Shape preserving aspects of FIF

Chapter 1 Introduction

2.1 FIF with Two Families of Shape Parameters

2.1.2 Shape preserving aspects of FIF

Since this inequality does not depend on the interval, we get kTαααC−T0Ck∞≤ |ααα|∞

[M + h

4M] + [M^∗+|I|

4 M∗]

. (2.14)

From (2.7) and (2.14), we get

kΦ−Ck∞=kTαααΦ−T0Ck∞≤ kTαααΦ−TαααCk∞+kTαααC−T0Ck∞, which implies that

kΦ−Ck_∞≤ 1 (1− |ααα|∞)

h|ααα|_∞

[M +h

4M] + [M^∗+ |I|

4 M_∗]i

. (2.15)

From [124], the error bound between the original function S and the classical rational cubic spline C is

kS−Ck∞≤ 1

2kS⁽³⁾k∞h³c^∗. (2.16) Using (2.15) and (2.16) with the following inequality

kS−Φk∞≤ kS−Ck∞+kC−Φk∞

we get the required bound for kS−Φk∞. This completes the proof.

Remark 2.1.5. Since|α_i|< a_i =h_i/(x_N−x₁), i= 1,2, . . . , N−1, from Theorem 2.1.1, it can be seen that kS−Φk∞=O(h).

• If |α_i|< a²_i, then kS−Φk_∞=O(h²).

• If |α_i|< a³_i, then kS−Φk∞=O(h³).

Positivity

Let {(x_i, y_i, d_i) : i = 1,2, . . . , N} be a positive data. In the following theorem, the sufficient conditions on the scaling factors and the shape parameters are obtained to ensure the positivity of rational cubic FIF. It can be seen that, the rational cubic FIF Φ is positive ifΦ(x)>0 for all x∈[x1, xN].

Theorem 2.1.2. Let {(x_i, y_i) : i = 1,2, . . . , N} be a data such that y_i > 0, i = 1,2, . . . , N. Let d_i, i = 1,2, . . . , N be the derivative value at the knot x_i. Then the following conditions on the scaling factors and the shape parameters are sufficient to the rational cubic FIF (2.4) to be positive.

0≤α_i <minn a_i, yi

y₁, yi+1

y_N o

, u_i >0andv_i >maxn

0, γ_1i, γ_2io , where γ_1i = −u_ih_id_i+α_iu_i(x_N −x₁)d₁

y_i−α_iy₁ , γ_2i = u_ih_id_i+1−α_iu_i(x_N −x₁)d_N

y_i+1−α_iy_N , i = 1, 2, . . ., N −1.

Proof. For each nodexj, j = 1,2, . . . , N, we obtain Φ(L_i(x_j)) = α_iΦ(x_j) + P_i(θ_j)

Qi(θj), θ_j = x_j −x₁ xN −x1

, i= 1,2, . . . , N −1.

Assume that α_i ≥ 0, i = 1,2, . . . , N −1. Also u_i > 0 and v_i >0 gives Q_i(θ_j) > 0. So Φ(L_i(x_j))>0, i= 1,2, . . . , N −1, j = 1,2, . . . , N, if P_i(θ_j)>0. Now, we have

Pi(θj) = A1,i(1−θj)³+A2,iθj(1−θj)²+A3,iθj2

(1−θj) +A4,iθj3

. It can be seen that P_i(θ_j)>0 ifA_1,i >0, A_2,i >0, A_3,i>0 and A_4,i >0. We get

A_1,i>0 ifα_i < y_i

y₁ andA_4,i >0 ifα_i < y_i+1 y_N . Let 0≤α_i <n

yi

y1, ^y_yⁱ⁺¹

N

o

. Then we have

A_2,i>0 if v_i > −u_ih_id_i+α_iu_i(x_N −x₁)d₁ y_i −α_iy₁ .

Also, we get

A3,i >0 ifvi > u_ih_id_i+1−α_iu_i(x_N −x₁)d_N y_i+1−α_iy_N .

