Abstract. This paper deals with the approximation properties of a rational cubic interpolation with linear denominator. The rational cubic spline based on function values only and the two families of parameters, in the description of the rational interpolant, modify the shape of the curve freely. Error expression of the values and the derivatives of interpolating functions are derived, which shows the interpaltion is stable

Hongling Geng

Meng Tian

CiteSeerX

Int. Journal of Math. Analysis, Vol. 5, 2011, no. 26, 1287 - 1294 
 
 
 
Error Analysis of a Rational Interpolation Spline  
 
 
Meng Tian and Hongling Geng 
 
 
Shandong University of Technology 
Zibo, 255049, P.R. China 
luckywalter@163.com   (Meng Tian) 
 
Abstract. This paper deals with the approximation properties of a rational cubic 
interpolation with linear denominator. The rational cubic spline based on function 
values only and the two families of parameters, in the description of the rational 
interpolant, modify the shape of the curve freely. Error expression of the values 
and the derivatives of interpolating functions are derived, which shows the 
interpaltion is stable. 
 
Keywords: Data visualization; Rational spline; Interpolation; Error analysis 
 
 
I. INTRODUCTION  
 
Smooth curve representation, to visualize the scientific data, is of great 
significance in the area of computer graphics. Smoothness is one of the very 
important requirements for a pleasing visual display. Since most of splines are 
polynomial splines, local modification of the interpolating curve is difficult under 
the conditions that the interpolating data are not changed. The cubic rational spline 
mentioned in this paper has two families of parameters which can change easily.  
The more important mathematical achievement of this method is that the 
uniqueness of the interpolating function for the given data would be replaced by  
the uniqueness of the interpolation for the given data and the selected parameters. 
For  the given interpolation data, the change of the parameters causes the change of 
the interpolation curve. When given suited parameters, the needed interpolation 
curve can be obtained.  
There are rational cubic splines with linear, quadratic or cubic denominator, 
which are effectively used in the design and modification of curves, such as region 
control, convexity control and monotonicity control[1-7]. So rational cubic spline 
is flexible in shape control, but it is hard to consider the approximation properties 
because of the appearance of the parameters. There are few paper in the literature 
on the approximation properties of the rational spline. In this paper, the 
approximation properties of a typical rational cubic spline restated in Section 2 are 
discussed. 
1288                                                                        Meng Tian and Hongling Geng 
 
 
II. THE RATIONAL CUBIC INTERPOLANT 
Let {( , ), 1,2, , }i ix f i n= L  be a given set of data points, where 1 2 nx x x< < <L , 
and  if   are  the function values at  the knots ix . For 1[ , ]i ix x x +∈ , Let 1 ,i i ih x x+= −       
( ) / ,i ix x hθ = −  (0 1)θ≤ ≤ .A piecewise rational cubic function ( )S x  is defined 
for 1[ , ]i ix x x +∈  as 
3 2
2 3
1 1 1
(1 ) (1 ) [(2 ) ]
(1 )[( 2 ) ]( ) .
(1 )
i i i i i i i i
i i i i i i i i
i
i i
u f u v f u h d
u v f v h d v fS x
u v
θ θ θ
θ θ θ
θ θ
+ + +
− + − + +
+ − + − +
=
− +
                (1) 
where id  are the derivative values at the knots ix , ,i iu v  are named shape 
parameters, and , 0i iu v > .The rational cubic interpolant has the following 
interpolatory properties: 
1 1 1 1( ) , ( ) , ( ) , ( ) ,i i i i i i i iS x f S x f S x d S x d+ + + +′ ′= = = =  
In most application, derivative parameters id  are not given and hence must be 
determined from the data ( , )i ix f . In [4], Duan etc. had dicussed, this kind of 
spline  where 1( ) /i i i id f f h+= −  , so ( ), max( )i i if d O h h h′− = = . Now we give 
another choice. An obvious choice is mentioned here:     
                                           
1 1 1
1 1 1 2 1 1 2
1 1 2 1 2 1
( )/( ), 2, , 1,
( ) /( ),
( ) /( ).
i i i i i i i
n n n n n n n
d h h h h i n
d h h h
d h h h
− − −
− − − − − −
= Δ + Δ + = −
=Δ + Δ −Δ +
=Δ + Δ −Δ +
L
                                 (2) 
    This method is based on three-point difference approximation for the id . A 
Newton series expansion analysis shows that 2( ), max( )i i if d O h h h′− = = [6]. For 
given bounded data, the derivative approximation using Eq.2 are bounded. Hence, 
for bounded values of the appropriate shape parameters ,i iu v , the interpolant Eq.1 
is bounded and unique.  
 
