How to save each guess / output as an array via Newton's Method?

Currently doing the Newton's Method for a homework problem.

Here's how the lecture demonstrated this.

x = 3; % Our initial Guess.
f = @(x) x.^2 - 191; % Our equation.
f_prime = @(x) 2*x; % Deriviative of Equation.
tol = 1e-4 % Our Tolerance.

for i = 1:1000 % Let this sucker go up to 1000.
    x = x - f(x)/f_prime(x);
    if abs(f(x)) < tol
        break
    end
end
if i == 1000
    warning('Newtons''s method did not converge.');
end
approx_root = x; % Root done by computation above.
true_root = sqrt(191); % Actual Root.

disp(abs(true_root - approx_root)); % This is the error.
disp(i) % This is the number of iterations it took.

Now, I don't know how to store each guess into a row vector.

If I wanted to store each guess into a row vector? How would you go about that? I'm certain there's a for loop component to it.

Answer 1

[madhan ravi]

x = zeros(1e3, 1);
x(1) = 3;
for ii = 2:1e3 % Let this sucker go up to 1000.
    x(ii) = x(ii-1) - f(x(ii-1))/f_prime(x(ii-1));
    if abs(f(x(ii-1))) < tol
        break
    end
end

[Rajdeep Mattu]

Can you explain this more in english on what's going on?

And this wouldn't be inside the original for loop right? But outside of it?

[madhan ravi]

 f = @(x) x.^2 - 191; % Our equation.
 f_prime = @(x) 2*x; % Deriviative of Equation.
 tol = 1e-4 % Our Tolerance.

 x = zeros(1e3, 1);

 x(1) = 3; % Our initial Guess.

 for ii = 2:1e3 % Let this sucker go up to 1000.
     x(ii) = x(ii-1) - f(x(ii-1))/f_prime(x(ii-1));
     if abs(f(x(ii-1))) < tol
        break
     end
 end

 if ii == 1000
     warning('Newtons''s method did not converge.');
 end
approx_root = x; % Root done by computation above.
true_root = sqrt(191); % Actual Root.

 disp(abs(true_root - approx_root)); % This is the error.
 disp(ii) % This is the number of iterations it took.