# Error Analysis for a Noisy Lacunary Cubic Spline Interpolation and a Simple Noisy Cubic Spline Quasi Interpolation

Authors: Feng-Gong Lang; Xiao-Ping Xu
Journal: Advances in Numerical Analysis
 (2014)

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## Abstract

We mainly present the error analysis for two new cubic spline based methods; one is a lacunary interpolation method and the other is a very simple
quasi interpolation method. The new methods are able to reconstruct a function and its first two derivatives from noisy function data. The explicit error bounds for the methods
are given and proved. Numerical tests and comparisons are performed. Numerical results verify the efficiency of our methods.


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## Body

## 1. Introduction

Cubic spline, as the most commonly used spline in practice, is a fundamental approximation tool [1–11]. Nowadays, it has been widely used in many fields such as numerical analysis, computer aided geometric design, mathematical modeling, and engineering problems. Essentially, cubic spline is a twice differentiable piecewise cubic polynomial defined over a partitioned interval.Mathematically, cubic spline interpolation is often introduced as follows. Lety(x) be a function defined over [a,b], let
(1)yj=y(xj),j=0,1,…,n,
be a set of given function data at the nodes
(2)a=x0<x1<⋯<xn-1<xn=b,
and let
(3)y0′=y′(x0),yn′=y′(xn)
be two boundary derivatives. Then, there exists a unique cubic spline s(x) satisfying
(4)s′(x0)=y0′,s′(xn)=yn′,s(xj)=yj,j=0,1,…,n.However, we often meet two troubles in the practical applications of cubic spline interpolation. The first trouble is that we cannot obtain the precise function values in (1). They generally involve some unavoidable measurement noise. The second trouble is that it often lacks the boundary derivatives in (3).To deal with the troubles, in this paper, we give two new effective cubic spline based methods for reconstructingy(x), y′(x), and y′′(x) from the given noisy data
(5)y~j=y(xj)+ɛj,j=0,1,…,n,
where ɛj is the measurement noise. The first one is a noisy lacunary interpolation method (Method I) and the second one is a very simple noisy quasi interpolation method (Method II). The error bounds of the methods, which have not been studied before and are important and useful for the users of cubic spline, are mainly studied in this paper.We organize the remainder of this paper as follows. In Section2, we present some useful preliminaries; in Section 3, we give the new methods; in Section 4, we present the theoretical results of the errors; in Section 5, we perform some numerical tests to verify the error analysis; finally, we conclude this paper in Section 6.


## 2. Preliminaries

### 2.1. Cubic B-Splines

We assume that the nodes in (2) are equidistant because this case is very common in practice. The nodes produce a uniform partition Δ for [a,b] with mesh size h=(b-a)/n. The dimension of the cubic spline space over Δ is n+3. The corresponding cubic B-splines are given below [4–8]. For i=2,3,…,n-2, let
(6)Bi(x)=16h3{(x-xi-2)3,ifx∈[xi-2,xi-1](x-xi-2)3-4(x-xi-1)3,ifx∈[xi-1,xi](xi+2-x)3-4(xi+1-x)3,ifx∈[xi,xi+1](xi+2-x)3,ifx∈[xi+1,xi+2]0,else.
The other six B-splines B-1(x), B0(x), B1(x), Bn-1(x), Bn(x), and Bn+1(x) are generated by the translation, where
(7)B-1(x)=16h3{(x1-x)3,ifx∈[x0,x1]0,else,B0(x)=16h3{(x2-x)3-4(x1-x)3,ifx∈[x0,x1](x2-x)3,ifx∈[x1,x2]0,else,B1(x)=16h3{(x-x0+h)3-4(x-x0)3,ifx∈[x0,x1](x3-x)3-4(x2-x)3,ifx∈[x1,x2](x3-x)3,ifx∈[x2,x3]0,else,Bn-1(x)=16h3{(x-xn-3)3,ifx∈[xn-3,xn-2](x-xn-3)3-4(x-xn-2)3,ifx∈[xn-2,xn-1](xn-x+h)3-4(xn-x)3,ifx∈[xn-1,xn]0,else,Bn(x)=16h3{(x-xn-2)3,ifx∈[xn-2,xn-1](x-xn-2)3-4(x-xn-1)3,ifx∈[xn-1,xn]0,else,Bn+1(x)=16h3{(x-xn-1)3,ifx∈[xn-1,xn]0,else.
They are linearly independent, nonnegative, and locally supported. Moreover,
(8)Bi(k)(x)=Bi+1(k)(x+h),k=0,1,2;i=2,3,…,n-3.
The values of Bi(x), Bi′(x), and Bi′′(x) at the nodes are listed in Table 1.Table 1
The values ofBi(k)(x)  (i=-1,0,…,n+1;k=0,1,2) at the nodes.
x
i
-
1
x
i
x
i
+
1
Else
B
i
(
x
)
1
6
4
6
1
6
0
B
i
′
(
x
)
1
2
h
0
-
1
2
h
0
B
i
′
′
(
x
)
1
h
2
-
2
h
2
1
h
2
0


### 2.2. Approximate Boundary Derivatives

Using two-point numerical differentiation formula, we have(9)y~0′=y~1-y~0h=y1-y0h+ɛ1-ɛ0h=y0′+h2y′′(ξ1)+ɛ1-ɛ0h=y0′+ɛ0′,y~n′=y~n-y~n-1h=yn-yn-1h+ɛn-ɛn-1h=yn′-h2y′′(ξ2)+ɛn-ɛn-1h=yn′+ɛn′.
Similar results can be obtained by using three-point and five-point numerical differentiation formulae. See Tables 2 and 3, where ξ1∈(x0,x1), ξ2∈(xn-1,xn), ξ3∈(x0,x2), ξ4∈(xn-2,xn), ξ5∈(x0,x4), and ξ6∈(xn-4,xn) and ɛ0′ and ɛn′ represent the computational truncated errors to y′(x0) and y′(xn). They arise from the used numerical differentiation formulae and the above-mentioned measurement noise.Table 2
The approximate boundary derivatives and their errors (I).
Two-point results
Three-point results
y
~
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0
y
~
1
-
y
~
0
h
-
3
y
~
0
+
4
y
~
1
-
y
~
2
2
h
ε
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′
0
h
2
y
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′
(
ξ
1
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ε
1
-
ε
0
h
-
h
2
3
y
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′
′
(
ξ
3
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+
-
3
ε
0
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1
-
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2
2
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y
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n
y
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n
-
y
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n
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1
h
y
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n
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2
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4
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1
+
3
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n
2
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ε
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′
n
-
h
2
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(
ξ
2
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n
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n
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1
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-
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2
3
y
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′
′
(
ξ
4
)
+
ε
n
-
2
-
4
ε
n
-
1
+
3
ε
n
2
hTable 3
The approximate boundary derivatives and their errors (II).
Five-point results
y
~
′
0
-
25
y
~
0
+
48
y
~
1
-
36
y
~
2
+
16
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3
-
3
y
~
4
12
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ε
~
′
0
-
h
4
5
y
(
5
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(
ξ
5
)
+
-
25
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0
+
48
ε
1
-
36
ε
2
+
16
ε
3
-
3
ε
4
12
h
y
~
′
n
3
y
~
n
-
4
-
16
y
~
n
-
3
+
36
y
~
n
-
2
-
48
y
~
n
-
1
+
25
y
~
n
12
h
ε
~
′
n
-
h
4
5
y
(
5
)
(
ξ
6
)
+
3
ε
n
-
4
-
16
ε
n
-
3
+
36
ε
n
-
2
-
48
ε
n
-
1
+
25
ε
n
12
h


