Newton-type methods of high order and domains of semilocal and global convergence

We present the geometric construction of some classical iterative methods that have global convergence and “infinite” speed of convergence when they are applied to solve certain nonlinear equations f(t)=0f(t)=0. In particular, for nonlinear equations with the degree of logarithmic convexity of f′f′, Lf′(t)=f′(t)f‴(t)/f″(t)2Lf′(t)=f′(t)f‴(t)/f″(t)2, is constant, a family of Newton-type iterative methods of high orders of convergence is constructed. We see that this family of iterations includes the classical iterative methods. The convergence of the family is studied in the real line and the complex plane, and domains of semilocal and global convergence are located.