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.
The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a novel extension of Newton's iteration regains its local quadratic convergence toward nonisolated solutions that are semiregular as properly defined regardless of whether the system is square, underdetermined or overdetermined while Jacobians can be rank-deficient . Furthermore, the iteration serves as a regularization mechanism for computing singular solutions from empirical data. When a system is perturbed , its nonisolated solutions can be altered substantially or even disappear. The iteration still locally converges to a stationary point that approximates a singular solution of the underlying system with an error bound in the same order of the data accuracy. Geometrically, the iteration approximately approaches the nearest point on the solution manifold. The method simplifies the modeling of nonlinear systems by permitting nonisolated solutions and enables a wide range of applications in algebraic computation.
R
nonisolated solutions of unregularized nonlinear systems . With
solutions of nonlinear systems with singular Jacobians. By
R
version
novel extension
R
to nonisolated zeros of smooth mappings assuming the solutions
toward nonisolated solutions that
A
while Jacobians can be rank-deficient
D
de facto
R
zeros
solutions
R
Even if the given
When a
R
so that the nonisolated solution disappears, the
, its nonisolated solutions can be altered substantially or even disappear. The
A
singular
R
converges to
approaches
R
This extension simplifies nonlinear system modeling by eliminating the zero isolation process
The method simplifies the modeling of nonlinear systems by permitting nonisolated solutions