## 2.1. Cubic B-Splines

We assume that the nodes in (2) are equidistant because this case is very common in practice. The nodes produce a uniform partition Δ for [a,b] with mesh size h=(b-a)/n. The dimension of the cubic spline space over Δ is n+3. The corresponding cubic B-splines are given below [4–8]. For i=2,3,…,n-2, let
(6)Bi(x)=16h3{(x-xi-2)3,ifx∈[xi-2,xi-1](x-xi-2)3-4(x-xi-1)3,ifx∈[xi-1,xi](xi+2-x)3-4(xi+1-x)3,ifx∈[xi,xi+1](xi+2-x)3,ifx∈[xi+1,xi+2]0,else.
The other six B-splines B-1(x), B0(x), B1(x), Bn-1(x), Bn(x), and Bn+1(x) are generated by the translation, where
(7)B-1(x)=16h3{(x1-x)3,ifx∈[x0,x1]0,else,B0(x)=16h3{(x2-x)3-4(x1-x)3,ifx∈[x0,x1](x2-x)3,ifx∈[x1,x2]0,else,B1(x)=16h3{(x-x0+h)3-4(x-x0)3,ifx∈[x0,x1](x3-x)3-4(x2-x)3,ifx∈[x1,x2](x3-x)3,ifx∈[x2,x3]0,else,Bn-1(x)=16h3{(x-xn-3)3,ifx∈[xn-3,xn-2](x-xn-3)3-4(x-xn-2)3,ifx∈[xn-2,xn-1](xn-x+h)3-4(xn-x)3,ifx∈[xn-1,xn]0,else,Bn(x)=16h3{(x-xn-2)3,ifx∈[xn-2,xn-1](x-xn-2)3-4(x-xn-1)3,ifx∈[xn-1,xn]0,else,Bn+1(x)=16h3{(x-xn-1)3,ifx∈[xn-1,xn]0,else.
They are linearly independent, nonnegative, and locally supported. Moreover,
(8)Bi(k)(x)=Bi+1(k)(x+h),k=0,1,2;i=2,3,…,n-3.
The values of Bi(x), Bi′(x), and Bi′′(x) at the nodes are listed in Table 1.Table 1
The values ofBi(k)(x)  (i=-1,0,…,n+1;k=0,1,2) at the nodes.
x
i
-
1
x
i
x
i
+
1
Else
B
i
(
x
)
1
6
4
6
1
6
0
B
i
′
(
x
)
1
2
h
0
-
1
2
h
0
B
i
′
′
(
x
)
1
h
2
-
2
h
2
1
h
2
0


## 2.2. Approximate Boundary Derivatives

Using two-point numerical differentiation formula, we have(9)y~0′=y~1-y~0h=y1-y0h+ɛ1-ɛ0h=y0′+h2y′′(ξ1)+ɛ1-ɛ0h=y0′+ɛ0′,y~n′=y~n-y~n-1h=yn-yn-1h+ɛn-ɛn-1h=yn′-h2y′′(ξ2)+ɛn-ɛn-1h=yn′+ɛn′.
Similar results can be obtained by using three-point and five-point numerical differentiation formulae. See Tables 2 and 3, where ξ1∈(x0,x1), ξ2∈(xn-1,xn), ξ3∈(x0,x2), ξ4∈(xn-2,xn), ξ5∈(x0,x4), and ξ6∈(xn-4,xn) and ɛ0′ and ɛn′ represent the computational truncated errors to y′(x0) and y′(xn). They arise from the used numerical differentiation formulae and the above-mentioned measurement noise.Table 2
The approximate boundary derivatives and their errors (I).
Two-point results
Three-point results
y
~
′
0
y
~
1
-
y
~
0
h
-
3
y
~
0
+
4
y
~
1
-
y
~
2
2
h
ε
~
′
0
h
2
y
′
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(
ξ
1
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+
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1
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ε
0
h
-
h
2
3
y
′
′
′
(
ξ
3
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+
-
3
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0
+
4
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1
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ε
2
2
h
y
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n
y
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n
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y
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n
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1
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2
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ξ
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n
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1
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h
2
3
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′
′
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(
ξ
4
)
+
ε
n
-
2
-
4
ε
n
-
1
+
3
ε
n
2
hTable 3
The approximate boundary derivatives and their errors (II).
Five-point results
y
~
′
0
-
25
y
~
0
+
48
y
~
1
-
36
y
~
2
+
16
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~
3
-
3
y
~
4
12
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ε
~
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0
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h
4
5
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5
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25
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48
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1
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36
ε
2
+
16
ε
3
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3
ε
4
12
h
y
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n
3
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-
16
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3
+
36
y
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2
-
48
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1
+
25
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~
n
12
h
ε
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n
-
h
4
5
y
(
5
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(
ξ
6
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+
3
ε
n
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4
-
16
ε
n
-
3
+
36
ε
n
-
2
-
48
ε
n
-
1
+
25
ε
n
12
h


## 3. Two New Methods

### 3.1. Method I

We study the following noisy lacunary cubic spline interpolation (NLCSI) problem. We hope to find a cubic spline s~(x) satisfying
(10)s~(xj)=y~j,j=0,1,2,…,n.
To make theNLCSI problem uniquely solvable, it requires using two approximate boundary derivatives in Section 2.2. Obviously, there also exists a unique noisy lacunary cubic spline s~(x) satisfying
(11)s~′(x0)=y~0′,s~′(xn)=y~n′,s~(xj)=y~j,j=0,1,2,…,n.Lets~(x)=∑i=-1n+1‍c~iBi(x) be the cubic spline determined by (11), where the unknown coefficients c~i(i=-1,0,…,n+1) can be obtained by solving the linear system
(12)(-101141141⋱⋱⋱141141-101)(c~-1c~0c~1⋮c~n-1c~nc~n+1)=(2hy~0′6y~06y~1⋮6y~n-16y~n2hy~n′)
followed from Table 1. Furthermore, we can use s~′(x)=∑i=-1n+1‍c~iBi′(x) and s~′′(x)=∑i=-1n+1‍c~iBi′′(x) to approximate y′(x) and y′′(x), respectively.


