IMPACT: International Journal of Research in Applied, Natural and Social Sciences (IMPACT: IJRANSS) ISSN(E): 2321-8851; ISSN(P): 2347-4580 Vol. 2, Issue 7, Jul 2014, 35-46 © Impact Journals
DISCRETE QUARTIC SPLINE INTERPOLATION Y. P. DUBEY & K. K. NIGAM Department of Mathematics, L. N. C. T, Jabalpur, Madhya Pradesh, India
ABSTRACT In this paper, we have obtained existence, uniqueness and error bounds for deficient discrete quartic spline interpolation.
KEYWORDS: Deficient, Discrete, Quartic Spline, Interpolation, Error Bounds Subject Classification Code: 41AO5, 65DO7
1. INTRODUCTION Let us consider a mesh on [0, 1] which is define by
0 = x0 < x1 < ...... < xn =1 with xi − xi −1 = Pi
for i = 1,2,....n.
and h > 0, will be a real number, consider a real continuous function s(x,h) defined over [0, 1] which is such that its restriction
[
]
si on xi−1 , x i is a polynomial of degree 4 or less for i = 1, 2,....n. Then s(x, h) defines a discrete
quartic spline if
Dh{1} si ( xi , h ) = Dh{ j} si +1 ( xi , h ) j = 0,1 Where the difference operator
(1.1)
Dh are defined as
Dh{0} f ( x) = f ( x)
Dh{1} f ( x) =
f ( x + h) − f ( x ) ) h
Dh{2} f ( x) =
f ( x + h ) − 2 f ( x ) + f ( x − h) ) 2h
Let
D (4,1, ∆, h) is the class of deficient discrete quartic spline interpolation of deficiency one, where in
D * (4,1, ∆, h) denotes the class of all discrete deficient quartic splines which satisfies the boundary condition s ( x 0 , h ) = f ( x0 , h ) s ( x n , h) = f ( x n , h)
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(1.2)
36
Y. P. Dubey & K. K. Nigam
Mangasarian and Schumaker [6, 7] introduced discrete quartic splines to find minimization problem. Existence, uniqueness and convergence properties of discrete cubic spline interpolation matching the given function at mesh point have been studied by Lyche [4, 5] which have been generalized by Dikshit and Power [1] (see also Dikshit and Rana [2]). It has been by Baneva, Kendall and Stefanov [3] that the local behaviour of the derivative of a cubic spline interpolator is some times used to smooth a histogram which has been estimated in [8], Rana [8] has obtained a precise estimate concerning the discrete cubic spline interpolating the given function at the mesh points (See also [9], [10]). In the present paper we obtain a similar precise estimate concerning the deficient quartic spline interpolant matching the given function at two intermediate points between successive mesh points and first difference of interior points of interval [0, 1].
2. EXISTENCE AND UNIQUENESS We introduced the following interpolating conditions for a given function f.
s (α i ) = f (α i )
(2.1)
s(β i ) = f (β i )
(2.2)
Dh{1} s (γ i ) = Dh{1} f (γ i )
(2.3)
Where
1 3
α i = xi −1 + Pi = γ i ,
for i = 1, 2,....n.
1 2
β1 = xi −1 + Pi we shall prove the following. Theorem 2.1 Let f be a 1-periodic, then for any h>0 then these exist a unique 1-periodic deficient discrete quartic spline s in the class
D * (4,1, ∆, h) which satisfies interpolatory condition (2.1) - (2.3).