From the above conditions, it is clear that Φ(L_i(x_j)) > 0 for all i = 1,2, . . . , N −1, j = 1,2, . . . , N if the scaling factors and the shape parameters satisfy the sufficient conditions given in the statement of Theorem 2.1.2. Since the rational cubic FIF has recursive nature, therefore, Φ(Li(xj)) > 0, for all i = 1,2, . . . , N −1, j = 1,2, . . . , N which in turn gives Φ(x)>0 for all x∈[x₁, x_N].

Remark 2.1.6. If α_i = 0, i = 1,2, . . . , N −1, then the sufficient conditions for the classical rational spline C to be positive are

u_i >0andv_i >maxn

0,−u_ih_id_i

y_i , u_ih_id_i+1 y_i+1

o . Monotonicity

Let{(x_i, y_i) :i= 1,2, . . . , N}be a monotonic data. Letd_i be the derivative value at the knotx_i. Without loss of generality assume that the data is monotonically increasing i.e., y₁ ≤y₂ ≤ . . .≤ y_N. Then ∆_i = (y_i+1−y_i)/(x_i+1 −x_i)≥ 0, i = 1,2, . . . , N −1. From calculus, Φis monotonically increasing in [x₁, x_N] ifΦ⁽¹⁾(x)≥0, for allx∈[x₁, x_N].We have

Φ⁽¹⁾(L_i(x)) = α_iΦ⁽¹⁾(x)

a_i + Ψ_i(θ) (Q_i(θ))², where Ψ_i(θ) =

5

X

k=1

B_k,iθ^k−1(1−θ)^5−k,

B_1,i =u²_id^∗_i,

B_2,i = 2u_i{(3u_i+v_i)∆^∗_i −u_id^∗_i+1},

B_3,i = 3u²_i∆^∗_i + (3u_i+v_i){(3u_i+v_i)∆^∗_i −u_id^∗_i+1−u_id^∗_i}, B_4,i = 2u_i{(3u_i+v_i)∆^∗_i −u_id^∗_i},

B_5,i =u²_id^∗_i+1, d^∗_i =d_i− α_id₁

ai

, d^∗_i+1 =d_i+1− α_id_N ai

and ∆^∗_i = ∆_i−α_iy_N −y₁ hi

.

Theorem 2.1.3. Let {(x_i, y_i, d_i) : i= 1,2, . . . , N} be a monotonically increasing data.

Let the derivative values satisfy the necessary condition for monotonicity, i.e.,sgn(di) = sgn(d_i+1) = sgn(∆_i).Then the following conditions on the scaling factors and the shape parameters are sufficient to the rational cubic FIF Φdefined in (2.4) to be monotonically increasing.

0≤α_i <

(

a_i, a_id_i d1

, a_id_i+1 dN

, y_i+1−y_i yN −y1

) , u_i >0andv_i >maxn

0, u_i(d^∗_i +d^∗_i+1)

∆^∗_i

o

, i= 1,2, . . . , N −1.

Proof. For each nodex_j, j = 1, 2,. . ., N, we have Φ⁽¹⁾(L_i(x_j)) = α_iΦ⁽¹⁾(x_j)

ai

+ Ψ_i(θ_j)

(Qi(θj))², θ_j = x_j −x₁ xN −x1

, i= 1,2, . . . , N −1.

Assume α_i ≥ 0, i = 1,2, . . . , N −1. It can be seen that (Q_i(θ_j))² > 0. Therefore, Φ⁽¹⁾(L_i(x_j))≥0, if Ψ_i(θ_j)≥0. We have

Ψ_i(θ_j) = B_1,i(1−θ_j)⁴+B_2,iθ_j(1−θ_j)³+B_3,iθ_j²(1−θ_j)²+B_4,iθ_j³(1−θ_j) +B_5,iθ_j⁴. It can be seen that Ψ_i(θ_j) ≥ 0, if B_1,i ≥ 0, B_2,i ≥ 0, B_3,i ≥ 0, B_4,i ≥ 0 and B_5,i ≥ 0.