### 3.2.  Method II

By using the given function data, we can directly get a cubic spline(13)s~~(x)=∑i=-1n+1‍y~iBi(x),
where y~-1=y(x0-h)+ɛ-1 and y~n+1=y(xn+h)+ɛn+1. We can also use s~~(x), s~~′(x), and s~~′′(x) to approximate y(x), y′(x), and y′′(x), respectively.The method is very simple and effective method for noisy data because it avoids using approximate boundary derivatives and also avoids solving the linear system (12).


## 3.1. Method I

We study the following noisy lacunary cubic spline interpolation (NLCSI) problem. We hope to find a cubic spline s~(x) satisfying
(10)s~(xj)=y~j,j=0,1,2,…,n.
To make theNLCSI problem uniquely solvable, it requires using two approximate boundary derivatives in Section 2.2. Obviously, there also exists a unique noisy lacunary cubic spline s~(x) satisfying
(11)s~′(x0)=y~0′,s~′(xn)=y~n′,s~(xj)=y~j,j=0,1,2,…,n.Lets~(x)=∑i=-1n+1‍c~iBi(x) be the cubic spline determined by (11), where the unknown coefficients c~i(i=-1,0,…,n+1) can be obtained by solving the linear system
(12)(-101141141⋱⋱⋱141141-101)(c~-1c~0c~1⋮c~n-1c~nc~n+1)=(2hy~0′6y~06y~1⋮6y~n-16y~n2hy~n′)
followed from Table 1. Furthermore, we can use s~′(x)=∑i=-1n+1‍c~iBi′(x) and s~′′(x)=∑i=-1n+1‍c~iBi′′(x) to approximate y′(x) and y′′(x), respectively.


## 3.2.  Method II

By using the given function data, we can directly get a cubic spline(13)s~~(x)=∑i=-1n+1‍y~iBi(x),
where y~-1=y(x0-h)+ɛ-1 and y~n+1=y(xn+h)+ɛn+1. We can also use s~~(x), s~~′(x), and s~~′′(x) to approximate y(x), y′(x), and y′′(x), respectively.The method is very simple and effective method for noisy data because it avoids using approximate boundary derivatives and also avoids solving the linear system (12).


## 4. Main Results

### 4.1. Error Analysis for Method I

We denote (12) by AC~=Y~.Lemma 1.
Ais invertible and ∥A-1∥∞≤2.Proof.
Add column one to column three and also add columnn+3 to column n+1, and we get a strictly diagonally dominant matrix
(14)A1=(-100142141⋱⋱⋱141241001)(n+3)×(n+3).
Obviously, A1 is invertible and A1=AP, where
(15)P=(101101⋱101101)(n+3)×(n+3).
We have
(16)A-1=PA1-1,∥A1-1∥∞≤1,∥P∥∞=2.
Hence, we have ∥A-1∥∞=∥PA1-1∥∞≤∥P∥∞∥A1-1∥∞≤2.Lets(x)=∑i=-1n+1‍ciBi(x) be the cubic spline determined by (4),
(17)C=(c-1,c0,…,cn,cn+1)T,Y=(2hy0′,6y0,…,6yn,2hyn′)T,
and then we have AC=Y. Let
(18)ɛ′=max⁡{|ɛ0′|,|ɛn′|},ɛ=max⁡0≤i≤n{|ɛi|},
and we have
(19)∥Y~-Y∥∞≤max⁡{2hɛ′,6ɛ};
see Table 4 for the results, where M5=∥y(5)(x)∥∞=max⁡a≤x≤b|y(5)(x)| and M4, M3, and M2 are defined similarly.Table 4
The bounds ofε′ and ∥Y~-Y∥∞.
Two-point results
Three-point results
Five-point results
ε
′
≤
1
2
M
2
h
+
2
ε
h
1
3
M
3
h
2
+
4
ε
h
1
5
M
5
h
4
+
32
ε
3
h
∥
Y
~
-
Y
∥
∞
≤
max
⁡
{
M
2
h
2
+
4
ε
,
6
ε
}
2
3
M
3
h
3
+
8
ε
2
5
M
5
h
5
+
64
ε
3Lemma 2.
Consider∥C~-C∥∞≤2∥Y~-Y∥∞.Proof.
Consider∥C~-C∥∞=∥A-1(Y~-Y)∥∞≤∥A-1∥∞∥Y~-Y∥∞≤2∥Y~-Y∥∞.Lemma 3.
Consider∑i=-1n+1‍|Bi(x)|≡1, ∑i=-1n+1‍|Bi′(x)|≤3/2h, and ∑i=-1n+1‍|Bi′′(x)|≤4/h2.Proof.
Because of property (8), we only need to check them over a typical subinterval [xj,xj+1]. By differentiating (6), for a general i, we have
(20)Bi′(x)=12h3{(x-xi-2)2,ifx∈[xi-2,xi-1](x-xi-2)2-4(x-xi-1)2,ifx∈[xi-1,xi]-(xi+2-x)2+4(xi+1-x)2,ifx∈[xi,xi+1]-(xi+2-x)2,ifx∈[xi+1,xi+2]0,else,Bi′′(x)=1h3{x-xi-2,ifx∈[xi-2,xi-1]-3x-xi-2+4xi-1,ifx∈[xi-1,xi]xi+2-4xi+1+3x,ifx∈[xi,xi+1]xi+2-x,ifx∈[xi+1,xi+2]0,else.
All of them are locally supported over four adjacent subintervals. (i)
By the nonnegativity and partition of unity of cubic B-splines, forx∈[xj,xj+1], we have
(21)∑i=-1n+1‍|Bi(x)|=∑i=-1n+1‍Bi(x)=∑i=j-1j+2‍Bi(x)≡1.(ii)
Forx∈[xj,xj+1], we have
(22)∑i=-1n+1‍|Bi′(x)|=|Bj-1′(x)|+|Bj′(x)|+|Bj+1′(x)|+|Bj+2′(x)|=12h3[(x-xj+1)2+|4(x-xj+1)2-(x-xj+2)2|+|(x-xj-1)2-4(x-xj)2|+(x-xj)2|4(x-xj+1)2-(x-xj+2)2|]=12h3[-4(x-xj)2+4h(x-xj)+2h2]≤32h.(iii)
Forx∈[xj,xj+1], we have
(23)∑i=-1n+1‍|Bi′′(x)|=|Bj-1′′(x)|+|Bj′′(x)|+|Bj+1′′(x)|+|Bj+2′′(x)|=1h3[(xj+1-x)+|3x+xj+2-4xj+1|+|-3x-xj-1+4xj|+(x-xj)]=1h3{4h-6(x-xj),x∈[xj,xj+h3]2h,x∈[xj+h3,xj+2h3]6(x-xj)-2h,x∈[xj+2h3,xj+1]≤4h2.Lemma 4.
Lets(x) and s~(x) be the cubic spline interpolants of y(x) determined by (4) and (11), respectively. Then we have
(24)|s(x)-s~(x)|≤2∥Y~-Y∥∞,|s′(x)-s~′(x)|≤3h∥Y~-Y∥∞,|s′′(x)-s~′′(x)|≤8h2∥Y~-Y∥∞.Proof.
First of all, fork=0,1,2, we have
(25)|s(k)(x)-s~(k)(x)|=|∑i=-1n+1‍(ci-c~i)Bi(k)(x)|≤∥C~-C∥∞∑i=-1n+1‍|Bi(k)(x)|.
And then using Lemmas 2 and 3, we get these results.Theorem 5.
Lets~(x) be the noisy lacunary cubic spline interpolant of y(x) determined by (11). Then one has
(26)|s~(x)-y(x)|≤5384M4h4+2∥Y~-Y∥∞,|s~′(x)-y′(x)|≤124M4h3+3h∥Y~-Y∥∞,|s~′′(x)-y′′(x)|≤38M4h2+8h2∥Y~-Y∥∞.Proof.
These results follow from the traditional cubic spline interpolation error theory [1, 3, 7], Lemma 4, and the following triangle inequality
(27)|s~(k)(x)-y(k)(x)|≤|s~(k)(x)-s(k)(x)|+|s(k)(x)-y(k)(x)|,k=0,1,2.


