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The textbook Newton's iteration is practically inapplicable on solutions ofnonlinear systems with singular Jacobians. By a simple modification, a novelextension of Newton's iteration regains its local quadratic convergence towardnonisolated solutions that are semiregular as properly defined regardless ofwhether the system is square, underdetermined or overdetermined while Jacobianscan be rank-deficient. Furthermore, the iteration serves as a regularizationmechanism for computing singular solutions from empirical data. When a systemis perturbed, its nonisolated solutions can be altered substantially or evendisappear. The iteration still locally converges to a stationary point thatapproximates a singular solution of the underlying system with an error boundin the same order of the data accuracy. Geometrically, the iterationapproximately approaches the nearest point on the solution manifold. The methodsimplifies the modeling of nonlinear systems by permitting nonisolatedsolutions and enables a wide range of applications in algebraic computation.

A Newton's Iteration Converges Quadratically to Nonisolated Solutions Too