The convergence of the perturbed Newton method and its application for ill-conditioned problems
The paper deals with the numerical solution to nonlinear equations, in particular the case of multiple or almost multiple roots. For this kind of ill-posed problem, the authors propose a modification of the classical Newton method called ``perturbed Newton method''. The idea of this algorithm is calculating the inverse derivative on every Newton iteration at a point given by a combination of the current approximation and the result of an action of some given linear operator. This operator can be selected depending on the equation features. The authors provide a theoretical investigation of convergence and practical recommendations for the use of the proposed method. The perturbed Newton behaves in a way similar to the classic Newton: when the iterates are far from the root it behaves like a bisection and if they are close to the root the sequence becomes second order. Presented numerical examples show the flexibility of perturbed Newton for ill-posed problems and demonstrate its better numerical performance when compared to the classical Newton for ill-posed equations.
scientific article; zbMATH DE number 6045127
scientific article