eleminating jacobian from newton method in matlab

To eliminate the use of the Jacobian matrix in the Newton method in MATLAB, one can use the built-in optimization function fsolve. fsolve allows you to find the roots of a system of nonlinear equations without explicitly calculating the Jacobian matrix.

Here is an example of how to use fsolve to solve a system of nonlinear equations:

main.m
% Define the function
f = @(x) [x(1)^2 + x(2)^2 - 1; 
          x(1)^2 - x(2)];

% Set initial guess
x0 = [1; 1];

% Solve for the roots
x = fsolve(f, x0);

% Display results
disp(x);
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In this example, we define a function f that takes a vector x and returns a vector of values that must be equal to zero in order to find a root. We set an initial guess x0 for the root and use fsolve to solve for it. Finally, we display the result.

Note that fsolve uses an iterative method similar to the Newton method, but it does not require the user to explicitly calculate the Jacobian matrix.