 
III. ERROR ESTIMATION OF THE INTERPOLATION FUNCTION  
 
   For the error estimation of the piecewise rational cubic interpolant Eq.1, we 
deal with the case when the knots are equally spaced, let 1i ih x x+= −  . Without loss 
of generality, it is necessary to consider the subinterval 
1[ , ]i ix x + .When
1
1( ) [ , ]nf x C x x∈ , and ( )S x is the rational cubic interpolatory 
function of ( )f x , using the Peano-Kernal Theorem [4] gives 
( ) ( ) ( ) ( ) ( )2
1
0
i
i
x
x xR f f t S t f R x dτ τ τ+− +⎡ ⎤′= − = ∫ −⎣ ⎦  
Error analysis of a rational interpolation spline                                              1289 
 
 
where 
( )
2
1
0
(1 ) ,                                         ;
2[(1 ) ]
(1 )[(1 )(2 ) (2 ) ] ,              ;
2[(1 ) ]
[(1 )(1 ) (3 ) ] ,           
2[(1 ) ]
i
i i
i i
i i
i
i i
x
i i
i i
u x x
u v
u v x x
u v
R x
u v
u v
θ θ τ
θ θ
θ θ θ θ θ τ
θ θ
τ
θ θ θ θ θ
θ θ
−
+
−
− < <
− +
− − + + −
< <
− +⎡ ⎤− =⎣ ⎦ − + + −
−
− + 1
2
1 1
          
           ;
(1 ) ,                                          .
2[(1 ) ]
i
i
i i
i i
x x
v x x
u v
τ
θ θ τ
θ θ
+
+ +
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪ < <
⎪
⎪
−⎪− < <⎪ − +⎩
 
1
1
1 2
( ), ;
( ), ;
                    
( ), ;
( ), .
i i
i
i
i i
p x x
q x x
r x x
k x x
τ τ
τ τ
τ τ
τ τ
−
+
+ +
< <⎧
⎪ < <⎪= ⎨ < <⎪
⎪ < <⎩
 
Obviously,  
1
1 1 2
( ) 0, ;               ( ) 0, ;
( ) 0, .                ( ) 0, .
i i i
i i i
p x x q x x
r x x k x x
τ τ τ τ
τ τ τ τ
−
+ + +
≤ < < ≥ < <
≤ < < ≥ < <
 
After complicated calculation, we obtain 
( )( ) ( ) ( )( ) ( )
2 1
1 1
2 1
1 1
( ) ( ) [ ( ) ( )  + ( ) ( ) ]
          ( ) { [ ( )] ( ) ( ) [ ( )] }
( ) (1 )
          { 1 1 3 2 5 2
2 1
i i i
i i i
i i i
i i i
i i i
i i
x x xxR f f x p d k d q d r dxx x x
x x xxf x p d k d q d r dxx x x
f x h
u u v
u v
τ τ τ τ τ τ τ τ
τ τ τ τ τ τ τ τ
θ θ
θ θ θ θ θ
θ θ
+ +
− +
+ +
− +
′≤ + +∫ ∫ ∫ ∫
′≤ − + + + −∫ ∫ ∫ ∫
′ −
≤ − + − + + −
− +
( )
( ) ( )( ) ( )
}
( ) 1
          { 1 2 3 }.
1
i
i i
i i
v
f x h
u v
u v
θ
θ θ
θ θ θ θ
θ θ
+
′ −
≤ − + + −⎡ ⎤⎣ ⎦− +
   
