ISSN 2307-7743 http://scienceasia.asia
_______________
2010 Mathematics Subject Classification: 41A05.
Key words and phrases: Deficient, Quartic Spline, Interpolation
© 2014 Science Asia 1 / 10
THE DEFICIENT QUARTIC SPLINE INTERPOLATION
Y.P.DUBEY,R.K.DUBEY AND S.S.RANA
Abstract: In the present paper, we have studied the existence, uniqueness and convergence properties of deficient quartic spline interpolation.
1. INTRODUCTION
Quartic and higher degree splines are still popular for best approximation (see de'Boor [1]). In the study of lower degree piecewise spline functions there is a disadvantage, that we have corners at joint of two lower degree pieces and therefore to achieve a prescribed accuracy more data than higher methods are needed. Rana and Dubey [9] generalized the result of Garry and Howell [7] and obtained best error bounds for quartic spline interpolation. Dubey and Shukla [4] have obtained error bound for quartic spline interpolation. For various aspects of cubic, quartic and spline of degree six reference may be given to Meir and Sharma [8], Hall and Meyer [6], Gemling-Meyling [5], Dubey [3]. In this paper, we have investigated the existence uniqeueness and convergence properties of deficient quartic spline interpolation which match the given function at values at mesh points, interior points with appropriate boundary conditions.
2. EXISTENCE AND UNIQUENESS
Consider a mesh P of [0,1] given by 0xx0... xn 1 such that
i i
i x h
x 1 i=0,1,...n-1 and hmax hi . Let S(4,P) denote the set of real algebraic polynomials of degree not greater than 4 i.e.
] 1 , 0 [ ,
{ ) , 4
( P s s C2
S i i , i = 1,2...n-1}.
where siis the restriction of s over [xi,xi1]. Let S* (4,P) denotes the class of all deficient quartic splines S(4,P) which satisfy the boundary conditions,
) ( " ) (
" x0 f x0
) ( " ) (
" xn f xn
s . (2.1)
We shall prove the following.
THEOREM 2.1 : Let f', f'' exist. Then there exist a unique deficient quartic spline in S* (4, P) which satisfies the interpolatory conditions,
) ( ) (xi f xi
s i = 0, 1, 2,...n (2.2)
) ( ) ( i f i
s i = 0, 1,...., n-1 (2.3)
where i xi hi if 0 3 6 and hi
is non increasing.
PROOF : Consider a quartic polynomial P(z)on [0,1]. We can easily verify that
) ( ) ( ) ( ) 1 ( ) ( ) 0 ( )
(z P q1 z P q2 z P q3 z
P
), ( ) 1 ( " ) ( ) 0 (
" q4 z P q5 z
P
(2.4)
where
, 6 / ] ) 1 ( 6 ) 1 ( 12 ) 12 6 6 [{( 1 )
( 4 3 3 4
1 z z z z A
q
A z z z z
q2( )[(6 4 12 3) 12 3 6 4]/6 ,
A z z z z
q3( )[6 12 3 6 4]/6 ,
2 3 4
2 3 4
4(z) [(2 5 3 )z 3( 2 )z
q
A z
z ( 3 2 ) ]/6
) 6 5
( 4 2 3 3 2 4
,
A z z
z z
q5( )[( 4 3) ( 4 ) 3 ( 3 ) 4]/6 ,
withA( 4 23)
Now writing
i i
h x x
t ( ) , (2.4) may be expressed in term of restriction siof s as follows :-
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
(x f x q1 t f x 1 q2 t f q3 t q4 t s"1 x q5 t s x
which clearly satisfies the condition (2.1) - (2.3) and si(x)is a quartic in [xi,xi1] for i=0,...,n. Since sC2[0,1], so applying continuity condition of first derivative of s in (2.5) we have
) 3
2 ( { ) 3
3
( 4 3 2 "1 1 4 3
1
i i
i s h
h
i i i
i
i s h s F
h
"
1 3 4 "
3 4 2
) (
)} 5 2 3
( (Say)
where
i i
i
i f
h h
F
1 2 3 3
4
) 12 2 ( ) 12 6 6 (
) ( 6 )
12 12 6 6 (
1 1
1
1 3
4
i
i i
i
f h h
f
6 ( ) 6 ( 2) ( 1)
3
i i
i i
x f h f
h
(2.6)
Clearly the coefficient matrix of the system of equation (2.6) diagonally dominant under the conditions of Theorem 2.1 and hence invertible. This complete proof of theorem 2.1.
