6. Applied-Discrete Quartic Spline Interpolation-Y.P.dubey

5/21/2018 6. Applied-Discrete Quartic Spline Interpolation-Y.P.dubey

Impact Factor(JCC): 1.4507 - This article can be downloaded from www.impactjournals.us

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

1......0 10 =


36 Y. P. Dubey & K. K. Nigam

Index Copernicus Value: 3.0 - Articles can be sent to [email protected]

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 issome 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.

)()( ii fs = (2.1)

)()( ii fs = (2.2)

)()( }1{}1{ihih

fDsD = (2.3)

Whereiiii Px =+=

3

11 , for i = 1, 2,....n.

ii Px2

111 +=

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 ),,1,4(* hD 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()()0()(3

1)(

2

1)(.

3

1)( 543

}1{

21 zqPzqPzqPDzqPzqPzP h ++

+

+

= (2.4)

Where

( )

+

++

+

+

=

2

4223222

1

3

1

81

2

]183246

29

2

92

2

3

6

1

)(

h

zhhzzhzh

zq


Discrete Quartic Spline Interpolation 37


++

+

+

+

=

2

242322

2

3

1

81

2

169

32

3

64

27

160

3

16

81

224

81

32

)(

h

hzhzzhz

zq

++

=

2

432

3

3

1

81

2

3

2

9

11

3

2

9

1

)(

h

zzzz

zq

++

+

++

++

=

3812

49

4

3

16

27

26

381

58

3

4

27

21

)(2

423222

2

4

h

zhzhzh

zh

zq

+

++

+

+

=

381

2

29

1

3

8

54

7

6

7

162

8

6

1

162

1

)(2

4232222

5h

zhzhzhzh

zq

Denoting 10,

= t

p

xxt

i

i, we can expressed (2.4) in the form of restriction ),( hxsi of the deficient discrete

quartic spline s(x, h) on [ ]11, + ii xx as follows :-

( ) )()()()()(),( 3}1{

21 xqDPxqfxqfhxs ihiii ++=

( ) )()()( 514 xqxsxqxs ii +++ (2.5)

Observing (2.5) it may easily be verify that ),( hxsi is a quartic on [ ]1, +ii xx for i=0, 1,...., n-1 satisfying

(2.1) - (2.3) and writing2),( bhabaH += , for real a, b we shall apply continuity of the first difference of ),( hxs at

ix in (2.5) to see that

]

+

1

22

1

3

13

32,

189

23262,

81

92ii

shHPHP

[

+

+

22

1

3

3

16,

54

17

6

13,

27

4hHPHP

ii

iii shHPHP

+

223

1 3

16

,27

26

3

4

,9

2




1

223

13

8,

54

7

6

1,

162

1+

+

+ iii shHPHP (2.6)

iF=

i = 1, 2,.....n-1.

Where

+

=

222

1

3

1 486

56959,

6

37hHhPHPF ii

( )

+

223

11 24,6

29

2

3,

6

1hHPHPf iii

( ) }

+

+

22

1

3

3

128,

27

224

9

736,

81

128hHPHPf iii

( ) ( )

+

22

13

64,

27

1600,

81

32hHPHff iii

( ) ]9

11,

9

1)(

9

13,

9

2 31

}1{

1

}1{

1

3

+

+

HPPfDfDHPP iiihihii

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 sidominates over the sum of the absolute values of the coefficient

of 1+is and 1is i.e. is positive.

+

=

22

13

16,

126

1511

6

1,

81

80),( hHPHhPT ii

+

+

22

3

8,

54

45

6

7,

162

35hHPH i

Thus the coefficient matrix of equation (2.6) is diagonally dominant and hence invertible.

Remark: In the case 0h 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

FhMhA =)()(

Where A(h) is coefficient matrix and )()( hshM i= . 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 :-

)(||)(|| 1 hyhA (3.1)




Where { } 1)(max)( = hThy i . For convenience we assure in this section that 1 = Nh when N is positive integer.

It is also assume that the mesh points }{ix are such that hix ]1,0[ for i = 0, 1.....n. Where discrete interval [ ]h1,0 is the

set of points { }Nhh......,0 for a function fand two distinct points 21,xx in it domain the first difference is defined by

[ ] [ ]

21

2121

)()(,

xx

xfxfxx

f

= (3.2)

For convenience, we write)1(f for fDh

}1{and ),( pfw for modules of continuity of f, the discrete norms of a

functionfover the interval [0, 1]his defined by

|)(|max||||]1,0[

xffhx

= (3.3)

We shall obtain in the following the bound of error function )(),()( xfhxsxe = over the discrete

intervalh]1,0[ .

