Generation of root finding algorithms via perturbation theory and some formulas

Abstract

Perturbation theory is systematically used to generate root finding algorithms. Depending on the number of correction terms in the perturbation expansion and the number of Taylor expansion terms, different root finding formulas can be generated. The way of separating the resulting equations after the perturbation expansion alters the root-finding formulas also. Well known cases such as Newton–Raphson and its second correction, namely the Householder’s iteration, are derived as examples. Moreover, higher order algorithms which may or may not be the corrections of well known formulas are derived. The formulas are contrasted with each other as well as with some new algorithms obtained by modified Adomian Decomposition Method proposed in Ref. [S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation 145 (2003) 887–893].