jacobian in newton method in matlab

To use the Newton method to solve a system of nonlinear equations in MATLAB, one needs to compute the Jacobian of the system.

Here is an example implementation of the Newton method in MATLAB, where F is the system of nonlinear equations, J is the Jacobian of F, x0 is the initial guess, tol is the tolerance for the solution, and max_iter is the maximum number of iterations:

main.m
function [x, iter] = newton(F, J, x0, tol, max_iter)
  iter = 0;
  x = x0;
  while norm(F(x)) > tol && iter < max_iter
    Jx = J(x);
    dx = -Jx\F(x);
    x = x + dx;
    iter = iter + 1;
  end
end
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To use this function, one needs to provide the function handle F, the function handle J, the initial guess x0, the tolerance tol, and the maximum number of iterations max_iter.

For example, if we want to solve the system of equations sin(x) + y^2 = 0, x^2 + y = 1, we can define the function handles as follows:

main.m
F = @(x) [sin(x(1)) + x(2)^2; x(1)^2 + x(2) - 1];
J = @(x) [cos(x(1)), 2*x(2); 2*x(1), 1];
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Then we can call the newton function as follows:

main.m
x0 = [0; 1];   % initial guess
tol = 1e-6;    % tolerance
max_iter = 100; % maximum number of iterations
[x, iter] = newton(F, J, x0, tol, max_iter); % solve the system
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After running this code, the solution x and the number of iterations iter will be returned.