi,1y = ∆1i,1+ (∆1i,1−∆1i,2) h∗1
h∗1 +h∗2, zi,jy = ∆1i,j+ ∆1i,j−1
2 , j = 2,3, . . . , N −1, zi,Ny = ∆1i,N−1+ (∆1i,N−1−∆1i,N−2) h∗N−1
h∗N−1+h∗N−2.