Theorem 1     Suppose 1 1( ) [ , ]nf x C x x∈ , and  1 2 nx x x< < <L  is an equal-knot 
spacing. For the given ,i iu v , ( )S x  is the corresponding rational cubic interpolation 
function given in Eq. 1, where  id  are replaced with Eq. 2. Then for  
1[ , ], ( 2, , 2)i ix x x i n+∈ = −L  there holds  
( )( ) iR f h f t c′≤  
where ( )
( ) ( )( ) ( )0 1
1
1 2 3
1maxi i ii i
c u v
u vθ
θ θ
θ θ θ θ
θ θ≤ ≤
−
= − + + −⎡ ⎤⎣ ⎦− +
, and ic is called the optimal 
error constant. 
As shown in Table1, although iu and iv vary in [0.01, 1000], the optimal error 
coefficients ic change little. This means that this kind of interpolation is stable.  
1290                                                                        Meng Tian and Hongling Geng 
 
 
TABLE I.  VALUES OF ic   FOR VARIOUS OF iu  AND iv  
i  iu  iv  θ  ic  
1 1000.0 1.0 0.5480 0.6311 
2 100.0 1.0 0.5470 0.6308 
3 10.0 1.0 0.5380 0.6289 
4 5.0 1.0 0.5300 0.6275 
5 1.5 1.0 0.5080 0.6252 
6 1.0 1.0 0.5000 0.6250 
7 1.0 1.5 0.4920 0.6252 
8 1.0 5.0 0.4700 0.6275 
9 1.0 10.0 0.4620 0.6289 
10 1.0 100.0 0.4530 0.6308 
11 1.0 1000.0 0.4520 0.6311 
 
Theorem 2   For given iu and iv ,  the optimal error constant ic is bounded ,  
and 
( )( )( ) ....968819176179.0
27
757417
8
5
=
+−−
≤≤ ic  
Proof.  Let 
( )
( ) ( )( ) ( )
1
( , , ) 1 2 3
1i i i ii i
u v u v
u v
θ θ
ω θ θ θ θ θ
θ θ
−
= − + + −⎡ ⎤⎣ ⎦− +
 and    i i iv uλ= , 
then       ( )[ ]1 (1 )(2 ) (3 )( , , ) ( , )
(1 )
i
i i i
i
u v U
θ θ θ θ θ θ λ
ω θ λ θ
θ θλ
− − + + −
= =
− +
. 
Since 
                                    0
))1((
)21()1(),(
2
22
≥
+−
−−
=
∂
∂
ii
iU
θλθ
θθθ
λ
θλ
, 
we can get when 1[0, ]
2
θ ∈ , ( )θλ ,iU is increasing about the parameter ),0( +∞∈iλ ,  
and ( )θλ ,iU is decreasing about the parameter ),0( +∞∈iλ when 
1[ ,1]
2
θ ∈  .  It is 
easy to known that  
1[0, )
2
θ ∈ , ( ) )2)(1(,lim
0
θθθθλ
λ
+−=
→ i
U
i
; ( ) )3)(1(,lim θθθθλ
λ
−−=
+∞→ i
U
i
.
1( ,1],
2
θ ∈ ( ) )3)(1(,lim
0
θθθθλ
λ
−−=
→ i
U
i
; ( ) )2)(1(,lim θθθθλ
λ
+−=
+∞→ i
U
i
. 
 
Error analysis of a rational interpolation spline                                              1291 
 
 
Next we only to look for the maximum and the minimum of )2)(1( θθθ +− and 
)3)(1( θθθ −−  on [0,1]θ ∈ , after complicate calculation, we obtain  
  ( )( ) ，
8
5,min
10
=
≤≤
θλ
θ i
U  where 5000.0=θ . 
( )( ) ( )( )( )
27
757417,max
10
+−−
=
≤≤
θλ
θ i
U , where 
3
74 −
=θ . 
 
