The rate of multiplicity of the roots of nonlinear equations and its application to iterative methods

Nonsimple roots of nonlinear equations present some challenges for classic iterative methods, such as instability or slow, if any, convergence. As a consequence, they require a greater computational cost, depending on the knowledge of the order of multiplicity of the roots. In this paper, we introduce dimensionless function, called rate of multiplicity, which estimates the order of multiplicity of the roots, as a dynamic global concept, in order to accelerate iterative processes. This rate works not only with integer but also fractional order of multiplicity and even with poles (negative order of multiplicity).