High-order convergence of the k-fold pseudo-Newton's irrational method locating a simple real zero

By extending the classical Newton's irrational method defined by an iterationxn+1=xn-f(xn)f′(xn)2-f(xn)f″(xn),we present a high-order k-fold pseudo-Newton's irrational method for locating a simple zero of a nonlinear equation. Its order of convergence is proven to be at least k + 3 and the convergence behavior of the asymptotic error constant is investigated near the corresponding simple zero. A root-finding algorithm is described as well as the introduction on the convergence of the fixed-point iterative method. Various numerical examples have successfully demonstrated a good agreement with the theory presented here.