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Construct Polynomial of Degree n by Using Repeated Linear Interpolation Construct Polynomial of Degree n by Using Repeated Linear Interpolation

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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.
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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
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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
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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
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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
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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
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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
Poly. Of 1st degree
Poly. Of 2nd degree
Poly. Of 3rd degree
-x
-2
x-2
 
+6
9x-18
 
 
23x-60
 
+60
43x-140
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
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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
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