3. ERROR BOUNDS
Following the method of Hall and Meyer [6] in this solution, we shall obtain bounds of error function e(x) f (x)s(x) for the spline interpolant of Theorem 2.1 which are best possible. Let s(x)be the second time continuously differentiable quartic spline function satisfying the conditions of theorem 2.1. Now considering
] 1 , 0 [ 5 C
f and writing Li[f,x] for the unique quartic which agree with
) ( " ) ( ) ( ),
(xi f xi 1 f i f xi
f and f"(xi1), we see that for x[xi,xi1], we have
| ) ( ) ( | | ) ( ) (
| f x s x f x si x
)] ( ] , [ | | ] , [ ) (
| f x hi f x Li f x si x
(3.1)
First we have to obtain the bounds of right hand side of (3.1) by Cauchy Theorem (See Davis [2]). Thus, we have
| ) ( ) 1 ( | ! 5 | ] , [ ) (
| 2 2
5
t t t
h x f L x f
i
where
i i
h x x
t and max | (5)( )|
1 0
x f F
x
.
To get the bounds of |Li[f,x]si(x)|,we have from (2.4)
)] ( ) ( " ) ( ) ( " [{ ) ( ] ,
[f x s x h21e x 1 q4 t h2e x q5 t
hi i i i i i (3.3)
Thus,
| ) {( ] , [
|Li f x i x
| ) ( ) ( " ) ( ) ( "
|hi21e xi 1 q4 t hi2e xi q5 t
(3.4)
Let max | ( )|
1
0 t i
x e
exists for i= j then the inequality (3.4).
| ) ( ] , [
|Li f x si x
|] ) ( )} ( [| | ) ( "
| 4 5
2
t q t q x e
hj j
Let
1 0 0 max ) (
t
k [| 2 6 3|3| 3 2 2 1| t6| 1|t2 3(1)t3].
Thus |Li[f,x]si(x)|h2j |e"(xj)|k(t) (3.6)
Now, we proceed to obtain |e"(xi)| replacing s"(xi)by e"(xi)in equation (2.6), we have
) 3
2 ( { ) 3
3 5
( 4 3 2 "1 1 4 3
1
i i
i e h
h
" 1 3 "
3 4 2
) 1 ( )}
5 2 3
(
hi ei hi ei
" 1 2
3 4
1(5 3 3 )
Fi hi fi
)} 5 2 3 ( ) 3
2 (
{ 1 4 3 2 4 3
hi hi
) ( )
1 ( 3 "1 "
f E f h
fi i i (say). (3.7)
In view of that E(f) is linear functional which is zero for polynomials of degree 4 or less we can apply Peano Theorem (see Davis [2]) to obtain
1
1
4 )
5 (
) ( ! 4
) ( )
( j
j
x
x
dy y x E y f f
Thus, from (3.8) we have
1
1
| ) ( | ! 4
1 | ) (
| 4
j
j
x
x
dy y x E F f
E (3.9)
Where F max | f 5(y)|.
Further, it can observed that from (3.9) in xi1x xj1, we get
4
1 2 3 3
4 4
) ( } ) 12 2 ( 12
6 6 [{ ] )
[(
x y
h h
y x
E i
i i
i
4 1 3
4 4
1 1
) (
) 2 ( 6 ) ( 6 ) (
6
x y
h y
h y
hi i i i i i
) 3 2 } {
12 4 2
hij
2 1 3 4 2
3 4 2
) (
) (
12 ) ( ) 5 2 3
(
hi xi y hi xi y . (3.10)
In order to estimate the internal of r.h.s. of (3.10) we rewrite the above expression in the following symmetric form about xi to see
)} 1 ( 2 ) )(
2 {(
63 xi y hi 2 hi2 [xi yhi]2 i yxi1
= 4 3 2 4 3 2
2 ) ( ) 8 4 ( ) ( ) 1 2 [( 6
x y h x y
hi i i i
] ) 3 5 2
( 2 hi2 [(xi y)2] xi yi
= 6 [( 4 2 3 1)( )2 (4 3 8 2 1)
1
x y
hi i
2 2
1 2
2
1(x y) 2 (2 5 3)h ](x y)
hi i i i , i1yxi
= 2
1 2
1 2 1 1
3
] [
)} 1 ( 2 ) (
) 2 {( 6
i i i i i
i
h y x h
h y x h
,
1
1
i
i y
x .
