Convergence Properties of Newton ' s Method in the Proximity ofa

We analyse the elementwise convergence in the Jacobian of the classical Newton method as the solution of a non-linear system f(x) = 0 is approached. It is shown that each Jacobian element approaches a constant value at a rate determined by the nonlinearity of the relevant function with respect to the appropriate scalar variable. For many practical problems, this means that in the last steps of the Newton iteration, only a low-rank update has to be applied to the Jacobian. The consequences of these results are examined for both direct and iterative solution methods, and appropriate algorithmic modiications stated. Finally, two numerical examples are presented to illustrate the theory.