Question

Predict the roots of the polynomial f(x)=x^3-6x^2+11x-6 writing a code. Show your code and print the iterative steps. Use a) Fixed Point Iteration and b) Newton Raphson. The initial values and the convergence criteria are up to you.

Answer 1

Fixed Point Iteration:

clear;
clc;

g=[1 -6 11 -6];

x1 = 0;
x2 = g(x1);

iterations = 0;% # iteration counter

ezplot(g,[0,1]);
hold on
ezplot('x',[[0,1]])

while (abs(x2-x1) > 1e-5 && iterations<100)
plot([x1 x1], [x1 x2], 'k-')
plot([x1 x2], [x2 x2], 'k--')
%pause
iterations = iterations + 1;
x1 = x2;
x2 = g(x1);
end
iterations
[x1 x2]

Newton Raphson:

clear all
close all
clc

f=[1 -6 11 -6]

df=@(x) -sin(x)-3

a=0; b=1;
x=a;
for i=1:1:100
x1=x-(f(x)/df(x));
x=x1;
end
sol=x;
fprintf('Approximate Root is %.15f',sol)
a=0;b=1;
x=a;
er(5)=0;
for i=1:1:5
x1=x-(f(x)/df(x));
x=x1;
er(i)=x1-sol;
end
plot(er)
xlabel('Number of iterations')
ylabel('Error')
title('Error Vs. Number of iterations')