Thus, it is clear from above expression that E[(x y)4] is non negative for
1
1
i
i x x
, 5) ( |
) ( |
1
1
4 4
1 1
4
j j
x
x i i
h h K dy y x
E (3.11)
where
24 124 400 543 84
2
2 3
4 5
3
1
K .
Combining (3.9) and (3.11), we have
4 4
1 1
! 5
) ( |
) (
|Ei f F K hi hi . (3.12)
Thus, from (3.8) and (3.11)
6 3
]) 2 3 3 [( ! 5
) (
) ( |
| | ) ( |
2 4 3 1 2
4
4 4
1 1
" "
i i
i i i
i
h h
h h FK e
x e
(3.13)
Now using (3.2), (3.7) along with (3.13) in (3.1) we have
F t k x e h F t
t t h x
e | (1 ) ( )| | ( i)| ( )
! 5 | ) (
| 2 2 2 "
5
=
) 2 6 2 6 (
) ( ) ( |
) ( ) ( |
5 3 4 2
5 1 2
2 5
t t F FK h k t
t h
(3.14)
| ) ( | ! 5
5
t C F h
(3.15)
where C(t)
t2(1t)2(t)K1*()K(t)
and
) 2 6 2 6 (
) ( )
(
2 4 3
1 *
1
K
K
Thus we prove the following.
Theorem 3.1. Suppose s(x) is the quartic spline of Theorem 2.1 interpolating a function f(x) and f C5[0,1]. Then
| ) ( | max ! 5 | ) (
| (5)
1 0 5
x f h
K x e
x
, (3.17)
where max | ( )| max | 2(1 2)( ) 1*( ) ( )|
1 0 1
0
t k K t
t t t
C K
t t
,
| ) ( | max ) ( ! 5 1 | ) ( "
| (5)
1 *
1 5
x f K
h x e
x i
. (3.18)
Further more, it can be seen easily that K in (3.17) can not be improved for an equally spaced mesh. Inequality (3.18) is also best possible. Also, we have
| ) ( | max ! 5 | ) ( "
| 5
1 5
1 f x
h K x e
x i
, (3.19)
where K1is positive constant.
Equation (3.15) proves inequality (3.17) where as inequality (3.18) is direct consequence of (3.13).
Now we turn to prove that the inequality (3.17) is best possible in the limit.
Considering
! 5 ) (
5 x x
f and using Cauchy formula Davis [2] we have
! 5
) ( ) 1 ( !
5 !* 5
2 2 5 5 5
x h t t t
x x
Li (3.20)
For the function with equally spaced knots from equation (3.7), the following relation holds
"1 2
3 4 3 1 5
3 3 5 !
5 ) ( !
5
i
e h
K x
"1 3 "
2 3
) 1 ( 3
8
ei ei (3.21)
Consider for a moment
" 1 "
1 2 2
3 1 "
) 1 ( ! 6
) (
j j
j e e
h K e
. (3.22)
We have from (3.4)
( ) ( )
) 1 ( ! 6
) ( ,
)
( 4 5
2 2
5 1
t q t q h K x f L x
s i
2 2 3
2 2 2 2
5 1
) 3 1 2 ( 2 ) 3 6 ( [ ) 1 ( ! 6
) (
t t
h K
] ) 1 ( 3 ) 1 (
6 t3 t4
Now combining (3.20) and (3.23) we get for
1
i
i x x
x
t K
h x f x
s { ( 6 3)
) 1 ( 6
) ( [ ! 5 ) ( )
( 2 2
2 2
1 5
3 2
2 3
) 1 ( 6 ) 3 1 2 (
2 t t
)] ( ) 1 ( } ) 1 (
3 4 2 2
t t t t . (3.24)
From (3.24) it is clearly observed that (3.17) is best possible, provided we would prove that
2 2
5 1 "
1 " "
1
) 1 ( ! 6
) (
h k e
e
ei i i (3.25)
In fact (3.25) is attained only in the limit, the difficulty will appear in the use of boundary condition i.e. e"(x0)e"(xn)0.