We have

B_1,i ≥0 ifα_i ≤ a_id_i

d₁ , B_5,i ≥0 ifα_i ≤ a_id_i+1 d_N . Let 0≤α_i <n

aidi

d1 , ^ai_d^di+1

N , ^y_y^i+1−yi

N−y1

o

.Then, we get B_2,i ≥0 ifv_i ≥ u_id^∗_i+1

∆^∗_i , B_3,i≥0 ifv_i ≥ u_i(d^∗_i +d^∗_i+1)

∆^∗_i , B_5,i≥0 ifv_i ≥ uid^∗_i

∆^∗_i . So according to the conditions prescribed in Theorem 2.1.3, it is clear that Φ⁽¹⁾(L_i(x_j))

≥0 for allj = 1,2, . . . , N, i= 1,2, . . . , N−1. SinceΦ⁽¹⁾ is also a fractal function and it has recursive nature, the conditionΦ⁽¹⁾(L_i(x_j))≥0,i= 1,2, . . . , N−1, j = 1,2, . . . , N gives Φ⁽¹⁾(x)≥0 for all x∈[x₁, x_N].

Remark 2.1.7. If α_i = 0, i= 1,2, . . . , N−1then the sufficient conditions for classical rational cubic spline (2.6) which preserves monotonicity are

u_i >0andv_i >maxn

0, u_i(d_i+d_i+1)

∆i

o

, i= 1,2, . . . , N −1.

Convexity

Assume that the data {(x_i, y_i) : i = 1,2, . . . , N} is strictly convex. To avoid the pos- sibility of straight line segments, assume that d₁ < ∆₁ < · · · < d_i < ∆_i < d_i+1 <

· · · < ∆_N−1 < d_N. Since Φ belongs to C¹, Φ is convex if Φ⁽²⁾(x⁺) or Φ⁽²⁾(x⁻) exists and non-negative for all x ∈ (x₁, x_N) [96]. In this section, sufficient conditions on the scaling factors and the shape parameters will be determined to ensure the convexity of the rational cubic FIF (2.4).

Theorem 2.1.4. Suppose {(x_i, y_i) :i= 1,2, . . . , N} is a strictly convex data. Let d_i be the derivative value at the knot x_i. Let the derivative values satisfy d₁ < ∆₁ < · · · <

d_i < ∆_i < d_i+1 < · · · < ∆_N−1 < d_N. Then sufficient conditions on the scaling factors and the shape parameters to ensure convexity of Φ in (2.4) are

0≤αi <min n

a²_i, hi(di+1−∆i)

dN(xN −x1)−(yN −y1), hi(∆i−di)

(yN −y1)−d1(xN −x1),ai(di+1−di) (dN −d1)

o ,

ui >0andvi >max n

0, u_i(d^∗_i+1−∆^∗_i)

(∆^∗_i −d^∗_i) , u_i(∆^∗_i −d^∗_i) (d^∗_i+1−∆^∗_i)

o . Proof. Informally, we have

Φ⁽²⁾(L_i(x)) = αiΦ⁽²⁾(x)

a²_i + Ψ^∗_i(θ) h_i(Q_i(θ))³, where Ψ^∗_i(θ) =

6

X

k=1

C_k,iθ^k−1(1−θ)^6−k,

C_1,i= 2u²_i{(2u_i +v_i)(∆^∗_i −d^∗_i)−u_i(d^∗_i+1−∆^∗_i)},

C_2,i= 2u²_i{7u_i(∆^∗_i −d^∗_i) + 2v_i(∆^∗_i −d^∗_i)−2u_i(d^∗_i+1−∆^∗_i)}, C_3,i= 2u_i{(6u²_i +u_iv_i)(∆^∗_i −d^∗_i) + 2u²_i(d^∗_i+1−d^∗_i)},

C4,i= 2ui{(6u²_i +uivi)(d^∗_i+1−∆^∗_i) + 2u²_i(d^∗_i+1−d^∗_i)},

C5,i= 2u²_i{7ui(d^∗_i+1−∆^∗_i) + 2vi(d^∗_i+1−∆^∗_i)−2ui(∆^∗_i −d^∗_i)}, C6,i= 2u²_i{(2ui +vi)(d^∗_i+1−∆^∗_i)−ui(∆^∗_i −d^∗_i)}.

