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IOP Conference Series: Materials Science and Engineering

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Construct Polynomial of Degree n by Using Repeated Linear

Interpolation

To cite this article: Mousa M. Kkhrajan and Yaseen Merzah Hemzah 2020 IOP Conf. Ser.: Mater. Sci. Eng. 928 042009

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2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

1

Construct Polynomial of Degree n by Using Repeated Linear

Interpolation

Mousa M. Kkhrajan1* ; Yaseen Merzah Hemzah2

Department of Mathematics and Computer Applications , College of Sciences , AL-

Muthanna University , samawa , Iraq

E-mail : mmkrady@gmail.com ; yaseenmerzah@mu.edu.iq

Abstract

In this paper the fundamental concept of repeated linear interpolation and its possible

applications in computer-aided geometric design, and start considering basic constructive

methods for curves and surfaces. We discuss here a repeated linear interpolation method that

we commonly find in computer graphics and geometric modelling. Repeated linear

Interpolation means to calculate a polynomial by using several points. For a given sequence

of points, this means to estimate a curve that passes through every single point. The purpose

of this paper is to construct a polynomial of degree less than or equal to n, by using repeated

linear Interpolation.

Keywords: Repeated linear Interpolation ; linear Interpolation ; Lagrange linear

Interpolation ; Lagrange interpolation.

1. Introduction

We know that any two points determine a straight line. More precisely; any two

points, (and ( with determine a unique first degree polynomial P1(

x) passing through these two points .(1-3).

In general; given (n+1) points in the plane ( , K= 0,1,

.....

,n with distinct xk ,s ;

there is a unique polynomial of degree less than or equal n , passing through this points ,

denoted by which is called the Interpolation polynomial and defined by :

P( K =0,1,…..,n

The most compact representation of the Interpolation polynomial is the Lagrange formula

, {see .(4-6)

2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

2

where are Lagrange basic polynomials defined by

and

We shall take three points at least

by using linear

twice – once using the first two points and then a second time using the last two points , which

will in general produce two different results for the Interpolated value of . This leads to

construct a polynomial of degree ≤ n is the aim of this paper.

2. Main Results

Now, We will study all cases in details, how to construct a polynomial of degree less than or

equal to n passing through the given points i= 0,1,2,3…,n

The linear formula is given as see (1, 2)

…………………… (1)

Let n=2 , to construct a poly. of degree two , passing through the points

,

First, take the points & then & and from (1), we

have (see Fig. 1)

}………………….. (2)

From (2) , we have a poly. Of second degree

……………….. (3) [Quadratic formula]

take n=3 , the points are

,,& , (see Fig. 2)

we have the following equations as:

………. (4) Linear interpolation

2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

3

……… (5) quadratic interpolation

From (4) & (5) , we have

………. (6) Cubic interpolation

a polynomial of degree 3.

If we have m=5 points, from repeated linear interpolation (1), we get a polynomial of degree n=4

as:

…………… (7) linear formula

….. (8) quadratic formula

(9) cubic formula

From (9),we get a poly. Of degree 4 as:

…………….. (10)

2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

4

If m=6 ,we have the following equations :

…………(11)

….. (12)

…… (13)

……. (14)

….(15) apolynomial of fifth degree.

2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

5

In general:

M= k , k+1 , k+2 , …..

N=2 , 3 , 4 ,……

Where M number of points and N degree of a polynomial. (see Fig. 3)

Figure 1- polynomial of degree (2)

Figure 2- polynomial of degree (3)

2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

6

Figure 3- polynomial of degree (n)

Illustrate Example1: Let y= f(x) defined as:

from (4) we have

;

;

From (5) we have

;

;

from(6) we get

Figure4- polynomial of degree (3)

Illustrate Example 2: the following data defined the function y=f(x) as:

3

2

1

0

xi

27

8

1

0

iy

2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

7

From (11) we have:

;

-2 ;

;

-60 ;

From (12):

;

+6 ;

;

+60

From(13):

;

;

And

, for k=0,1.

Then the function (polynomial) that corresponds to the data is:

y== .

We can rearrangement the result of example (2) as in the following table.

Table 1- Rearragment the result of ex.2 1

No. of point

Poly. Of 1st degree

Poly. Of 2nd degree

Poly. Of 3rd degree

0

-x

1

-2

x-2

2

+6

9x-18

3

23x-60

4

+60

43x-140

5

5

4

3

2

1

0

xi

75

32

9

0

-1

0

iy

2nd International Scientific Conference of Al-Ayen University (ISCAU-2020)

IOP Conf. Series: Materials Science and Engineering 928 (2020) 042009

IOP Publishing

doi:10.1088/1757-899X/928/4/042009

8

Figure 5- polynomial of degree (3)

3. Conclusions

1- Through the results that we obtained, a method was found to construct a polynomial of degree

less than or equal to n where n is a positive integer and that n> = 2 using the recursive linear

interpolation formula

2-Through the examples data were taken for a specific function and over an interval and according

to the required points (conditions) and by using the iterative method of linear insertion the same

function of the indicated data was obtained with an amount of error equal to zero

References

1. Watson WA, Oates PJ, Philipson T. Numerical analysis: the mathematics of

computing1969.

2. K.saeed R. Introduction to Numerical analysis. 1st ed. Iraq: sulaimani; 2015.

3..Bibliography rady MM. Extension of Lagrange Interpolation. International Journal of

Scientific & Technology Research. 2015;4(1).

4. Steffensen JF. Interpolation: Courier Corporation; 2006.

5. Salomon D. The computer graphics manual: Springer Science & Business Media; 2011.

6. Davis PJ. Interpolation and approximation: Courier Corporation; 1975.