Proof Let P(z) be a quartic polynomial on [0, 1], then we can show that
1 1 1 P ( z ) = P .q1 ( z ) + P q 2 ( z ) + Dh{1} P q3 ( z ) + P (0) q 4 ( z ) + P (1) q5 ( z ) 3 2 3
(
(2.4)
)
1 3 2 9 2 2 29 3 2 2 4 6 + 2 h z + 2 − 2 h z + z − 6 + 24h + 3 − 18h z ] Where q1 ( z ) = 2 1 2 − h 81 3
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37
Discrete Quartic Spline Interpolation
32 64 2 4 32 − 224 16 2 2 3 160 2 81 z + 81 + 3 h z + z 27 − 3 h + z − 9 + 16h q2 ( z ) = 2 1 2 − h 81 3 − 1 2 2 11 3 2 4 9 z + 3 z − 9 z + 3 z q3 ( z ) = 2 1 2 − h 81 3
2 4 2 58 h 2 2 26 16 2 3 4 + h z + + z − + h z + + 4h 2 z 4 1 + − 27 3 81 3 27 3 9 q4 ( z ) = 2 2 h − 81 3
1 1 2 8 7 2 2 7 8 2 3 1 2 4 − 162 + 6 h z + 162 − 6 h z + − 54 + 3 h z + 9 − 2h z q5 ( z ) = 2 2 h − 81 3 Denoting
t=
x − xi , 0 ≤ t ≤1 , we can expressed (2.4) in the form of restriction si ( x, h) of the deficient discrete pi
quartic spline s(x, h) on
[xi−1 , xi+1 ] as follows :-
s ( x, h) = f (α i ) q1 ( x) + f ( β i ) q2 ( x)+ Pi Dh{1} (γ i ) q3 ( x) + si ( x) q4 ( x ) + si +1 ( x) q5 ( x)
(2.5)
si ( x, h) is a quartic on [xi , xi +1 ] for i=0, 1,...., n-1 satisfying
Observing (2.5) it may easily be verify that (2.1) - (2.3) and writing
H (a, b) = a + bh 2 , for real a, b we shall apply continuity of the first difference of s ( x, h) at
xi in (2.5) to see that
]
92 2326 − 32 2 P13 H , 2 Pi −21 + H , h si −1 3 189 81
[
− 4 13 2 − 17 16 2 + Pi 3 H , Pi −1 + H , h 54 3 27 6
−2 4 2 26 16 , Pi − H , h 2 si + Pi −31 H 27 3 9 3 Impact Factor(JCC): 1.4507 - This article can be downloaded from www.impactjournals.us
38
Y. P. Dubey & K. K. Nigam
1 1 2 −7 8 2 + Pi −31 H − , Pi + H , h si +1 54 3 162 6 = Fi Where
(2.6)
i = 1, 2,.....n-1.
− 37 − 569 F1 = Pi 3 H , 59 Pi −21 + h 2 H − 48h 2 6 6
−1 − 3 2 − 29 f (α i −1 ) − Pi −31 H , , 24 h 2 Pi + H 6 6 2 128 736 2 − 224 128 2 f (α i ) + Pi 3 − H , , Pi −1 + H h 3 81 9 27
}
32 160 − 64 2 f (β i −1 ) − f (β i ) − H ,0 Pi 2 + H , h 27 3 81 2 13 1 11 + Pi 3 Pi−1 H , Dh{1} f (γ i−1 ) + Dh{1} f (γ i ) Pi −31 Pi H , ] 9 9 9 9 Existence, uniqueness of s(x, h) depend on the existence of a unique solution of set of equation (2.6). It is easy to observe that in (2.6) absolute value of the coefficient of si dominates over the sum of the absolute values of the coefficient of
si +1 and si −1 i.e. is positive. 80 1 1511 16 2 35 7 2 45 8 Ti ( P, h) = H , Pi −21 + H , h + H , Pi + H , h 2 126 3 162 6 54 3 81 6 Thus the coefficient matrix of equation (2.6) is diagonally dominant and hence invertible. Remark: In the case
h → 0 theorem 2.1 gives the corresponding result for continuous quartic spline
interpolation under condition (2.1) - (2.3).
3. ERROR BOUNDS Now system of equation (2.8) may be written as
A ( h) M ( h ) = F Where A(h) is coefficient matrix and
M (h) = si (h) . However, as already shown in the proof of theorem 2.1.
