Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations

Necessary and sufficient conditions under which two- and three-point iterative methods have the order of convergence $p$ ($2\leqslant p\leqslant 8$) are formulated for the first time. These conditions can be effectively used to prove the convergence of iterative methods. In particular, the order of convergence of some known optimal methods is verified using the proposed sufficient convergence tests. The optimal set of parameters making it possible to increase the order of convergence is found. It is shown that the parameters of the known iterative methods with the optimal order of convergence have the same asymptotic behavior. The simplicity of choosing the parameters of the proposed methods is an advantage over the other known methods.