Course No.: EEE 212 Group No.:06

Course Title: Numerical Technique Laboratory

Name of the Assignment:

Name: Md. Tahsin Mostafiz

Student ID: 201106109

Department: EEE

Level:2 Term: 2

Partner Student’s ID: 2011061110

EXPLANATION OF THE ALGORITHM OF NEWTON’S INTERPOLATION. GENERATION OF A SMOOTH CURVE FOR SOME GIVEN DATA POINTS, USING NEWTON'S INTERPOLATION.

Objective:

The objective of this experiment is to

Explanation of the algorithm of Newton's interpolation.

For the following data points using Newton's interpolation a smooth curve

has to be generated:

(-1, 2), (0, 1), (1,-2), (2, 3), (3, 6).

Figures and Code

Newton polynomial Interpolation:

To construct and evaluate the Newton polynomial of degree ≤ n that passes

through the n +1 points (xk , y k ) = (xk , f (xk )) for k = 0,1…n : This polynomial Pn(x) is said to be a Newton Polynomial with n centers x0, x1 to

xn-1. It involves sums of products of linear factors upto:

an(x-x0)(x-x1)(x-x2)……………(x-xn-1)

To construct and evaluate the Newton polynomial of degree ≤ n that passes

through the n +1 points, let us take (xk , y k ) = (xk , f (xk )) for k = 0,1…n :

where

CODE:

clc;

clear all;

close all;

x = [-1 0 1 2 3];

y = [2 1 -2 3 6];

n = length(x);

%calculating the divided difference table. The first column is

done separately to maintain the next loop's initializations.

for i=1:n

d(i,1) = y(i);

end

for j=2:n

for k=j:n

d(k,j)=(d(k,j-1)-d(k-1,j-1))/(x(k)-x(k-j+1));

%calculating values of d

end

end

%creating a 2D array to keep track of the product of divided

difference and polynomial before that difference.

s=zeros(n,n);

for i=1:n

p=1;

if i~=1

for k=1:i-1

a = [1 -x(k)]; %Here 1 is the co-effecient of the

variable x.

p = conv(a,p); %polynomials which will be multiplied

by co-efficients d.

end

end

p=p*d(i,i);

f = fliplr(p); % p has been flipped left to right to keep

the constants in the leftmost column.

for j =1:length(p)

s(i,j) = s(i,j)+f(j); %creating the array of p*d

end

end

A = fliplr(s); % flipped again to keep the higher degree of x’s

co-efficients in the leftmost coloumn.

for j = 1:n

res(j) = sum(A(:,j)); %adding the co-efficients of same

column

end

disp(res);

x1 = x(1):.1:x(n); % value of x has been incremented from first

element of x to last with .1 increment.

plot(x,y,'o',x1,polyval(res,x1)); %plotting values

grid on

RESULT:

The following data was shown in command window after running the code:

So the given data points follow the polynmial provided below:

−0.8333𝑥4 + 3.3333𝑥3 − 0.1667𝑥2 − 5.3333𝑥 + 1.0000

And the graph for the following data passes through given points is shown below:

DISCUSSION:

MATLAB has successfully been used in this assignment to interpolate data by

using Newton polynomial .