A Newton method for systems of m equations in n variables

The classical Newton-Kantorovich method for solving systems of equations f(x) = 0 uses the inverse of the Jacobian of f at each iteration. If the number of equations is different than the number of variables, or if the Jacobian cannot be assumed nonsingular, a generalized inverse of the Jacobian can be used in a Newton method whose limit points are stationary points of ||f(x)||². We study conditions for local convergence of this method, prove quadratic convergence, and implement an adaptive version of this iterative method, allowing a controlled increase of the ranks of the {2}-inverses used in the iterations.