A(h) is invertible. Denoting the inverse of A(h) by A-1(h) we note that row max norm A-1(h) satisfies the following inequality :-
|| A −1 (h) || ≤ y (h)
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(3.1)
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Discrete Quartic Spline Interpolation
y (h) = max {Ti (h)} . For convenience we assure in this section that 1 = Nh when N is positive integer. −1
Where
{xi } are such that xi ∈[0,1]h for i = 0, 1.....n. Where discrete interval [0,1]h is the
It is also assume that the mesh points set of points
{0, h......Nh} for a function f and two distinct points
x1 , x2 in it domain the first difference is defined by
[x1 , x2 ] f = [ f ( x1 ) − f ( x2 )]
(3.2)
x1 − x2
For convenience, we write
f (1) for Dh{1} f and w ( f , p ) for modules of continuity of f, the discrete norms of a
function f over the interval [0, 1]h is defined by
|| f || = max | f ( x) |
(3.3)
x∈[ 0 ,1]h
We shall obtain in the following the bound of error function interval
e( x) = s ( x, h) − f ( x) over the discrete
[0,1]h .
Theorem 3.1 Suppose
s ( x, h) is the discrete quartic spline interpolant of Theorem 2.1. Then
|| e( x) || ≤ k ( P, h) w ( f , p )
(3.4)
|| e( xi ) || ≤ y (h) K * ( P, h) w ( f , p )
(3.5)
|| e (1) ( x) || ≤ K1 ( P, h) w ( f , p )
(3.6)
Where K (p, h), K*(P,h) and
K1 ( P, h) are some positive functions of p and h.
Proof Writing
f ( xi ) = f i equation 3.1 may be written as
A(h) . (e( xi ) ) = Fi (h) − A(h) ( f i ) = Li ( f ) When we replace
(Say)
si (h) by ei ( xi ) = s ( xi , h) − f i
We need the following Lemma due to Lyche [4, 5], to estimate inequality (3.5). Lemma 3.1 Let
{ai }im=1 and {b j }nj=1 be given sequence of non negative real numbers such that
m
n
i =1
j =1
∑ ai = ∑ b j Impact Factor(JCC): 1.4507 - This article can be downloaded from www.impactjournals.us
(3.7) (3.8)
40
Y. P. Dubey & K. K. Nigam
Then for any real valued function f, defined on discrete interval [0, 1]h, we have
[
m
∑ ai ne0 , xe1 ,.......xik i =1
(
≤ w f ( k ) , | 1 − Kh | Where
] − ∑ b [y
]
n
f
) ∑Ka!
j =1
j
i0
, y j1 ..... y jk f
i
(3.9)
x jk , y jk ∈[0,1]h for relevant value of i, j and k. We can write the equation (3.9) is of the form of error
function as follows
