Analysis and modification of Newton's method at singularities

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 a certain regularity condition Q-linear convergence is shown to be almost sure from all initial points that are sufficiently close to x* . The possibility of significantly better performance by other nonlinear 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 underdetermined system is known to have a nullspace of a certain dimension at a solution of interest, an overdetermined system based on the QR or LU decomposition of Vf is used to obtain superlinear convergence.