### 4.2. Error Analysis for Method II

From (13) and Table 1, for j=0,1,…,n, we have
(28)s~~(xj)=y~j-1+4y~j+y~j+16,(29)s~~′(xj)=y~j+1-y~j-12h,(30)s~~′′(xj)=y~j-1-2y~j+y~j+1h2.
It is a surprise to find that s~~′(xj) and s~~′′(xj) are the same as the well-known central numerical differentiation formulae.Lemma 6.
Lets~~(x) be the noisy cubic spline quasi interpolant of y(x) determined by (13). Then for j=0,1,…,n, one has
(31)|s~~(xj)-y(xj)|≤16M2h2+ɛ,(32)|s~~′(xj)-y′(xj)|≤16M3h2+ɛh,(33)|s~~′′(xj)-y′′(xj)|≤112M4h2+4ɛh2.Proof.
By (30), we have
(34)|s~~′′(xj)-y′′(xj)|=|y~j-1-2y~j+y~j+1h2-y′′(xj)|=|yj-1-2yj+yj+1h2-y′′(xj)+ɛj-1-2ɛj+ɛj+1h2|≤112M4h2+4ɛh2.
The proofs of (31) and (32) are similar, which are omitted.Theorem 7.
Lets~~(x) be the noisy cubic spline quasi interpolant of y(x) determined by (13). Then we have
(35)|s~~(x)-y(x)|≤1384M4h4+124M3h3+16M2h2+54ɛ,(36)|s~~′(x)-y′(x)|≤548M4h3+16M3h2+3hɛ,(37)|s~~′′(x)-y′′(x)|≤524M4h2+4h2ɛ.Proof.
We first prove (37). s~~(x) is a cubic spline; hence s~~′′(x) is a piecewise continuous linear function over [a,b] with respect to the partition Δ. Let L(x) be the piecewise linear interpolant to y′′(x) with respect to Δ. For j=1,2,…,n, let s~~j′′(x) and Lj(x) be the restriction of s~~′′(x) and L(x) over [xj-1,xj]. Then we have
(38)s~~j′′(x)=s~~′′(xj-1)xj-xh+s~~′′(xj)x-xj-1h,Lj(x)=y′′(xj-1)xj-xh+y′′(xj)x-xj-1h.
For j=1,2,…,n, by (38) and (33), we get
(39)|s~~j′′(x)-Lj(x)|=|(s~~′′(xj-1)-y′′(xj-1))xj-xh+(s~~′′(xj)-y′′(xj))x-xj-1h|≤112M4h2+4h2ɛ.
Hence, for all x∈[a,b], we have
(40)|s~~′′(x)-L(x)|≤112M4h2+4h2ɛ.
Moreover, by the piecewise linear polynomial interpolation theory [1, 3, 7], for all x∈[a,b], we have
(41)|y′′(x)-L(x)|≤18M4h2.
Then (37) follows immediately from (40) and (41).
Next, we prove (36). For j=1,2,…,n, let s~~j′(x) be the restriction of s~~′(x) over [xj-1,xj]. Then for x∈[xj-1,xj], by (32) and (37), we have(42)|s~~j′(x)-y′(x)|={|∫xj-1x‍(s~~j′′(t)-y′′(t))dt+(s~~j′(xj-1)-y′(xj-1))|,x∈[xj-1,xj-1+h2]|∫xjx‍(s~~j′′(t)-y′′(t))dt+(s~~j′(xj)-y′(xj))|,x∈[xj-1+h2,xj]≤{∫xj-1xj-1+h/2‍|s~~j′′(t)-y′′(t)|dt+|s~~j′(xj-1)-y′(xj-1)|,x∈[xj-1,xj-1+h2]∫xj-1+h/2xj‍|s~~j′′(t)-y′′(t)|dt+|s~~j′(xj)-y′(xj)|,x∈[xj-1+h2,xj]≤548M4h3+16M3h2+3hɛ.
Finally, we prove (35). For every subinterval [xj-1,xj], j=1,2,…,n, we give
(43)αj1(x)=2(x-xj-1)3-3h(x-xj-1)2+h3h3,αj2(x)=-2(x-xj-1)3+3h(x-xj-1)2h3,βj1(x)=(x-xj)2(x-xj-1)h2,βj2(x)=(x-xj)(x-xj-1)2h2.
They are very useful in cubic Hermite interpolation, and we also have
(44)αj1(x)≥0,αj2(x)≥0,βj1(x)≥0,βj2(x)≤0,αj1(x)+αj2(x)=1,βj1(x)-βj2(x)=(xj-1-x)(x-xj)h≤h4.
Let s~~j(x)=s~~(x)|[xj-1,xj] be the restriction of s~~(x) over [xj-1,xj]; then it can also be written as
(45)s~~j(x)=s~~(xj-1)αj1(x)+s~~(xj)αj2(x)+s~~′(xj-1)βj1(x)+s~~′(xj)βj2(x).
Let
(46)Hj(x)=y(xj-1)αj1(x)+y(xj)αj2(x)+y′(xj-1)βj1(x)+y′(xj)βj2(x)
be the cubic Hermite interpolant of y(x) over [xj-1,xj]; then for x∈[xj-1,xj], by using (31), (32), (44), (45), and (46), we have
(47)|y(x)-Hj(x)|≤1384M4h4,|s~~j(x)-Hj(x)|=|(s~~(xj-1)-y(xj-1))αj1(x)(s~~(xj)-y(xj))+(s~~(xj)-y(xj))αj2(x)+(s~~′(xj-1)-y′(xj-1))βj1(x)+(s~~′(xj)-y′(xj))βj2(x)(s~~(xj)-y(xj))|≤(16M2h2+ɛ)+h4(16M3h2+ɛh)=124M3h3+16M2h2+54ɛ.
By the triangle inequality, we get (35).


