Extension Of Lagrange Interpolation

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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 4, ISSUE 01, JANUARY 2015 ISSN 2277-8616

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Extension Of Lagrange Interpolation

Mousa Makey Krady

Abstract: In this paper is to present generalization of Lagrange interpolation polynomials in higher dimensions by using Gramer's formula .The aim of this paper is to construct a polynomials in space with error tends to zero.

Keywords: Lagrange interpolation ;multivariable interpolation.

————————————————————

1. Introduction:

More than two hundred years ,in 1796 in his Lecous elementaries sure les mathematics that the French mathematician J.L.Lagrange formulated the interpolation polynomial called after him .He fitted on n points of the space

R

2 a polynomial of (n-1) degree and constructed it a suitable linear combination at basic polynomial

l

_i

(

x

)

giving in the

i

th point one, in the more point is zero. The Lagrangian interpolation polynomial and the Newtonian one are equivalent, but Lagrange interpolation has advantage ,its wanted polynomial can be written immediately, without solving a system of (n+1) equations, [4],[5]. Polynomial interpolation is a classical topic of numerical analysis which is useful in various area of applied mathematics. Polynomial are among the mathematical objects which can be handled most easily in practice, they can be represented by finite in formation and can be easily integrated and differentiated symbolically. Therefore, there is a wide area of applications for polynomial interpolation in several variables which range from surface reconstruction to cubature, finite elements and even optimization,[1] . Interpolation, a fundamental topic in numerical analysis, is the problem of constructing function which goes through a given set of data points.these data points are obtained by sampling of a function, the values of the

f

n

.

can be used to construct an interpolation ,which must be agree with the interpolated function at the data points. Multivariate interpolation has application in computer graphics , numerical quadrature ,and numerical solution of differential . [2][3]

2. polynomial interpolation

Lagrange gave the following interpolation polynomial p(x) of degree n given at (n+1) points

(

x

_i

,

y

_i

)

,

i



0 ,

1 ,..,

n

.

such that





n

i i i

l

x

y

x

p

y

0

)

(

)

(

Where

l

_i

(

x

)

are Lagrange basic polynomials defined by

)

2 ....(

...

0

1 )

(

)

1 (

)

(

)

(

)

(

0



















j

i

j

i

x

l

x

l

j i

n

j i

j i j

i



The benefit of Lagrange interpolation is that we can find and write down the interpolation immediately ,without computing the coefficients in the function.

3. Multivariable Lagrange interpolation:

Let

f



f

(

X

1

,

X

2

,..,

X

n

)

be an m-variable multinomial

function of degree n, since there are n points ,











 



n

m

n

p

, it is a necessary condition that we have p

distinct points

)

,...,

(

,

1 ,

)

,

,...,

(

X

₁

X

_m_,_i

f

_i



R

m1



i



p

f

_i



f

X

₁

X

_m_,_i for f be uniquely defined

)

3 ...(

...

)

,...,

(

1







n e

e e m

i

i i

X

f



i

e

where



are the coefficients in f





m j

e j

ei

_X

ij

X

1

hence we write f in the form





p

i i i

l

X

f

1

,

)

(

where

l

_i

(

X

)

multinomial function in

independent variables

X

1

,...,

X

_m ,consider the linear

equations







n e

j e i

j

j e j

X

f

1



,where

1 

i



p

we can

construct the matrix

________________________

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)

4 ...(

...

]

[

1 1 1

1 1















p p p i

e p e

p

e i e

i

e e

e i

X

M

X

M



Provided that

det(

m

)



0

(non singular move) and square matrix. Let





det(m

)

, substitutions

X

_j



X

in M

)) ( det( ) (

) 5 ..( ...

.. ...

. ... ...

) (

1 1

X m X

row j

X X

X M

j j

th

e p e

p

e j e

j

e e

j

p p i

p

 



      

 

      

 

  

Make another substitutions

X



X

_i

in

M

_j

(

X

)

,

(

i



j

)

row j

row i

X X

M _th

th

e p e

p

e i e

i

e i e

i

e e

i j

p p p p

 

         

 

         

 



. ...

. ... ... ...

) (

1 1 1 1

1 1

  

That is mean

det(

M

_j

)

_i



0

) , ( ...

) , ( )

, (

) , (

) 6 ....( ) ( ....

) ( )

( )

, (

) ( )

( )

, (

) , ( , ) ( ) (

2 2 1

1

2 2 1

1

1 1

y x l Z y

x l Z y x l Z Z

y x X

X Z

X Z y x z

X f X

l f y

x f

y x X x X

l

p p

p p p

i

p i

i i i

  

 

         

  



 

 



 

Experiment 1:

Suppose (1,0,0),(0,1,0) and (0,0,1) are there points are given and lie on z=f(x,y) . to construct polynomial of degree n=1, m=2 with

3

1

3 

























 



n

m

n

p

Hence

z

_i





₁

x

_i





₂

y

_i





₃

,

1 

i



3 ,



₁

,



₂

,



₃

are coefficients

3 3 2

3 1

1 0 0

  

 



  

 

From (4) we have:

From (5) we get:

x

y

M

y

M

x

M

Y

X

M































1 )

det(

)

det(

)

det(

,

1

0

1

0

1

3 3

2 2

1 1 1

By (6) we get

x

y

x

f

then

x

y

z





















1 )

