On the Derivation of Higher Order Root-Finding Methods

High order root-finding algorithms are constructed based on some canonical conditions and a generalized Taylor series. The convergence order is automatically determined using these canonical conditions. The proposed approaches resulted in deriving methods of any desired order including the Newton, Halley, and Ostrowski iterations. It is also shown that, when zeros are simple, higher order methods may be obtained by applying lower order methods such as Newton Iteration to new functions which have same zeros as the original function. These functions are constructed so that the first few derivatives beyond the first vanish. Several examples are given for constructing methods of higher order for computing the zeros of the entire function sin(z).