## 4.1. Error Analysis for Method I

We denote (12) by AC~=Y~.Lemma 1.
Ais invertible and ∥A-1∥∞≤2.Proof.
Add column one to column three and also add columnn+3 to column n+1, and we get a strictly diagonally dominant matrix
(14)A1=(-100142141⋱⋱⋱141241001)(n+3)×(n+3).
Obviously, A1 is invertible and A1=AP, where
(15)P=(101101⋱101101)(n+3)×(n+3).
We have
(16)A-1=PA1-1,∥A1-1∥∞≤1,∥P∥∞=2.
Hence, we have ∥A-1∥∞=∥PA1-1∥∞≤∥P∥∞∥A1-1∥∞≤2.Lets(x)=∑i=-1n+1‍ciBi(x) be the cubic spline determined by (4),
(17)C=(c-1,c0,…,cn,cn+1)T,Y=(2hy0′,6y0,…,6yn,2hyn′)T,
and then we have AC=Y. Let
(18)ɛ′=max⁡{|ɛ0′|,|ɛn′|},ɛ=max⁡0≤i≤n{|ɛi|},
and we have
(19)∥Y~-Y∥∞≤max⁡{2hɛ′,6ɛ};
see Table 4 for the results, where M5=∥y(5)(x)∥∞=max⁡a≤x≤b|y(5)(x)| and M4, M3, and M2 are defined similarly.Table 4
The bounds ofε′ and ∥Y~-Y∥∞.
Two-point results
Three-point results
Five-point results
ε
′
≤
1
2
M
2
h
+
2
ε
h
1
3
M
3
h
2
+
4
ε
h
1
5
M
5
h
4
+
32
ε
3
h
∥
Y
~
-
Y
∥
∞
≤
max
⁡
{
M
2
h
2
+
4
ε
,
6
ε
}
2
3
M
3
h
3
+
8
ε
2
5
M
5
h
5
+
64
ε
3Lemma 2.
Consider∥C~-C∥∞≤2∥Y~-Y∥∞.Proof.
Consider∥C~-C∥∞=∥A-1(Y~-Y)∥∞≤∥A-1∥∞∥Y~-Y∥∞≤2∥Y~-Y∥∞.Lemma 3.
Consider∑i=-1n+1‍|Bi(x)|≡1, ∑i=-1n+1‍|Bi′(x)|≤3/2h, and ∑i=-1n+1‍|Bi′′(x)|≤4/h2.Proof.
Because of property (8), we only need to check them over a typical subinterval [xj,xj+1]. By differentiating (6), for a general i, we have
(20)Bi′(x)=12h3{(x-xi-2)2,ifx∈[xi-2,xi-1](x-xi-2)2-4(x-xi-1)2,ifx∈[xi-1,xi]-(xi+2-x)2+4(xi+1-x)2,ifx∈[xi,xi+1]-(xi+2-x)2,ifx∈[xi+1,xi+2]0,else,Bi′′(x)=1h3{x-xi-2,ifx∈[xi-2,xi-1]-3x-xi-2+4xi-1,ifx∈[xi-1,xi]xi+2-4xi+1+3x,ifx∈[xi,xi+1]xi+2-x,ifx∈[xi+1,xi+2]0,else.
All of them are locally supported over four adjacent subintervals. (i)
By the nonnegativity and partition of unity of cubic B-splines, forx∈[xj,xj+1], we have
(21)∑i=-1n+1‍|Bi(x)|=∑i=-1n+1‍Bi(x)=∑i=j-1j+2‍Bi(x)≡1.(ii)
Forx∈[xj,xj+1], we have
(22)∑i=-1n+1‍|Bi′(x)|=|Bj-1′(x)|+|Bj′(x)|+|Bj+1′(x)|+|Bj+2′(x)|=12h3[(x-xj+1)2+|4(x-xj+1)2-(x-xj+2)2|+|(x-xj-1)2-4(x-xj)2|+(x-xj)2|4(x-xj+1)2-(x-xj+2)2|]=12h3[-4(x-xj)2+4h(x-xj)+2h2]≤32h.(iii)
Forx∈[xj,xj+1], we have
(23)∑i=-1n+1‍|Bi′′(x)|=|Bj-1′′(x)|+|Bj′′(x)|+|Bj+1′′(x)|+|Bj+2′′(x)|=1h3[(xj+1-x)+|3x+xj+2-4xj+1|+|-3x-xj-1+4xj|+(x-xj)]=1h3{4h-6(x-xj),x∈[xj,xj+h3]2h,x∈[xj+h3,xj+2h3]6(x-xj)-2h,x∈[xj+2h3,xj+1]≤4h2.Lemma 4.
Lets(x) and s~(x) be the cubic spline interpolants of y(x) determined by (4) and (11), respectively. Then we have
(24)|s(x)-s~(x)|≤2∥Y~-Y∥∞,|s′(x)-s~′(x)|≤3h∥Y~-Y∥∞,|s′′(x)-s~′′(x)|≤8h2∥Y~-Y∥∞.Proof.
First of all, fork=0,1,2, we have
(25)|s(k)(x)-s~(k)(x)|=|∑i=-1n+1‍(ci-c~i)Bi(k)(x)|≤∥C~-C∥∞∑i=-1n+1‍|Bi(k)(x)|.
And then using Lemmas 2 and 3, we get these results.Theorem 5.
Lets~(x) be the noisy lacunary cubic spline interpolant of y(x) determined by (11). Then one has
(26)|s~(x)-y(x)|≤5384M4h4+2∥Y~-Y∥∞,|s~′(x)-y′(x)|≤124M4h3+3h∥Y~-Y∥∞,|s~′′(x)-y′′(x)|≤38M4h2+8h2∥Y~-Y∥∞.Proof.
These results follow from the traditional cubic spline interpolation error theory [1, 3, 7], Lemma 4, and the following triangle inequality
(27)|s~(k)(x)-y(k)(x)|≤|s~(k)(x)-s(k)(x)|+|s(k)(x)-y(k)(x)|,k=0,1,2.


