  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. Furthermore, the iteration serves as a de facto regularization
mechanism for computing singular zeros from empirical data. Even if the given
system is perturbed so that the nonisolated solution disappears, the iteration
still locally converges to a stationary point that approximates a solution of
the underlying system with an error bound in the same order of the data
accuracy. Geometrically, the iteration approximately converges to the nearest
point on the solution manifold. This extension simplifies nonlinear system
modeling by eliminating the zero isolation process and enables a wide range of
applications in algebraic computation.
