create newton's method for finding zeroes in matlab

Newton's method is a root-finding algorithm that uses linear approximation to find a zero of a function. The basic idea is to start with an initial estimate of the zero and iterate using the formula:

main.m
x_{n+1} = x_n - f(x_n)/f'(x_n)
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where f(x) is the function and f'(x) is its derivative.

Here's how to implement Newton's method in MATLAB:

main.m
function [x, iter] = newton(f, df, x0, tol, max_iter)
% NEWTON Newton's method for finding zeros of a function
%   [x, iter] = newton(f, df, x0, tol, max_iter) finds a zero of the
%   function f using Newton's method, starting from the initial guess x0.
%   The derivative of f, given by df, is also required. The function stops
%   when the absolute error is less than tol or the maximum number of
%   iterations is reached. Returns the estimated zero x and the number of
%   iterations iter.

% Initializations
x = x0;
iter = 0;

% Iterate
while abs(f(x)) > tol && iter < max_iter
    x = x - f(x)/df(x);
    iter = iter + 1;
end

% Check for convergence
if abs(f(x)) > tol
    warning('Newton''s method failed to converge');
end
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To use this function, you need to define the function f and its derivative df as separate MATLAB functions. For example, to find a zero of the function f(x) = x^2 - 2, you can define:

main.m
f = @(x) x^2 - 2;
df = @(x) 2*x;
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Then, you can call the newton function with an initial guess x0 and the desired tolerance and maximum number of iterations, like this:

main.m
x0 = 1;
tol = 1e-6;
max_iter = 100;
[x, iter] = newton(f, df, x0, tol, max_iter);
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This will return an estimated zero x and the number of iterations iter required to find it.

Note that Newton's method can fail to converge (i.e., enter an infinite loop) if the initial guess is too far from the zero, or if the function or its derivative are undefined or have a singularity near the zero. Therefore, it's important to choose a reasonable initial guess and to test the function for different inputs.