An infinite family of one-step iterators for solving nonlinear equations to increase the order of convergence and a new algorithm of global convergence

In this paper we present an infinite family of one-step iterative formulas for solving nonlinear equations (Present Method One), from now on PMI, that can be expressed as xn+1=Fm(xn)xn+1=Fm(xn), with 1≤m<∞1≤m<∞, integer, FmFm being functions to be built later, in such a way that the velocity of convergence of such iterations increases more and more as mm goes to infinity; in other words: given an arbitrary integer m0≥1m0≥1, we will prove that the corresponding iteration formula of the family, xn+1=Fm0(xn)xn+1=Fm0(xn), has an order of convergence m0+1m0+1.The increment of the velocity of convergence of the sequence of the iterator family xn+1=Fm+1(xn)xn+1=Fm+1(xn) with respect to the previous one xn+1=Fm(xn)xn+1=Fm(xn) is attained at the expense of one derivative evaluation more.Besides, we introduce a new algorithm (Present Method Two), from now on PMII, that plays the role of seeker for an initial value to guarantee the local convergence of the PMI.Both of them can be combined as an algorithm of global convergence, included the case of singular roots, that does not depend on the chosen initial value, and that allows to find all the roots in a feasible interval in a general and complete way. These are, in my opinion, the main results of this work.