Theorem 3.1

Suppose ),( hxs is the discrete quartic spline interpolant of Theorem 2.1. Then

),(),(||)(|| pfwhPkxe (3.4)

),(),(*)(||)(|| pfwhPKhyxe i (3.5)

),(),(||)(|| 1)1( pfwhPKxe (3.6)

Where K (p, h), K*(P,h) and ),(1 hPK are some positive functions of p and h.

Proof

Writing ii fxf =)( equation 3.1 may be written as

( ) )()()()()(.)( fLfhAhFxehA iiii == (Say) (3.7)

When we replace )(hsi by iiii fhxsxe = ),()( (3.8)

We need the following Lemma due to Lyche [4, 5], to estimate inequality (3.5).

Lemma 3.1

Let { }mii

a1=and { }n

jjb

1=be given sequence of non negative real numbers such that

==

=

n

j

j

m

i

i ba11




Then for any real valued function f, defined on discrete interval [0, 1]h, we have

[ ] [ ] = =

m

i

n

j

jjijfieei fyyybxxna

kk

1 1

.....,,.......,1010

( )!

|1|,)(

K

aKhfw

ik (3.9)

Where hjj kkyx ]1,0[, for relevant value of i, j and k. We can write the equation (3.9) is of the form of error

function as follows

+

1

22

1

3

13

32,

189

23262,

8

92ii ehHPHP

+

+

22

1

3

13

16,

54

17

6

13,

27

4hHPHP i

iii ehHPHP

+

223

13

16,

27

26

3

4,

9

2

1

223

13

8,

54

7

6

1,

162

1+

+

+ iii ehHPHP )(fRi=

Where 122

1

3

3

32

189

23262,

81

92)(

+

= iiiii fhHPHPFfR

+

22

1

3

3

16,

54

17

6

13,

27

4hHPHP ii

iii fhHPHP

223

13

16,

27

26

3

4,

9

2

1

223

13

8,

54

7

6

1,

162

1+

+

iii fhHPHP (3.10)

Writing equation (3.9) is the form of divided different and using Lemma 3.1 given by Lyche [5]

[ [ ]

+

=

8,

36

43

4

7,

9

2,)( 221111

3HhPHPPfR ifiiiii

[ ] [ ]

+

12

1,

81

10,,

3

1,

81

1111

33

1

3HxPPPPxH

fiiiiiiii i




[ ] 1322

11

22

13

32,

27

222,

81

8,

3

8,

108

61

+

+

+ iiifiii PPHhPHxxHhP

[ ] [ ]fiiiiiiii hhHPphhPP +++ 111

331

3 ,9

13,92,

91

+

+

223

118

64,

81

800,

243

16hHPHPP iii

[ ]

+

22

9

16,

81

26

9

1,

81

5, hHPH ifii

[ ] ]

+

+

22

19

16,81

7

9

1,243

1, hHpHx fii

[ ]

+

2

19

11, hx

fii [ ]

fii hh + ,

[ ] [ ] = =

=5

1

5

11010

,,)(i j

fjjjfiiii yybxxafL (3.11)

( )

==

= =

5

1

5

1

)1(

, i j ji baPfw

+

=

8,

36

43

12

25,

81

20 2211

3HhPHPP iii

+

+

223

19

16,

81

73

9

1,

81

5hHPHPP iii (3.12)

Where

+

=

13

8,

108

61

12

1,

81

10 221

3

11 HhPHPPa iji

+

=

22

11

3

123

32,

27

222,

81

8hHPHPPa ii

=

9

13,

9

21

3

13 HPPa i




+

=

22

1

3

1418

64,

81

800,

243

16hHPHPPa ii

+

=

221

215

916,

817

91,

2431 hHPHPPa ii

+

=

22

1

3

11 8,36

43

4

7,

9

2hHPHPPb jii

=

3

1,

81

113112 HPPb i

3

1

3

3 9

1

=ii ppb

+

=

2223

149

16,

81

26

9

1,

81

5hhHpHppb iii

ii pphb3

1

2

59

11

=

and11110 4211111

, yxxxyx i =====

hyxxx iji == 132 00 , ,

hxi += 131

00100 544211, xxyyy ii =====

1100 5353, =+=== hyyhy ii

154 11,

+== ii xxx

Now using the equation (3.10) and (3.9) in (3.8).

