A class of iterative methods with third-order convergence to solve nonlinear equations

Algebraic and differential equations generally co-build mathematical models. Either lack or intractability of their analytical solution often forces workers to resort to an iterative method and face the likely challenges of slow convergence, non-convergence or even divergence. This manuscript presents a novel class of third-order iterative techniques in the form Of x(k+l) = g(u) (x(k)) = x(k) + f(x(k))u(x(k)) to solve a nonlinear equation f with the aid of a weight function u. The class currently contains an invert-and-average (g(Kia)), an average-and-invert (g(Kai)), and an invert-and-exponentiate (g(Ke)) branch. Each branch has several members some of which embed second-order Newton's (g(N)), third-order Chebychev's (g(C)) or Halley's (g(H)) solvers. Class members surpassed stand-alone applications of these three well-known methods. Other methods are also permitted as auxiliaries provided they are at least of second order. Asymptotic convergence constants are calculated. Assignment of class parameters to non-members carries them to a common basis for comparison. This research also generated a one-step "solver" that is usable for post-priori analysis, trouble shooting, and comparison. (C) 2007 Elsevier B.V. All rights reserved.