92 2326 − 32 2 , P13 H , 2 Pi −21 + H h ei −1 189 3 8 − 4 13 2 17 16 + P13 H , Pi −1 + H , h 2 54 3 27 6 2 4 26 16 + Pi −31 H , Pi 2 − H , h 2 ei 27 3 9 3 1 1 2 −7 8 2 + Pi −31 H , Pi + H , h ei +1 = Ri ( f ) 54 3 162 6 Where
92 2326 32 2 Ri ( f ) = Fi − Pi 3 H , 2 Pi −21 + H − h f i −1 3 189 81
− 4 13 2 − 17 16 2 − Pi 3 H , Pi −1 + H , h 54 3 27 6 − 2 4 2 26 16 − Pi −31 H , Pi − H , h 2 f i 27 3 9 3 −1 1 2 7 8 − Pi −31 H , Pi + H , h 2 f i+1 54 3 162 6
(3.10)
Writing equation (3.9) is the form of divided different and using Lemma 3.1 given by Lyche [5]
[
2 −7 2 43 2 Ri ( f ) = − Pi 3 Pi −1 [α i −1 , β i −1 ] f H , Pi−1 + h H , − 8 36 9 4
[
11 − 1 − H , [α i , xi ] Pi 3 Pi −31 + Pi 3 Pi −1 β i −1i , xi 81 3
]
f
10 − 1 H , 81 12
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41
Discrete Quartic Spline Interpolation
61 8 Pi −21 + h 2 H , + [xi −1 , xi ] f 108 3 −
−8 − 22 − 32 3 , − 2 Pi −21 + h 2 H , Pi Pi−1 H 3 27 81
1 3 3 2 13 Pi Pi −1 [γ i − h, γ i + h] + pi3 Pi −1 H , [γ i −1 − h, γ i−1 + h ] f 9 9 9
− 16 2 − 80 64 2 + Pi −31 Pi H , 0 Pi + H , h 81 18 243
[α i , β i ] f − H − 5 , − 1 Pi 2 + H 26 , 16 h 2 81 9
81 9
[xi , α i ] f ]+ H
1 −1 2 7 − 16 2 , p1 + H , h 81 9 243 9
[α i , xi+1 ] f − − 11 h 2 [γ i − h, γ i + h] f ] 9 5
[
⇒ Li ( f ) = ∑ ai xi0 , xi1 i =1
] − ∑ b [y 5
f
j =1
j
j0
, y j1
]
f
(3.11)
5 5 ≤ w f (1) , P = ∑ ai = ∑ b j j =1 i =1
(
)
20 − 25 2 43 2 = Pi 3 Pi −1 H , Pi −1 + h H , − 8 36 81 12 − 5 −1 2 − 73 − 16 2 , Pi + H , + Pi Pi −31 H h 81 9 81 9 Where
10 − 1 61 8 a1 = P13 Pi − j H , Pi −21 + h 2 H , 108 13 81 12 −8 − 22 − 32 2 a2 = P13 Pi −1 H , − 2 Pi −21 + H , h 3 27 81 2 13 a3 = P13 Pi −1 H , 9 9
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(3.12)
42
Y. P. Dubey & K. K. Nigam
− 16 2 − 80 64 2 a 4 = Pi −31 Pi H , 0 P1 + H , h 81 18 243 1 −1 2 7 − 16 2 a5 = Pi −21 Pi H , P1 + H , h 81 9 243 9 2 −7 2 43 2 b1 = P13 Pi −1 H , Pi − j + H , − 8 h 36 9 4 11 − 1 b2 = P1 Pi −31 H , 81 3 b3 = −
1 3 3 pi pi −1 9
− 5 −1 2 26 16 b4 = pi3−1 pi H , pi + H , h 2 h 2 81 9 81 9 b5 = − and
11 2 3 h pi −1 pi 9
x10 = β i −1 = y11 , x11 = x1 = x21 = y 41 x20 = xi − j , x30 = yi −1 − h , x31 = γ i−1 + h y10 = α i −1 , y 20 = α i = y41 = x40 = x50 y30 = γ i − h = y50 , y31 = γ i + h = 51 x41 = β i , x51 = xi +1
Now using the equation (3.10) and (3.9) in (3.8).