( )pfhhPKhyxei ,),(*)(||)(||

)1( (3.13)

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:

)()()()( 541 fMtQetQexe iii ++=

Where )()()()()( 21 tQftQffM iii +=




)()()()()( 5413)1(

1 xftQftQftQfp iiii +++ (3.14)

Again, we write )(fMi in form of Divided difference and using Lemma 3.1, we get

( ) = =

=

3

1

2

1

)1( ,|)(|i j

jii bapfwfM

+

+

+=

222

14

3

3

1

4

1

36

1ththPa i

+

+

+

4232 32

14

36

29thth

+

++

+

=

4232222

2 29

5

3

8

27

13

6

1

18

19

3

2

9

1ththththPa i

+++= 4323

3

2

9

11

3

2

9

1 ttttPq i

+

+

+

+

+=

4232222

118

1

3

4

108

7

36

59

108

5

12

1

324

1ththhthPb i

=

2

23

1

27

8htPb i

and ii xx ==

10 11 ,

ii xxx == 10 212 ,

hhx i +== 133 10 ,

ii nyy == 10 11 ,

xyxy ji = 10 22 ,

Thus = =

==3

1

2

1i j

iji Pba

+

+

32222

3

4

108

7

12

7

81

2

12

5

36

1 ththth

+

+

42

18

5th

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 )(}1{ xe

( ) )()()()( }1{2}1{1}1{ tQftQfxsP iiii +=




( ) )()()()( }1{51}1{

4

}1{

3

}1{tQstQxstQfDP iiihi ++++ (3.15)

)()()()(}1{

5

}1{

41

}1{fUtQetQexePA

iiiii ++= (3.16)

Where )()()()()()()(}1{

3

}1{}1{

2

}1{

1 tQyfDPtQftQffU ihiiii ++=

)()()(}1{}1{

5

}1{

41 xfAPtQftQf iii ++

By using Lemma 3.1, we get

( ) ( )

==

= = 12

7,

324

1,

3

1

2

1

)1(HPbaPfwfU i

i j

jii

( ) ( )

+

++

+

+ 22222 43,

2134,

3629

23,

32 httHhtHthH (3.17)

Where ( )

++

=

22

1 33

1,

81

58

3

2,

9

1httHHPa i

( )

+

222,4

24

3

8,

27

157httHH

( )

++

+

=

22

2 36

7

,81

4

12

1

,324

1

httHHpa i

( )

++

1,

18

14

3

4,

108

23 22HhttH

( ) ( )

++++

=

2222

33

83

4

1

3

4

9

1htthttpa i

+

+

=

3

12,

36

29

2

3,

3

2

4

1,

36

11 HtHHpb i

( ) ( )]2222 3,2

143 httHht +

++

=

3

1,

81

22 Hpb i and

110 211, xxxx ii ===

hxhxxx iii +=== + 102 3312 ,,




ii yy ==

10 11,

hxyhxy +==10 22

,

By using (3.5), (3.13) and (3.17) in (3.16), we get inequality (3.6) of Theorem 3.1.

REFERENCES

1.

Dikshit, H.P. and Powar, P.L. Area Matching interpolation by discrete cubic splines, Approx Theory and

Application Proc.Ind. Cont.St. Jhon's New Foundland (1984), 35-45.

2.

Dikshit, H.P. and Rana, S.S. Discrete Cubic Spline Interpolation with a non-uniform mesh, Rockey Mount Journal

Maths 17 (1987), 700-718.

3. Boneva, K. Kandall D, and Stefanov, I. Spline transformation. J.Roy. Statis, Soc, Ser B 33 (1971), 1-70.

4.

Lyche, T. Discrete Cubic Spline Interpolation, Report RRS, University of Cyto 1975.

5. Lyche, T. Discrete cubic spline interpolation, BIT 16, (1978), 281-290.

6.

Mangasarian, O.L. and Schumaker, L.L. Discrete spline via Mathematical Programming SIAMJ Controla (1971),

174-183.

7. Mangasarian, O.L. and Schumaker L.L. Best Summation formula and Discrete Splines Num. 10 (1973).

8.

Rana, S.S. Local Behavior of the first difference of discrete cubic spline Interpolation, Approx. Theory and Appl.

8(4), 120-127.

9. Rana, S.S. and Dubey, Y.P. Local behavior of the deficient discrete cubic spline Interpolation, J. Approx. Theory

86 (1996), 120-127.

10.

Rana, S.S. and Dubey, Y.P. Best error bounds for quantic spline interpolation, Indian Journal of Pure and Appl.

Mathematics 28 (2000), 1337-1344.

Documents

6. Applied-Discrete Quartic Spline Interpolation-Y.P.dubey