implement the multidimensional newton method for computing f(x) = 0 (with x, 0 r n ) in a matlab function mynewton with the jacobian not given in matlab

To implement the multidimensional Newton Method in MATLAB, we need to first define the function f(x) = 0 (with x, 0 r n ) in a separate MATLAB function. Then, we need to write another function mynewton that will solve for the root of f(x) using the Newton Method.

However, the Newton Method requires the Jacobian of f(x), which is not provided in MATLAB. Therefore, we need to approximate the Jacobian using finite difference equations.

The following code shows how to implement the multidimensional Newton Method with the Jacobian approximated by finite difference equations:

main.m
function [x, info] = mynewton(f, x0, tol, maxiter)
% Implementation of the Multidimensional Newton Method
% to solve f(x) = 0, with x, 0 r n
% Where f is a MATLAB function that returns the value of the function and the Jacobian at x
% x0 is the initial guess for x
% tol is the tolerance for the stopping criteria
% maxiter is the maximum number of iterations allowed

x = x0;
fx = f(x);
info(1) = norm(fx);
iter = 1;

while (norm(fx) > tol) && (iter < maxiter)
    J = zeros(length(x));
    h = 1e-8;

    % Approximate the Jacobian using finite differences
    for i = 1:length(x)
        xplus = x;
        xminus = x;
        xplus(i) = x(i) + h;
        xminus(i) = x(i) - h;
        fxplus = f(xplus);
        fxminus = f(xminus);
        J(:, i) = (fxplus - fxminus) / (2*h);
    end

    % Solve for the increment delta_x
    delta_x = -J \ fx;

    % Update x and fx
    x = x + delta_x;
    fx = f(x);

    % Update iteration count and residual info
    iter = iter + 1;
    info(iter) = norm(fx);
end

if iter == maxiter
    fprintf('Maximum number of iterations reached \n');
end

1093 chars
45 lines

This implementation is flexible enough to handle any function f(x) without knowing the Jacobian a priori, as long as it's provided as an input argument to mynewton.

gistlibby LogSnag