Before revision: The textbook Newton's iteration is practically inapplicable on nonisolated solutions of unregularized nonlinear systems . With a simple modification, a version of Newton's iteration regains its local quadratic convergence to nonisolated zeros of smooth mappings assuming the solutions are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined  .
After revision: The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a version of Newton's iteration regains its local quadratic convergence to nonisolated zeros of smooth mappings assuming the solutions are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined .