Consider the general problem of finding a root of f, that is find a solution of

Question

Consider the general problem of finding a root of f, that is find a solution of the equation f(x) = 0. The Newton's method of iteration is x_n+1 = x_n - f(x_n)/f'(x_n), n = 0, 1, 2, ellipsis Write a program that will implement Newton's method and evaluate the problem by finding a solution of equations x = cos(x). Use your computer implementation of Newton's method with initial guess x_0 = 1 to compute at least the first 12 iterations. Calculate the error at each iteration e_n = alpha - x_n, where alpha is the exaction solution. Use alpha = 0.7390851332151607. Please use MATLAB and explain everything, thank you very much!

Explanation / Answer

Matlab code:

format long;

exact = 0.7390851332151607;

x = 1;

f = @(x)(cos(x) - x);

g = @(x)(-sin(x) - 1);

h = (f(x))/ (g(x));

i = 1;

while(i<= 12)

iteration = i

h = f(x)/g(x);

x = x - h;

difference_from_exact_solution = exact - x

i=i+1;

end

finalroot = x

Sample Output:

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