iteration newton raphson

i am having difficulty trying to find an iteration formula using newton-raphson for solving the equation x=cos(2x) and write an m-file for finding the solution whereby x matches cos(2x) to atleast 8 decimal places using format long. can anybody help me?

[arushi]

Hi Sukhraj,

Here is the one example of implemtention for your reference:

function x = solve_cos2x()
    % Initial guess
    x = 0.5; % You might need to adjust this based on the problem
    tolerance = 1e-8;
    maxIterations = 1000;
    iteration = 0;
    
    while true
        % Calculate the function and its derivative
        fx = x - cos(2 * x);
        dfx = 1 + 2 * sin(2 * x);
        
        % Newton-Raphson update
        x_new = x - fx / dfx;
        
        % Check for convergence
        if abs(x_new - x) < tolerance
            break;
        end
        
        % Update x
        x = x_new;
        
        % Increment iteration count
        iteration = iteration + 1;
        if iteration >= maxIterations
            warning('Maximum iterations reached without convergence.');
            break;
        end
    end
end

Hope this helps.

Answer 1

[Rahul]

Hi @sukhraj Randhawa,

I understand that you are trying to find an iteration formula using Newton-Raphson for solving the equation x=cos(2x) where x matches cos(2x) to atleast 8 decimal places using format long.

As similarly mentioned by @arushi, you can use a loop for iterating through the values and update the variables using the Newton-Raphson method. An example is as follows:

format long % Initially set the format as long
x_solution = solve_cos_2x();

function x = solve_cos_2x()
   
    syms z % Setting this to get equations
    fx = z - cos(2*z);
    diff_fx = diff(fx);
    
    x = 0.5; % You can choose a different starting point if needed
    tol = 1e-8; 
    max_iter = 100; % You can adjust max iters
    
    for iter = 1:max_iter
        
        % Newton-Raphson update
        x_next = x - double(subs(fx, z, x)) / double(subs(diff_fx, z, x));
        
        if abs(x_next - x) < tol
            break;
        end

        x = x_next;
    end
    
    if iter == max_iter
        warning('Maximum number of iterations reached without convergence.');
    else
        fprintf('Converged to x = %.10f in %d iterations\n', x, iter);
    end
end

Here I have set 'fx' and 'diff_fx' using symbolic variable 'z'. Hence I was able to use 'diff' function for differentiation.

Using the loop we are updating the Newton-Raphson equation to reach convergence.

You can refer to the following documentations to know more about these functions:

'diff': https://www.mathworks.com/help/releases/R2024a/symbolic/differentiation.html?searchHighlight=Differentiation%5D&s_tid=doc_srchtitle

'subs': https://www.mathworks.com/help/releases/R2024a/symbolic/subs.html?searchHighlight=subs&s_tid=doc_srchtitle

'format': https://www.mathworks.com/help/releases/R2024a/matlab/ref/format.html?searchHighlight=format%20long&s_tid=doc_srchtitle

Hope this helps! Thanks.