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iteration newton raphson

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sukhraj Randhawa
sukhraj Randhawa il 21 Mar 2012
Risposto: Rahul il 26 Ago 2024
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?
  1 Commento
arushi
arushi il 26 Ago 2024
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.

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Risposte (1)

Rahul
Rahul il 26 Ago 2024
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:
Hope this helps! Thanks.

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