Discretization of implicit ODEs for singular root-finding problems

This paper addresses the use of dynamical system theory to tackle singular root-finding problems. The use of continuous-time methods leads to implicit differential systems when applied to singular nonlinear equations. The analysis is based on a taxonomy of singularities and uses previous stability results proved in the context of quasilinear implicit ODEs. The proposed approach provides a framework for the systematic formulation of quadratically convergent iterations to singular roots. The scope of the work includes also the introduction of discrete-time analysis techniques for singular problems which are based on continuous-time stability and numerical stability. Some numerical experiments illustrate the applicability of the proposed techniques.