## 4.2. Error Analysis for Method II

From (13) and Table 1, for j=0,1,…,n, we have
(28)s~~(xj)=y~j-1+4y~j+y~j+16,(29)s~~′(xj)=y~j+1-y~j-12h,(30)s~~′′(xj)=y~j-1-2y~j+y~j+1h2.
It is a surprise to find that s~~′(xj) and s~~′′(xj) are the same as the well-known central numerical differentiation formulae.Lemma 6.
Lets~~(x) be the noisy cubic spline quasi interpolant of y(x) determined by (13). Then for j=0,1,…,n, one has
(31)|s~~(xj)-y(xj)|≤16M2h2+ɛ,(32)|s~~′(xj)-y′(xj)|≤16M3h2+ɛh,(33)|s~~′′(xj)-y′′(xj)|≤112M4h2+4ɛh2.Proof.
By (30), we have
(34)|s~~′′(xj)-y′′(xj)|=|y~j-1-2y~j+y~j+1h2-y′′(xj)|=|yj-1-2yj+yj+1h2-y′′(xj)+ɛj-1-2ɛj+ɛj+1h2|≤112M4h2+4ɛh2.
The proofs of (31) and (32) are similar, which are omitted.Theorem 7.
Lets~~(x) be the noisy cubic spline quasi interpolant of y(x) determined by (13). Then we have
(35)|s~~(x)-y(x)|≤1384M4h4+124M3h3+16M2h2+54ɛ,(36)|s~~′(x)-y′(x)|≤548M4h3+16M3h2+3hɛ,(37)|s~~′′(x)-y′′(x)|≤524M4h2+4h2ɛ.Proof.
We first prove (37). s~~(x) is a cubic spline; hence s~~′′(x) is a piecewise continuous linear function over [a,b] with respect to the partition Δ. Let L(x) be the piecewise linear interpolant to y′′(x) with respect to Δ. For j=1,2,…,n, let s~~j′′(x) and Lj(x) be the restriction of s~~′′(x) and L(x) over [xj-1,xj]. Then we have
(38)s~~j′′(x)=s~~′′(xj-1)xj-xh+s~~′′(xj)x-xj-1h,Lj(x)=y′′(xj-1)xj-xh+y′′(xj)x-xj-1h.
For j=1,2,…,n, by (38) and (33), we get
(39)|s~~j′′(x)-Lj(x)|=|(s~~′′(xj-1)-y′′(xj-1))xj-xh+(s~~′′(xj)-y′′(xj))x-xj-1h|≤112M4h2+4h2ɛ.
Hence, for all x∈[a,b], we have
(40)|s~~′′(x)-L(x)|≤112M4h2+4h2ɛ.
Moreover, by the piecewise linear polynomial interpolation theory [1, 3, 7], for all x∈[a,b], we have
(41)|y′′(x)-L(x)|≤18M4h2.
Then (37) follows immediately from (40) and (41).
Next, we prove (36). For j=1,2,…,n, let s~~j′(x) be the restriction of s~~′(x) over [xj-1,xj]. Then for x∈[xj-1,xj], by (32) and (37), we have(42)|s~~j′(x)-y′(x)|={|∫xj-1x‍(s~~j′′(t)-y′′(t))dt+(s~~j′(xj-1)-y′(xj-1))|,x∈[xj-1,xj-1+h2]|∫xjx‍(s~~j′′(t)-y′′(t))dt+(s~~j′(xj)-y′(xj))|,x∈[xj-1+h2,xj]≤{∫xj-1xj-1+h/2‍|s~~j′′(t)-y′′(t)|dt+|s~~j′(xj-1)-y′(xj-1)|,x∈[xj-1,xj-1+h2]∫xj-1+h/2xj‍|s~~j′′(t)-y′′(t)|dt+|s~~j′(xj)-y′(xj)|,x∈[xj-1+h2,xj]≤548M4h3+16M3h2+3hɛ.
Finally, we prove (35). For every subinterval [xj-1,xj], j=1,2,…,n, we give
(43)αj1(x)=2(x-xj-1)3-3h(x-xj-1)2+h3h3,αj2(x)=-2(x-xj-1)3+3h(x-xj-1)2h3,βj1(x)=(x-xj)2(x-xj-1)h2,βj2(x)=(x-xj)(x-xj-1)2h2.
They are very useful in cubic Hermite interpolation, and we also have
(44)αj1(x)≥0,αj2(x)≥0,βj1(x)≥0,βj2(x)≤0,αj1(x)+αj2(x)=1,βj1(x)-βj2(x)=(xj-1-x)(x-xj)h≤h4.
Let s~~j(x)=s~~(x)|[xj-1,xj] be the restriction of s~~(x) over [xj-1,xj]; then it can also be written as
(45)s~~j(x)=s~~(xj-1)αj1(x)+s~~(xj)αj2(x)+s~~′(xj-1)βj1(x)+s~~′(xj)βj2(x).
Let
(46)Hj(x)=y(xj-1)αj1(x)+y(xj)αj2(x)+y′(xj-1)βj1(x)+y′(xj)βj2(x)
be the cubic Hermite interpolant of y(x) over [xj-1,xj]; then for x∈[xj-1,xj], by using (31), (32), (44), (45), and (46), we have
(47)|y(x)-Hj(x)|≤1384M4h4,|s~~j(x)-Hj(x)|=|(s~~(xj-1)-y(xj-1))αj1(x)(s~~(xj)-y(xj))+(s~~(xj)-y(xj))αj2(x)+(s~~′(xj-1)-y′(xj-1))βj1(x)+(s~~′(xj)-y′(xj))βj2(x)(s~~(xj)-y(xj))|≤(16M2h2+ɛ)+h4(16M3h2+ɛh)=124M3h3+16M2h2+54ɛ.
By the triangle inequality, we get (35).


