Question

4 [3 points] Use the function f(x on the interval [5, 5] and the 11 points below to find a N-10 Lagrange polynomial, using MATLAB. You will need to write code for and plot the polynomial as a function of x, but you do not have to write the polynomials down by hand 5 0 2 3 4 5 f(x) | 0.0399 | 00623 | 0.1098 | 0.2353 | 0.5 | 0 | 0.5 | 0.2343 | 0.1098 |0.0623 | 0.0399 [2 points] Plot the target function f(x), and interpolating polynomial on the same graph over the entire range [-5, 5] [1 point] Plot the error between each interpolating polynomial and the target function over [-5, 5] using the same graph a. b.

Please use matlab, post the code, and dont just write the code by hand. Thank you!

Answer 1

For, Lagrange interpolation of nth degree n+1 ,points are required which is given in question.

The formula for calculating the value through Lagrange interpolation is

f(x) =( (x-x1)*(x-x2)*......(x-xn)/(x0-x1)*(x0-x2)....(x0-xn))*f0 + .......+( (x-x1)*(x-x2)*......(x-xn-1)/(xn-x0)*(xn-x2)....(xn-xn-1))*fn

Here, the function value for all the points are given. So, we can use the values or save the value in array and use to calculate the value through matlab code.

(a) % Matlab code to calculate value using Lagrange polynomial

x = linspace(-5,5,11) ;% it will create array and save it in work-space of matlab having the values as shown in table, with % the variable x

y = (x.^2)/(1+ x.^4) ; % it will calculate the function f(x) for every value of x and will save in work-space with variable y

% here both x and y will have 11 points.

% for calculating the Lagrange interpolation ,there is direct command in matlab

xx = linspace(-5,5,10);

yy = lagrange(xx,x,y); % it will determine the lagrange polynomial values

plot(x, y, 'o', xx, yy, '.'); %It will plot both function and interpolating polynomial on same graph

(b) % consider error =e

for i =1:11

e(i) = (y(i)-yy(i))/y(i) % it wil calculate and save the error as a variable e in workspace

end

hold on

plot(x,e) % it will plot the error on same graph