Analysis and modification of Newton's method at singularities

Abstract

For systems of nonlinear equations f=0 with singular Jacobian Vf(x*) at some solution x* E F-1(0) the behaviour of Newton's method is analysed. Under certain regularity condition Q-linear convergence is shown to be almost sure from all initial points that are sufficiently c,lose to x*. The possibility of significantly better performance by other nonlienar equation solvers is ruled out. Instead convergence acceleration is achieved by variation of the stepsize or Richardson extrapolation. If the Jacobian Vf of a possibly undetermined system is know to have a nullspace of a certain dimensional a solution of interest, and overdetermined system based on the QR or LU decomposition of Vf is used to obtain superlinear convergence.