create a function that performs newton's method without the user inputting the derivative in matlab in matlab

To create a function that performs Newton's method without the user inputting the derivative in MATLAB, you can use numerical differentiation to approximate the derivative. Here is an example code:

main.m
function root = newtonsMethod(func, x0, tolerance, maxIterations)
    % Initialize variables
    x = x0;
    iteration = 0;
    
    % Perform Newton's method iterations
    while abs(func(x)) > tolerance && iteration < maxIterations
        % Approximate the derivative using numerical differentiation
        fprime = (func(x + 1e-6) - func(x)) / 1e-6;
        
        % Calculate the next approximation of the root
        x = x - func(x) / fprime;
       
        % Increment the iteration counter
        iteration = iteration + 1;
    end
    
    % Return the approximate root
    root = x;
end

% Example usage
func = @(x) x^2 - 9;
x0 = 5;
tolerance = 1e-6;
maxIterations = 100;

root = newtonsMethod(func, x0, tolerance, maxIterations);
disp(root);
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In this code, the newtonsMethod function takes four inputs:

• func: the function for which you want to find the root
• x0: the initial starting point for the iteration
• tolerance: the desired tolerance for the root approximation
• maxIterations: the maximum number of iterations to perform

The function uses numerical differentiation to approximate the derivative of the function at each iteration. It continues the iterations until either the tolerance is met or the maximum number of iterations is reached.

You can replace the func and x0 values with your own function and starting point. The root variable will contain the approximate root calculated by the function.

Note that this method may not always be accurate for all functions, especially if the derivative is steep or undefined at some points. In those cases, it is better to manually input the derivative of the function.