 
IV. ERROR ESTIMATION  OF  THE DERIVATIVE  
 
       Since the interpolation is local, without generality it is necessary only to 
consider the error in the subinterval 1[ , ]i ix x + .Using the Peano-Kernel Theorem 
gives the following  
( ) ( ) ( ) ( ) ( )2
1
0
i
i
x
x xR f f t S t f R x dτ τ τ+− +⎡ ⎤′= − = ∫ −⎣ ⎦  
then  
( ) ( ) ( )1 2
1 1
( ) ( ) ( )
         ( ) ( ) ( ) ( )i i i
i i i
t x x
x x xx x x
R f f x P x
f p d f q d f k d f xτ τ τ τ τ τ τ τ τ+ +
− +
′ ′ ′= −
′ ′ ′ ′ ′ ′ ′= + + +∫ ∫ ∫
 
where   
2(1 ) [(1 )(1 2 ) 2 ]( ) 22[(1 ) ]
i i i
x
i i
u u vp
u v h
θ θ θ θτ
θ θ
− − − −′ = −
− +
 
                
2 2 2 2
2
(1 ) (1 2 ) (3 2 ) 6 (1 )( ) ,
2[(1 ) ]
i i i i
x
i i
u v u vq
u v h
θ θ θ θ θ θτ
θ θ
− + + − + −′ = −
− +
 
2
2
[2(1 ) (1 2 ) ]( ) .
2[(1 ) ]
i i i
x
i i
v u vk
u v h
θ θ θ θτ
θ θ
− + −′ =
− +
 
   Consider ( )xp τ′ , 1[ , ]i ix xτ +∈ , we found that we should first consider the 
equation  
2(1 )(1 2 ) 2i iu vθ θ θ− − − =0 
It can be proved that for any arbitrary positive parameters iu and iv , the 
equation above has one and only one root in [0,1], that  is 
( )
2
*
3 8
4
i i i i
i i
u u u v
u v
θ
− +
=
−
.It  
1292                                                                        Meng Tian and Hongling Geng 
 
 
is easy to shown that, when *θθ ≤ , ( ) 0xp τ′ ≤ and when *θθ ≥ , ( ) 0xp τ′ ≥ . Thus, 
when *θθ ≤ , 
   
1 1
2
2
(1 ) [(1 )(1 2 ) 2 ]| ( ) | ( ) .
2[(1 ) ]
i i
i i
x x i i i
x xx x
i i
u u vp d p d
u v
θ θ θ θτ τ τ τ
θ θ− −
− − − −′ ′= − =
− +∫ ∫             
And when *θθ ≥ , we get  
1 1
2
2
(1 ) [(1 )(1 2 ) 2 ]| ( ) | ( ) .
2[(1 ) ]
i i
i i
x x i i i
x xx x
i i
u u vp d p d
u v
θ θ θ θτ τ τ τ
θ θ− −
− − − −′ ′= =
− +∫ ∫  
Secondly, we consider the property of ( )xq τ′ , 1[ , ]i ix xτ +∈ as a function of τ . It 
is obvious that for all [0,1]θ ∈ , ( ) 0xq τ′ ≤ . So we get   
1 1
2 2 2 2
2
(1 ) (1 2 ) (3 2 ) 6 (1 )( ) ( )
2[(1 ) ]
i i
i i
x x i i i i
x xx x
i i
u v u vq d q d
u v
θ θ θ θ θ θτ τ τ τ
θ θ
+ + − + + − + −′ ′= − =
− +∫ ∫   .     
      Thirdly, in the similar way, let 22(1 ) (1 2 ) 0i iu vθ θ θ− + − =  , *θ  is   the only 
root on  [0,1], and  
2
* 4 8
4( )
i i i i i
i i
u v v u v
u v
θ
− − +
=
−
 
Obviously, when *θθ ≤ , ( ) 0xk τ′ ≥ and when
*θθ ≥ , ( ) 0xk τ′ ≤ . 
And when *θθ ≤ , 
                  2
1 1
2
2
[2(1 ) (1 2 ) ]| ( ) | ( ) .
2[(1 ) ]
i i
i i
x x i i i
x xx x
i i
v u vk d k d
u v
θ θ θ θτ τ τ τ
θ θ
+
+ −
− + −′ ′= =
− +∫ ∫                 
 when *θθ ≥ , we can get  
2
1 1
2
2
[2(1 ) (1 2 ) ]| ( ) | ( ) .
2[(1 ) ]
i i
i i
x x i i i
x xx x
i i
v u vk d k d
u v
θ θ θ θτ τ τ τ
θ θ
+
+ −
− + −′ ′= − = −
− +∫ ∫                 
By the analysis above, the following is easily to get. 
( )
2 2
*
*
3 8 4 8
0 1
4 4( )
i i i i i i i i i
i i i i
u u u v u v v u v
u v u v
θ θ
− + − − +
< = < = <
− −
. 
Theorem 2 Suppose 1 1( ) [ , ]nf x C x x∈ , and  1 2 nx x x< < <L  is an equal-knot 
spacing. For the given ,i iu v , ( )S x  is the corresponding rational cubic  
Error analysis of a rational interpolation spline                                              1293 
 