,

(

)

1 (

3 3 2 2 1 1

Fig.1 Experiment (1) 1 )

det( ,

1 0 0

1 1 0

1 0 1

   

 

 

  

 

 M

(3)

www.ijstr.org 2. if n=2 , and m=2

6 2

2 2

6 1

) , (

     

 

 

     

 

 



  

n m n p

i y

x f

zi i i

Experiment 2:

Suppose the initial points of surface are (0,0,-1), (1,1,1),(1,0,-1),(2,1,5), ,(-2,-1,3),(3,2,9). Hence

1 5 3 2 2

2 4 2 4

2 2 2 4 2

6 6 2 2

4 8 2 8 2

8 8 4 12 4

) det(

4 8 12 4 20 8

) det(

4 ) det(

2 2

2 6

2 2

5

2 2

4

2 2

3

2 2

1 1

    

    

      

     

      

   

  



  

 

 



  



y y x x z and

x xy x y

y y x xy x

y y x x

y y x xy x

M

y y x xy x

M M

3. if n=3 and m=2 , p=10

z_i  f(x_i,y_i) 1i10

Experiment 3:

Suppose we are given set of point as:

) 9 , 2 , 2 ( , ) 6 , 1 , 2 ( ), 1 , 2 , 0 ( , ) 29 , 1 , 3 (

, ) 1 , 1 , 1 ( , ) 11 , 2 , 2 ( , ) 8 , 1 , 2 ( , ) 1 , 1 , 1 ( , ) 2 , 0 , 1 ( , ) 1 , 0 , 0 (

10 9

8 7

6 5

4 3

2 1

      

   



p p

p p p p

First ,we construct 11 matrices of size 10 x10 , and by MATLAB program , we calculate all determinate of them for example,

Fig.2 graph of experiment (2)

0 0 0 0 0 0 0 0 0 1

1 0 0 0 0 1 0 1 0 1

-1 1 -1 1 1 1 -1 -1 1 1

8 -4 2 -1 1 4 -2 2 -1 1

Det (M)= 8 8 8 8 4 4 4 2 2 1 = 87552

-1 -1 -1 -1 -1 1 1 -1 -1 1

27 9 3 1 1 9 3 3 1 1

0 0 0 -8 4 0 0 0 -2 1

-8 -2 -2 1 1 4 -2 -2 1 1

-8 8 -8 -8 4 4 4 -2 -2 1

x^3 x^2*y x*y^2 y^3 y^2 x^2 x*y x y 1

1 0 0 0 0 1 0 1 0 1

-1 1 -1 1 1 1 -1 -1 1 1

8 -4 2 -1 1 4 -2 2 -1 1

Det (M1)=8 8 8 8 4 4 4 2 2 1

-1 -1 -1 -1 -1 1 1 -1 -1 1

27 9 3 1 1 9 3 3 1 1

0 0 0 -8 4 0 0 0 -2 1

-8 -2 -2 1 1 4 -2 -2 1 1

-8 8 -8 -8 4 4 4 -2 -2 1

=-171648x3_+812160_x2_y_+162432_x2_+499968x_y2

-1169280xy-78336x+35712y3- 639360y2 -1377792y+87552,

and so on

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1 ))

det(

/

)

det(

9 )

det(

6 )

det(

)

det(

29 )

det(

)

det(

11 )

det(

8 )

det(

)

det(

2 )

(det(

109440

32832

10944

76608

21888

10944

54720

10944

)

det(

110592

82944

13824

9216

122112

69120

34560

85248

25344

)

det(

140544

50688

20736

768 101376

27648

13824

69120

13056

)

det(

20736

26496

8064

5376

31104

18432

9216

24192

3840

)

det(

382464

177408

6912

4608

354816

140544

48384

241920

52992

)

det(

16512

17856

4800

768 24768

5760

2880

14400

2112

)

det(

153600

27648

24576

3072

99072

23040

11520

57600

8448

)

det(

476928

259200

10368

6912

408960

20506

80640

293760

73728

)

det(

755712

304128

36864

92160

608256

253440

82944

414720

78336

)

det(

87552

1377792

639360

35712

78336

1169280

499968

162432

812160

171648

)

det(

87552

004 7552

.

8 )

det(

3

10 9

8 7

6 5

4 3

2 1

2 3

2

2 2

3 10

10

2

3 2

2 2

3 9

9

2

3 2

2 2

3 8

8

2

3 2

2 2

3 7

7

2

3 2

2 2

3 6

6

2 3

2

2 2

3 5

5

2

3 2

2 2

3 4

4

2

3

2 2

3 3

3

2

3 2

2 2

3 2

2

3 2

2 2

3 1

1











































































































































y

x

z

M

z

y

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

y

x

xy

x

y

x

M

e

M

4. Conclusion:

We tried to build polynomials of different degrees in R3_and

used MATLAB program from several different angles.

References

[1] Boor, C.Dc. (1995).A Multivariate divided difference in c. k. chui and L.L.schumaker ,editors. Approximation theory vlll ,vol l: Approximation and interpolation, p.87-96. World scientific publishing co.

[2] Olver ,P.J.( 2006) . on multivariate interpolation. Studies in Applied mathematics , 116(4),201-240 ,.

[3] steffensen , J.F.( 2006) .interpolation., Dover publical, lnc, newyork ,second edition,.