Now, we get

Φ⁽²⁾(x⁺₁) = C_1,1 h₁u³₁

h 1− α₁

a²₁ i−1

, (2.17)

Φ⁽²⁾(x⁻_N) = C6,N−1

hN−1u³_N−1 h

1− αN−1

a²_N−1 i−1

, (2.18)

Φ⁽²⁾(x⁺_n) = α_n

a²_nΦ⁽²⁾(x⁺₁) + C_1,n

u³_nh_n, n= 2,3, . . . , N −1. (2.19) Let 0≤α_i < a²_i, i= 1,2, . . . , N −1.From (2.17), (2.18) and (2.19), it is evident that if C_1,i ≥ 0, i = 1,2, . . . , N −1 and C_6,N−1 ≥ 0, then the right-handed second derivatives at the knots x_i, i = 1,2, . . . , N −1 and the left-handed second derivative at x_N are non-negative. For a knot point x_n, n= 1,2, . . . , N −1, we get

Φ⁽²⁾(L_i(x⁺_n)) = α_iΦ⁽²⁾(x⁺_n)

a²_i +R_i(x⁺_n), i= 1,2, . . . , N −1,

where R_i(x) =R_i(x₁ +θ(x_N −x₁)) = Ψ^∗_i(θ)/(h_i(Q_i(θ))³). Assuming that C_1,i ≥0, i= 1,2, . . . , N −1, we get Φ⁽²⁾(L_i(x⁺_n)) ≥ 0 if R_i(x⁺_n) ≥ 0. Note that R_i(x⁺_n) ≥ 0 if the coefficients C_j,i ≥0, for j = 1,2, . . . ,6. From the Three Chords Lemma for the convex functions [145], it follows that for a strictly convex interpolant, the end point derivatives should necessarily satisfy d1 <(yN −y1)/(xN −x1)< dN. One can see that

(∆^∗_i −d^∗_i)>0 ifα_i < h_i(∆_i−d_i)

(y_N −y₁)−d₁(x_N −x₁), (d^∗_i+1−∆^∗_i)>0 ifα_i < h_i(d_i+1−∆_i)

d_N(x_N −x₁)−(y_N −y₁), (d^∗_i+1−d^∗_i)>0 ifα_i < ai(di+1−di)

(d_N −d₁) . Let

0≤α_i <minn h_i(d_i+1−∆_i)

dN(xN −x1)−(yN −y1), h_i(∆_i−d_i)

(yN −y1)−d1(xN −x1),a_i(d_i+1−d_i) (dN −d1)

o

. (2.20)

It can be seen that

C_1,i ≥0 ⇔ (2u_i+v_i)(∆^∗_i −d^∗_i)−u_i(d^∗_i+1−∆^∗_i)≥0.

(2u_i+v_i)(∆^∗_i −d^∗_i)−u_i(d^∗_i+1−∆^∗_i)≥0 ifv_i ≥ u_i(d^∗_i+1−∆^∗_i) (∆^∗_i −d^∗_i) . C2,i ≥0 ⇔ 7ui(∆^∗_i −d^∗_i) + 2vi(∆^∗_i −d^∗_i)−2ui(d^∗_i+1−∆^∗_i)≥0.

7u_i(∆^∗_i −d^∗_i) + 2v_i(∆^∗_i −d^∗_i)−2u_i(d^∗_i+1−∆^∗_i)≥0 if v_i ≥ ui(d^∗_i+1−∆^∗_i) (∆^∗_i −d^∗_i) . C_3,i≥0 ⇔ (6u²_i +u_iv_i)(∆^∗_i −d^∗_i) + 2u²_i(d^∗_i+1−d^∗_i)≥0.