## 5. Numerical Tests and Discussions

### 5.1. Numerical Tests

In this section, we perform numerical tests by Matlab. The following examples(48)f1(x)=51+x2,x∈[-1,1],f2(x)=ex,x∈[-1,1],
are considered.In every numerical test, the mesh sizeh and the measurement noise bound ɛ are both given. Because the measurement noises ɛi(i=0,1,…,n) are random, we let ɛi=ɛ·ri, where ri(i=0,1,…,n) are random numbers and satisfy -1≤ri≤1.In Tables5 and 6, Methods I-1, I-2, and I-3 represent Method I with two-point, three-point, and five-point approximate boundary derivatives, respectively. CSM represents the cubic spline method in [11]. E0, E1, and E2 are the maximum absolute error of the function, the first order derivative, and the second order derivative, respectively.Table 5
Numerical results off1(x).
Method
I-1
I-2
I-3
II
CSM [11]
ε
=
1
0
-
4
h
=
0.2
E
0
7.8
×
1
0
-
3
1.6
×
1
0
-
3
5.9
×
1
0
-
4
6.0
×
1
0
-
2
4.4
×
1
0
-
3
E
1
2.4
×
1
0
-
1
4.7
×
1
0
-
2
1.3
×
1
0
-
2
1.4
×
1
0
-
1
2.1
×
1
0
-
2
E
2
4.16
0.75
0.41
0.38
0.45
ε
=
1
0
-
4
h
=
0.1
E
0
2.0
×
1
0
-
3
1.3
×
1
0
-
4
1.2
×
1
0
-
4
1.6
×
1
0
-
2
3.4
×
1
0
-
4
E
1
1.2
×
1
0
-
1
4.7
×
1
0
-
3
3.0
×
1
0
-
3
3.8
×
1
0
-
2
3.1
×
1
0
-
3
E
2
4.29
0.15
0.14
0.10
0.15
ε
=
1
0
-
3
h
=
0.2
E
0
8.0
×
1
0
-
3
1.8
×
1
0
-
3
1.3
×
1
0
-
3
6.0
×
1
0
-
2
4.5
×
1
0
-
3
E
1
2.4
×
1
0
-
1
4.7
×
1
0
-
2
1.8
×
1
0
-
2
1.5
×
1
0
-
1
2.3
×
1
0
-
2
E
2
4.17
0.75
0.43
0.39
0.45
ε
=
1
0
-
3
h
=
0.1
E
0
2.2
×
1
0
-
3
1.0
×
1
0
-
3
9.1
×
1
0
-
4
1.6
×
1
0
-
2
9.4
×
1
0
-
4
E
1
1.2
×
1
0
-
1
1.6
×
1
0
-
2
1.0
×
1
0
-
2
4.0
×
1
0
-
2
1.1
×
1
0
-
2
E
2
4.29
0.65
0.42
0.29
0.44Table 6
Numerical results off2(x).
Method
I-1
I-2
I-3
II
CSM [11]
ε
=
1
0
-
5
h
=
0.2
E
0
8.0
×
1
0
-
3
9.8
×
1
0
-
4
1.7
×
1
0
-
5
1.3
×
1
0
-
2
8.2
×
1
0
-
5
E
1
6.8
×
1
0
-
2
8.3
×
1
0
-
3
2.3
×
1
0
-
4
1.4
×
1
0
-
2
1.1
×
1
0
-
4
E
2
1.17
0.13
6.8
×
1
0
-
3
7.3
×
1
0
-
3
7.3
×
1
0
-
3
ε
=
1
0
-
5
h
=
0.1
E
0
2.0
×
1
0
-
3
1.3
×
1
0
-
4
1.0
×
1
0
-
5
3.9
×
1
0
-
3
1.2
×
1
0
-
5
E
1
3.5
×
1
0
-
2
2.2
×
1
0
-
3
2.7
×
1
0
-
4
4.1
×
1
0
-
3
1.1
×
1
0
-
4
E
2
1.21
7.5
×
1
0
-
2
8.3
×
1
0
-
3
3.6
×
1
0
-
3
8.4
×
1
0
-
3
ε
=
1
0
-
4
h
=
0.2
E
0
8.0
×
1
0
-
3
9.8
×
1
0
-
4
1.0
×
1
0
-
4
1.3
×
1
0
-
2
1.2
×
1
0
-
4
E
1
6.8
×
1
0
-
2
8.3
×
1
0
-
3
1.4
×
1
0
-
3
1.4
×
1
0
-
2
4.8
×
1
0
-
4
E
2
1.17
0.13
0.019
0.010
0.021
ε
=
1
0
-
4
h
=
0.1
E
0
2.0
×
1
0
-
3
1.5
×
1
0
-
4
7.3
×
1
0
-
5
3.9
×
1
0
-
3
8.5
×
1
0
-
5
E
1
3.5
×
1
0
-
2
2.6
×
1
0
-
3
2.9
×
1
0
-
3
4.0
×
1
0
-
3
1.0
×
1
0
-
3
E
2
1.22
0.08
0.08
0.03
0.09


### 5.2. Discussions

Generally, the maximum absolute errorsE0, E1, and E2 vary if one of ɛ and h does. If h is fixed and ɛ decreases, then the maximum absolute errors E0, E1, and E2 will decrease. But if ɛ is fixed while h decreases, the errors will not decrease necessarily; they maybe increase sometimes. See the theoretical results in Theorem 5 and Theorem 7 and the numerical results in Tables 5 and 6.Whenh and ɛ are both fixed in a specific test, it is easy to find that E0 and E1 of Method I-2 and Method I-3 are better than those of Method I-1 and Method II, while E2 of Method I-3 and Method II are better than Method I-1 and Method I-2. See Tables 5 and 6.It is very reasonable to compare our methods with the cubic spline method (CSM) in [11] because our methods are also based on cubic spline. From Tables 5 and 6, we find that the errors of Method I-3 are overall better than CSM in [11]. At the same time, E2 of Method II are better than CSM in [11]. In summary, when approximating a function, we advise using Method I-3, Method I-2, and CSM [11]; when approximating its first order derivative, we advise using Method I-3 and CSM [11]; when approximating its second order derivative, we advise using Method I-3, Method II, and CSM [11].Whenɛ=0, Method I-1 and Method II are O(h2) methods, Method I-2 is an O(h3) method, and Method I-3 is an O(h4) method. The cubic spline method (CSM) [11] is also an O(h4) method, while the method in [12] is an O(h2) method; the method in [13] is an O(h2.5log⁡h) method conditionally, only if the shape parameter c=O(h) therein. Obviously, the approximation orders of Method I-2 and Method I-3 are higher than the methods in [12, 13], the approximation orders of Method I-1 and Method II equal that of the method in [12], and the approximation order of Method I-3 equals that of CSM [11]. Undoubtedly, our methods are full of approximation ability. Furthermore, [12, 13] have not studied first order and second order derivative approximations. At the same time, our methods are more suitable for noisy data than the methods in [12, 13]. Hence, Method I-2, Method I-3, and CSM [11] are more preferable than others.


