newton method in matlab

The Newton method, also known as Newton-Raphson method, is a numerical method commonly used to find the roots of a given equation. Here is an example of how to implement the Newton method in MATLAB:

main.m
% Define the function to find root of
f = @(x) x^3 - 2*x - 5;

% Define the derivative of the function
df = @(x) 3*x^2 - 2;

% Set initial guess for the root
x0 = 2;

% Set tolerance for convergence
tol = 1e-6;

% Set maximum number of iterations
max_iter = 100;

% Start iterations
for i = 1:max_iter
    % Calculate the Newton step
    dx = -f(x0)/df(x0);
    
    % Update the solution
    x1 = x0 + dx;
    
    % Check for convergence
    if abs(x1 - x0) < tol
        fprintf('Solution converged after %d iterations.\n', i);
        fprintf('Root is: %f\n', x1);
        break;
    end
    
    % Update solution for next iteration
    x0 = x1;
end

% If the loop finished without finding a root
if i == max_iter
    fprintf('Method failed after %d iterations.\n', max_iter);
end
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In this example, we define the function we want to find the roots of as f, and its derivative as df. We set an initial guess for the root x0, a tolerance for convergence tol, and a maximum number of iterations max_iter.

Then, we start iterating using a loop. At each iteration, we calculate the Newton step by dividing the function value at x0 by its derivative at x0. We update the solution by adding the Newton step to x0. We check for convergence by checking if the difference between x1 and x0 is smaller than the tolerance. If so, the root has been found and we exit the loop. Otherwise, we continue iterating with the updated solution x0.

Finally, we print the converged root and the number of iterations it took to find it. If the loop finished without finding a root, we print an error message.