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

Abstract

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 xk + 1 = gu (xk) = xk + f (xk) u (xk) to solve a nonlinear equation f with the aid of a weight function u. The class currently contains an invert-and-average(gKia), an average-and-invert(gKai), and an invert-and-exponentiate(gKe) branch. Each branch has several members some of which embed second-order Newton's (gN), third-order Chebychev's (gC) or Halley's (gH) 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. © 2007 Elsevier B.V. All rights reserved.