 
interpolation function given in Eq. 1, where  id  are replaced with Eq. 2. Then for  
1[ , ], ( 2, , 2)i ix x x i n+∈ = −L , the error of the derivative ( )S x′  satisfies  
( )( ) ( ) ( ) 1if x S x f x c′ ′ ′− ≤ + , 
Where   
0 1
( , , )maxi i ic W u v
θ
θ
≤ ≤
= .                                    
2
*2
3 2 3 2
*
*2
2
[2(1 ) (1 2 ) ]1 ,                     0 ;
[(1 ) ]
2[(1 ) ]( , , ) 2 ,                                   ;
[(1 ) ]
(1 ) [ (1 )(1 2 ) 2 ]1
[(1 )
i i i
i i
i i
i i
i i
i i i
i i
v u v
u v
u vW u v
u v
u u v
u v
θ θ θ θ θ θ
θ θ
θ θθ θ θ θ
θ θ
θ θ θ θ
θ θ
− + −
+ ≤ ≤
− +
− +
= − ≤ ≤
− +
− − − − +
+
− +
*
2 ,            1.]
θ θ
⎧
⎪
⎪
⎪⎪
⎨
⎪
⎪
⎪ ≤ ≤
⎪⎩
                       
Some values of ic  for different 0, 0i iu v> > are given in Table 2 .As shown in 
Table 2, although iu and iv  varies, the optimal error coefficients ic change little. 
This means that this kind of interpolation is stable. 
 
Table  2 : Values of ic for  various of  iu and iv  
i  iu  iv  θ  ic  
1 10.0 5.0 0.6132 1.5195 
2 5.0 10.0 0.3868 1.5195 
3 1.5 1.2 0.5971 1.5021 
4 1.1 0.9 0.5335 1.5017 
5 0.9 1.1 0.4665 1.5017 
6 0.5 0.4 0.5371 1.5021 
7 0.3 0.2 0.5670 1.5068 
8 0.2 0.3 0.4530 1.5068 
 
 
REFERENCES 
 
[1] J.A.Gregory, R.Delbourgo, Piecewise rational quadratic interpolation to 
monotonic data, IMA Journal of Numerical Analysis, 2(1982),123-130. 
 
[2] K.W. Brodlie, S.Butt, Preserving convexity using piecewise cubic 
interpolation, Computers and Graphics,15(1991),15-23. 
 
[3] Q.Duan, K.Djidjeli, W.G.Price, E.H.Twizell, A rational cubic spline based on 
function values, Computers and Graphics,22(1998),479-486. 
1294                                                                        Meng Tian and Hongling Geng 
 
 
 
[4] Q.Duan, K.Djidjeli, W.G.Price, E.H.Twizell, The approximation properties of 
some rational cubic splines, Jouanal of Computers Mathematics, 
72(1999),155-166. 
 
[5] R.Delbourgo, Shape preserving interpolation to convex data by rational 
functions with quadratic numerator and linear denominator,IMA 
J.Numer.Anal., 9(1989), 123-136. 
 
[6] R.H.Wang, G.Q.Zhu, Rational function approximation and its application. 
Science Press, Beijing(2004), 290-309. 
 
[7] Tian Meng, Li Shi-long, Convexity-preserving piecewise rational cubic 
interpolation,  Journal of Shandong University, 10(2007),80-83.  
 
Received: January, 2011 


Error Analysis of a Rational Interpolation Spline

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=69D5B860206672F7D9BA9E6CFBD463AA?doi=10.1.1.1046.6128&rep=rep1&type=pdf

Error Analysis of a Rational Interpolation Spline

Abstract

Similar works

Full text

Available Versions

CiteSeerX