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Generation of root finding algorithms via perturbation theory and some formulas. (English) Zbl 1115.65056

Summary: 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 compared with each other as well as with some new algorithms obtained by the modified Adomian decomposition method proposed by S. Abbasbandy [Appl. Math. Comput. 145, No. 2–3, 887–893 (2003; Zbl 1032.65048)].

65H05Single nonlinear equations (numerical methods)