Question

Write a program to compute the root of the function f(x) = x³ + 2 x² + 10 x - 20 by Newton method ( x₀ =2 _{). Stop computation when the successive values differ by not more than 0.5 * 10^-5 . Evaluate f(x) and f '(x) using nested multiplication. The output should contain:}

_{(1) A table showing at each step the value of the root , the value of the function,and the error based upon successive} approximation values and percentage error for each method.

(2) A plot showing the variation of percentage error with iteration number for each method on the same graph so as to compare the method.

Answer 1

We have solved the given problem using MATLAB code. (Code attached).

MATLAB code:

function newton()

fprintf('\n');
prompt1='Enter the Initial guess x0: ';
prompt2='Enter the accuracy percentage: ';
x0= input(prompt1);
tol= input(prompt2);
  
f = @(x) x.^3+2*x.^2+10*x-20;
dfdx= @(x) 3*x.^2+4*x+10;
error=1e10;
x_old=x0;
A(1,:)=[0 x_old f(x_old) 0 0];
  
fprintf('\n\n\nIteration Root Function Value Absolute error Percentage error\n');
fprintf('--------- ----------- ---------------- ---------------- -----------------\n');
fprintf('%3d %12.8f %12.8f\n',A(1,1),A(1,2),A(1,3));
n=1;
  
while error>tol
n=n+1;
x_new=x_old-f(x_old)/dfdx(x_old);
error=abs(x_new-x_old);
A(n,:)=[n-1 x_new f(x_new) abs(x_new-x_old) abs(x_new-x_old)/x_old];
fprintf('%3d %12.8f %12.8f %12.8f %12.8f\n',A(n,1),A(n,2),A(n,3),A(n,4),A(n,5));
x_old=x_new;
end
fprintf('\n\n');
  
figure(1)
plot(A(2:end,1),A(2:end,5))
xlabel('Iterations')
ylabel('Percentage error')
title('Iterations vs Percentage error graph')
end