However, it can be shown that as we move many sub-intervals away from the boundaries
2 2
5 1
) 1 ( ! 6
) ( )
( "
K h
x
e i . For that we shall apply (3.20) indutively to move away
from the end conditions e"(x0)e"(xn)0.
The first step in this direction is to establish that e"(xi)0 for some i=1,2,...n which can be shown by contradictory result. Let e"(xi)0 for i=1,2,....n-1.
Now making use of (3.18) we have
" 1 3 "
1 2
3 4 1
2 2
3 1
) 1 ( ) 3
3 5 [( 2 1 | ) ( " | max ) 1 ( ! 6
) (
i i i
x
e e
x e h
K
" 1 3 "
2 3 "
1 2
3 4
) 1 ( ) 3 8 ( ) 3
3 5 [( 2 1
ei ei ei
! 5
) ( 2
1 K1 h3 .
Now from (3.21)
4 3 2
3 1 " 2 3
3 3 5 !
5 ) ( )
3 8
( ei K hi
" 1 3 "
1 1
i
i e
e
! 5
) ( )
3 8 (
3 1 " 2
3 K h
ei
. (3.26)
! 5
) ( ) 3 8 (
1 1 3
2 3
" K h
ei
.
Now again using (3.26) in (3.21) we have
! 5 ) 3 8 (
) ( )
( "
2 3
3 1
K h
x
e i
...
3 8
3 4 6 3
8 (
4 6 1
2
2 3
2 3 4 2
3 2 4
(3.28)
Now it can be seen easily that r.h.s. of (3.28), 2
3 1
) 1 ( 6
) (
K h and hence in the
limiting case 2
3 1
) 1 ( 6
) ( )
( "
K h
x
e i which verify proof of inequality (3.19).
Thus, corresponding to the function
! 5 ) (
5 x x
f (3.27) and (3.28) tends to
2 3 1
) 1 ( 6
) (
h K
REFERENCE
[1] Deboor, C.A. Practical Guide to Splines, Applied Mathematical Science, Vol. 27 Spoinger, Varlag, New York 1979.
[2] Davis, P.J. Interpolation and approximation, New York, 1969.
[3] Dubey, Y.P. Best Error Bounds of Spline of degree six. Int.Jour. of Mathematical Ana. Vol. 5 (2011), pp. 21-24.
[4] Dubey, Y.P. and Shukla, A. The Deficient C1 Quartic Spline Interpolation Research Inventy, International Journal of Engineering & Sci., Vol. 2, Issue 9 (April 2013), pp. 24-30.
[5] Gemlling, R.H.J. and Meyling, G. in Interpolation by Bivartate Quintic Splines of Class Construction of Theory of function 87 (ed) Sendor et al (1987) 152-61.
[6] Hall, C.A. and Meyer, W.W., J. Approximation Theory 16 (1976), pp 105-122.
[7] Howell, G. and Verma, A.K. Best Error Bound of Quartic Spline Interpolation, J. Approx. Theory 58 (1989), 58-67.
[8] Meir, A. and Sharma, A. Convergence of a class of interpolatory spline J. Approx. Theory (1968), pp. 243-250.
[9] Rana, S.S. and Dubey, Y.P. Best Error Bounds of Quintic Spline Interpolation J. Pune and App. Math 28 (10) 1937-44 (1997).
[10] Rana, S.S. and Dubey, Y.P. Best Error Bounds of deficient quartic spline interpolation, Indian Journal Pune and Appl. Math 30(4) (1999), 385-393.
Y.P.DUBEY, DEPARTMENT OF MATHEMATICS, L.N.C.T. JABALPUR 482 003
R.K. DUBEY, DEPARTMENT OF MATHEMATICS, GLOBAL NATURE CARE SANGATHAN'S GROUP OF INSTITUTIONS, JABALPUR
