Acceleration of Newton's method using nonlinear Jacobi preconditioning

For mildly nonlinear systems, involving concave diagonal nonlinearities, semi-global monotone convergence of Newton's method is guarantied provided that the Jacobian of the system is an M-matrix. However, regardless this convergence result, the efficiency of Newton's method becomes poor for stiff nonlinearities. We propose a nonlinear pre-conditioning procedure inspired by the Jacobi method and resulting in a new system of equations, which can be solved by Newton's method much more efficiently. The obtained preconditioned method is shown to exhibit semi-global convergence