The assumption on the scaling factors given in (2.20) ensures that C_3,i ≥0.

C_4,i ≥0 ⇔ (6u²_i +u_iv_i)(d^∗_i+1−∆^∗_i) + 2u²_i(d^∗_i+1−d^∗_i)≥0.

The assumption on the scaling factors given in (2.20) shows that C4,i ≥ 0. Similarly, one can observe that

C5,i ≥0 ifvi ≥ u_i(∆^∗_i −d^∗_i) (d^∗_i+1−∆^∗_i), C_6,i ≥0 ifv_i ≥ u_i(∆^∗_i −d^∗_i)

(d^∗_i+1−∆^∗_i).

Thus the conditions on the scaling factors and the shape parameters given in Theorem 2.1.4 ensure thatC_j,i ≥0, i= 1,2, . . . , N−1, j = 1,2, . . . ,6 and hence the non-negativity of Φ⁽²⁾(L_i(x⁺_n)) for i, n = 1,2, . . . , N −1, Φ⁽²⁾(x⁻_N). The non-negativity of Φ⁽²⁾(L_i(x⁺_n)) for i, n = 1,2, . . . , N −1, and Φ⁽²⁾(x⁻_N) ensure that the non-negativity of Φ⁽²⁾(x⁺) or Φ⁽²⁾(x⁻) for x∈(x₁, x_N).

Remark 2.1.8. If αi = 0, then the conditions u_i >0andv_i >maxn

0, u_i(d_i+1−∆_i)

(∆_i−d_i) , u_i(∆_i−d_i) (d_i+1−∆_i)

o ,

are the sufficient conditions for the classical rational cubic spline C to be convex.

Remark 2.1.9. If ∆_i−d_i = 0 or d_i+1−∆_i = 0, then take α_i = 0, d_i =d_i+1 = ∆_i. In this case, Φbecomes a straight lineΦ(L_i(x)) =y_i(1−θ) +y_i+1θ in the interval [x_i, x_i+1].

Constrained Interpolation

Let {(x_i, y_i) : i = 1,2, . . . , N} be a data such that it lies above the straight line p = mx+c, i.e.,y_i > p_i wherep_i =mx_i+c. Then, in general, the rational cubic FIFΦmay not lie above the linep=mx+cin [x₁, x_N]. In this section, sufficient conditions on the scaling factors and the shape parameters are derived such that Φ would lie above the

straight line p=mx+c. The straight linep=mx+ccan be written asp₁(1−θ) +p_Nθ, θ = (x−x1)/(x1 −xN), x∈ [x1, xN]. The rational cubic FIF Φ lies above the straight line p if Φ(x) > p₁(1− θ) +p_Nθ, for all x ∈ [x₁, x_N]. Since the graph of Φ is the attractor of the IFS (2.5), it is evident that Φ(x)> p1(1−θ) +pNθ, for all x∈[x1, xN] if F_i(x, y)> p_i(1−θ) +p_i+1θ for all (x, y) such that x∈ [x₁, x_N], y > p₁(1−θ) +p_Nθ, i= 1,2, . . . , N −1.

Theorem 2.1.5. Let {(x_i, y_i) : i = 1,2, . . . , N} be the data such that it lies above the straight line p=mx+c, i.e., y_i > p_i where p_i =mx_i+c. Then the rational cubic FIF Φ defined in (2.4) lies above the straight line p=mx+cin [x₁, x_N], if the scaling factors and the shape parameters satisfy the following conditions:

0≤α_i <min (

a_i, y_i−p_i y1−p1

, y_i+1−p_i+1 yN −pN

) ,

u_i >0andv_i >maxn

0, δ_1,i, δ_2,io , where

δ_1,i = −u_i[(y_i−p_i+1)−α_i(y₁−p_N)]−u_i[h_id_i−α_i(x_N −x₁)d₁] (yi−pi)−αi(y1−p1) , δ2,i = −u_i[(y_i+1−p_i)−α_i(y_N −p₁)]−u_i[−h_id_i+1+α_i(x_N −x₁)d_N]

(y_i+1−p_i+1)−α_i(y_N −p_N) , i= 1,2, . . . , N −1.

