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 + 1 = 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 Newtonʹs ( g N ) , third-order Chebychevʹs ( g C ) or Halleyʹs ( 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.