## 5.1. Numerical Tests

In this section, we perform numerical tests by Matlab. The following examples(48)f1(x)=51+x2,x∈[-1,1],f2(x)=ex,x∈[-1,1],
are considered.In every numerical test, the mesh sizeh and the measurement noise bound ɛ are both given. Because the measurement noises ɛi(i=0,1,…,n) are random, we let ɛi=ɛ·ri, where ri(i=0,1,…,n) are random numbers and satisfy -1≤ri≤1.In Tables5 and 6, Methods I-1, I-2, and I-3 represent Method I with two-point, three-point, and five-point approximate boundary derivatives, respectively. CSM represents the cubic spline method in [11]. E0, E1, and E2 are the maximum absolute error of the function, the first order derivative, and the second order derivative, respectively.Table 5
Numerical results off1(x).
Method
I-1
I-2
I-3
II
CSM [11]
ε
=
1
0
-
4
h
=
0.2
E
0
7.8
×
1
0
-
3
1.6
×
1
0
-
3
5.9
×
1
0
-
4
6.0
×
1
0
-
2
4.4
×
1
0
-
3
E
1
2.4
×
1
0
-
1
4.7
×
1
0
-
2
1.3
×
1
0
-
2
1.4
×
1
0
-
1
2.1
×
1
0
-
2
E
2
4.16
0.75
0.41
0.38
0.45
ε
=
1
0
-
4
h
=
0.1
E
0
2.0
×
1
0
-
3
1.3
×
1
0
-
4
1.2
×
1
0
-
4
1.6
×
1
0
-
2
3.4
×
1
0
-
4
E
1
1.2
×
1
0
-
1
4.7
×
1
0
-
3
3.0
×
1
0
-
3
3.8
×
1
0
-
2
3.1
×
1
0
-
3
E
2
4.29
0.15
0.14
0.10
0.15
ε
=
1
0
-
3
h
=
0.2
E
0
8.0
×
1
0
-
3
1.8
×
1
0
-
3
1.3
×
1
0
-
3
6.0
×
1
0
-
2
4.5
×
1
0
-
3
E
1
2.4
×
1
0
-
1
4.7
×
1
0
-
2
1.8
×
1
0
-
2
1.5
×
1
0
-
1
2.3
×
1
0
-
2
E
2
4.17
0.75
0.43
0.39
0.45
ε
=
1
0
-
3
h
=
0.1
E
0
2.2
×
1
0
-
3
1.0
×
1
0
-
3
9.1
×
1
0
-
4
1.6
×
1
0
-
2
9.4
×
1
0
-
4
E
1
1.2
×
1
0
-
1
1.6
×
1
0
-
2
1.0
×
1
0
-
2
4.0
×
1
0
-
2
1.1
×
1
0
-
2
E
2
4.29
0.65
0.42
0.29
0.44Table 6
Numerical results off2(x).
Method
I-1
I-2
I-3
II
CSM [11]
ε
=
1
0
-
5
h
=
0.2
E
0
8.0
×
1
0
-
3
9.8
×
1
0
-
4
1.7
×
1
0
-
5
1.3
×
1
0
-
2
8.2
×
1
0
-
5
E
1
6.8
×
1
0
-
2
8.3
×
1
0
-
3
2.3
×
1
0
-
4
1.4
×
1
0
-
2
1.1
×
1
0
-
4
E
2
1.17
0.13
6.8
×
1
0
-
3
7.3
×
1
0
-
3
7.3
×
1
0
-
3
ε
=
1
0
-
5
h
=
0.1
E
0
2.0
×
1
0
-
3
1.3
×
1
0
-
4
1.0
×
1
0
-
5
3.9
×
1
0
-
3
1.2
×
1
0
-
5
E
1
3.5
×
1
0
-
2
2.2
×
1
0
-
3
2.7
×
1
0
-
4
4.1
×
1
0
-
3
1.1
×
1
0
-
4
E
2
1.21
7.5
×
1
0
-
2
8.3
×
1
0
-
3
3.6
×
1
0
-
3
8.4
×
1
0
-
3
ε
=
1
0
-
4
h
=
0.2
E
0
8.0
×
1
0
-
3
9.8
×
1
0
-
4
1.0
×
1
0
-
4
1.3
×
1
0
-
2
1.2
×
1
0
-
4
E
1
6.8
×
1
0
-
2
8.3
×
1
0
-
3
1.4
×
1
0
-
3
1.4
×
1
0
-
2
4.8
×
1
0
-
4
E
2
1.17
0.13
0.019
0.010
0.021
ε
=
1
0
-
4
h
=
0.1
E
0
2.0
×
1
0
-
3
1.5
×
1
0
-
4
7.3
×
1
0
-
5
3.9
×
1
0
-
3
8.5
×
1
0
-
5
E
1
3.5
×
1
0
-
2
2.6
×
1
0
-
3
2.9
×
1
0
-
3
4.0
×
1
0
-
3
1.0
×
1
0
-
3
E
2
1.22
0.08
0.08
0.03
0.09


## 5.2. Discussions

Generally, the maximum absolute errorsE0, E1, and E2 vary if one of ɛ and h does. If h is fixed and ɛ decreases, then the maximum absolute errors E0, E1, and E2 will decrease. But if ɛ is fixed while h decreases, the errors will not decrease necessarily; they maybe increase sometimes. See the theoretical results in Theorem 5 and Theorem 7 and the numerical results in Tables 5 and 6.Whenh and ɛ are both fixed in a specific test, it is easy to find that E0 and E1 of Method I-2 and Method I-3 are better than those of Method I-1 and Method II, while E2 of Method I-3 and Method II are better than Method I-1 and Method I-2. See Tables 5 and 6.It is very reasonable to compare our methods with the cubic spline method (CSM) in [11] because our methods are also based on cubic spline. From Tables 5 and 6, we find that the errors of Method I-3 are overall better than CSM in [11]. At the same time, E2 of Method II are better than CSM in [11]. In summary, when approximating a function, we advise using Method I-3, Method I-2, and CSM [11]; when approximating its first order derivative, we advise using Method I-3 and CSM [11]; when approximating its second order derivative, we advise using Method I-3, Method II, and CSM [11].Whenɛ=0, Method I-1 and Method II are O(h2) methods, Method I-2 is an O(h3) method, and Method I-3 is an O(h4) method. The cubic spline method (CSM) [11] is also an O(h4) method, while the method in [12] is an O(h2) method; the method in [13] is an O(h2.5log⁡h) method conditionally, only if the shape parameter c=O(h) therein. Obviously, the approximation orders of Method I-2 and Method I-3 are higher than the methods in [12, 13], the approximation orders of Method I-1 and Method II equal that of the method in [12], and the approximation order of Method I-3 equals that of CSM [11]. Undoubtedly, our methods are full of approximation ability. Furthermore, [12, 13] have not studied first order and second order derivative approximations. At the same time, our methods are more suitable for noisy data than the methods in [12, 13]. Hence, Method I-2, Method I-3, and CSM [11] are more preferable than others.


## 6. Conclusions

The explicit error bounds for a noisy lacunary cubic spline interpolation and a simple noisy cubic spline quasi interpolation are well studied in this paper; see Theorems5 and 7. These new results are very useful in numerical approximation and related practical fields. Moreover, these results are also verified by some numerical examples. In a word, both theoretical analysis and numerical tests show that our methods are well behaved. We end the paper with the following remarks.(i)
The main contributions of the paper include (i) studying two new methods to approximate a function and its first order and second order derivatives from the given noisy data and (ii) analyzing the explicit error bounds for the methods.(ii)
The main advantages of our new methods include the following: (i) they are very simple; (ii) they are not only applicable to noisy data but also applicable to exact data; (iii) Method I-2 and Method I-3 have better performance in function approximation and first order derivative approximation than other methods; Method I-3 and Method II have better performance in second order derivative approximation than other methods.



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*Source: 353194-2014-09-10.xml*