(
|| e( xi ) || ≤ y (h) K * ( P, h) h f (1) , p
)
This is the inequality (3.5) of Theorem 3.1. To obtain inequality (3.4) of Theorem 3.1. Writing equation (2.5) in the form of error function as follows:
e( x) = ei −1 Q4 (t ) + ei Q5 (t ) + M i ( f ) Where
M i ( f ) = f (α i ) Q1 (t ) + f ( β i ) Q2 (t )
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(3.13)
43
Discrete Quartic Spline Interpolation
+ pi−1 f (1) (γ i ) Q3 (t ) + f i−1 Q4 (t ) + f i Q5 (t ) − f ( x) Again, we write
M i ( f ) in form of Divided difference and using Lemma 3.1, we get
(
| M i ( f ) | ≤ w f (1) , p
)∑ a = ∑ b 3
i =1
2
i
j =1
j
1 1 − 1 3 −1 29 a1 = Pi + h 2 t + + h 2 t 2 + − 4h 2 t 3 + + 3h 2 t 4 2 36 36 4 3 4 1 2 − 19 1 2 2 13 8 2 3 5 a 2 = Pi − h 2 t + − h t + + h t + − 2h 2 t 4 27 3 9 9 3 18 6 11 2 1 2 q3 = Pi − t + t 2 + t 3 + t 4 9 3 9 3 1 1 − 5 59 2 − 7 4 2 3 1 b1 = Pi h − + h2 + t 2 + + h t + − h 2 t 4 108 36 108 3 18 324 12 8 1 b2 = Pi t − h 2 27 3 and
x10 = α i , x11 = β i x20 = xi −1 , x21 = β i x30 = γ i − h, γ 31 = γ 1 + h y10 = β i , y11 = ni y 20 = xi − j , y 21 − x 3
Thus
2
∑ a = ∑b i =1
i
j =1
j
1 5 2 7 7 4 2 3 = Pi − h 2 t + − h 2 t 2 + − h t 108 3 36 12 81 12
−5 + + h2 t 4 18 using (3.5) and(3.13) in (3.14), we get inequality (3.4) of theorem 3.1. We now proceed to obtain bound of
e{1} ( x)
Pi si{1} ( x) = f (α i ) Q1{1} (t ) + f (β i )Q2{1} (t ) Impact Factor(JCC): 1.4507 - This article can be downloaded from www.impactjournals.us
(3.14)
44
Y. P. Dubey & K. K. Nigam
+ Pi Dh{1} f (γ i ) Q3{1} (t ) + si ( x) Q4{1} (t ) + si +1Q5{1} (t )
(3.15)
Ai Pi e{1} ( x) = ei−1 Q4{1} (t ) + ei Q5{1} (t ) + U i ( f )
(3.16)
Where
U i ( f ) = f (α i ) Q1{1} (t ) + f ( β i ) Q2{1} (t ) + Pi Dh{1} f ( yi ) Q3{1} (t ) + f i−1 Q4{1} (t ) + f i Q5{1} (t ) − APi f {1} ( x)
By using Lemma 3.1, we get
(
U i ( f ) ≤ w f (1) , P
)∑ a = ∑ b 3
i =1
2
i
j =1
j
−1 − 7 , = Pi H 324 12
2 −3 2 − 29 2 1 h t + H ,4 3t + h 2 + H ,−3 4t t 2 + h 2 +H , 3 2 36 2
(
Where
)
(
1 −2 58 1 2 2 a1 = Pi H , − H , t + 3t + h 81 3 9 3
(
(
)
)
)
157 8 2 H , − 4 H ,2 t t 2 + h 2 27 3 4 −1 1 4 7 a 2 = pi H , + tH , + 3t 2 + h 2 81 6 324 12
(
)
− 23 4 1 H , + 4t t 2 + h 2 H ,−1 108 3 18
(
)
8 −1 4 1 a3 = pi + t − 3t 2 + h 2 + t t 2 + h 2 3 9 3 4
(
)
(
)
−1 −1 2 −3 − 29 12 b1 = pi H , + H , , t + H 3 2 36 3 36 4
(3t
2
)
(
1 + h 2 + 4 H , −3 t t 2 + h 2 2
)]
2 −1 b2 = pi H , and 81 3 x10 = xi , x11 = β i = x21 x22 = xi +1 , x30 = γ i − h, x31 = γ i + h Index Copernicus Value: 3.0 - Articles can be sent to editor@impactjournals.us
(3.17)
Discrete Quartic Spline Interpolation
45
y10 = α i , y11 = β i y 20 = x − h, y 21 = x + h By using (3.5), (3.13) and (3.17) in (3.16), we get inequality (3.6) of Theorem 3.1.
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