Proof. For each fixedi= 1,2, . . . , N−1, let 0≤α_i < a_i and (x, y) such thatx∈[x₁, x_N] and y > p₁(1−θ) +p_Nθ. It is evident that yα_i >[p₁(1−θ) +p_Nθ]α_i. We have

α_iy+ P_i(θ)

Q_i(θ) >[p₁(1−θ) +p_Nθ]α_i+ P_i(θ) Q_i(θ). To prove F_i(x, y)> p_i(1−θ) +p_i+1θ, it is enough to prove that

[p₁(1−θ) +p_Nθ]α_i+ P_i(θ)

Qi(θ) > p_i(1−θ) +p_i+1θ. (2.21) Now (2.21) can be written as

Q_i(θ)[(α_ip₁−p_i)(1−θ) + (α_ip_N −p_i+1)θ] +P_i(θ)

Q_i(θ) >0. (2.22)

Since Q_i(θ)>0, to prove (2.22), it is sufficient to show that Q_i(θ)[(α_ip₁−p_i)(1−θ) + (αipN−pi+1)θ]+Pi(θ)>0. The numeratorQi(θ)[(αip1−pi)(1−θ)+(αipN−pi+1)θ]+Pi(θ) can be written as Q_i(θ)[(α_ip₁ −p_i)(1 −θ) + (α_ip_N −p_i+1)θ] +P_i(θ) = D_1,i(1−θ)³ + D2,iθ(1−θ)²+D3,iθ²(1−θ) +D4,iθ³, where

D_1,i=u_i[(y_i−p_i)−α_i(y₁−p₁)], D_4,i =u_i[(y_i+1−p_i+1)−α_i(y_N −p_N)], D_2,i= (2u_i+v_i)[(y_i−p_i)−α_i(y₁−p₁)] +u_i[(y_i−p_i+1)−α_i(y₁−p_N)]

+u_i[h_id_i−α_i(x_N −x₁)d₁],

D_3,i= (2u_i+v_i)[(y_i+1−p_i+1)−α_i(y_N −p_N)] +u_i[(y_i+1−p_i)−α_i(y_N −p₁)]

+u_i[−h_id_i+1+α_i(x_N −x₁)d_N].

It can be seen that Q_i(θ)[(α_ip₁ − p_i)(1 −θ) + (α_ip_N −p_i+1)θ] +P_i(θ) > 0 if all the coefficients D_j,i>0, j = 1,2, . . . ,4. We have

D_1,i>0 ifα_i < y_i−p_i y1−p1

, D_4,i >0 ifα_i < y_i+1−p_i+1 yN −pN

. Let 0 ≤α_i <minn

yi−pi

y1−p1, ^yi+1_y ^−pi+1

N−pN

o

. The coefficient D_2,i >0 if

v_i > −ui[(yi−pi+1)−αi(y1−pN)]−ui[hidi−αi(xN −x1)d1] (y_i−p_i)−α_i(y₁−p₁) . The coefficient D_3,i >0 if

vi > −u_i[(y_i+1−p_i)−α_i(y_N −p₁)]−u_i[−h_id_i+1+α_i(x_N −x₁)d_N] (y_i+1−p_i+1)−α_i(y_N −p_N) .

Thus the sufficient conditions on the scaling factors and the shape parameters to satisfy (2.21) are

0≤α_i <minn

a_i, y_i−p_i

y₁−p₁, y_i+1−p_i+1 y_N −p_N

o , ui >0 andvi >max

n

δ1,i, δ2,i

o , i= 1,2, . . . , N −1.