% Newton's to solve nonlinear system of 3 equations in 3 unknowns

NN = 6;
x = [1 1 1]';

for n = 1:NN
  f = [x'*x - 4,    sin(2*pi*x(1)) - x(3),    x(1)^2 - x(2)]';
  residuals(:,n) = f;
  J = [2*x(1)               2*x(2)       2*x(3); 
       2*pi*cos(2*pi*x(1))  0            -1;
       2*x(1)               -1           0];
  s = J \ (-f);
  x = x + s;
  estimates(:,n) = x;
end

estimates, residuals