A Newton's Iteration Converges Quadratically to Nonisolated Solutions Too

Abstract

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.