On the semi-local convergence of a sixth order method in Banach space

High convergence order methods are important in computational mathematics, since they generate sequences converging to a solution of a non-linear equation. The derivation of the order requires Taylor series expansions and the existence of derivatives not appearing on the method. Therefore, these results cannot assure the convergence of the method in those cases when such high order derivatives do not exist. But, the method may converge. In this article, a process is introduced by which the semi-local convergence analysis of a sixth order method is obtained using only information from the operators on the method. Numerical examples are included to complement the theory

A new family of twentieth order convergent methods with applications to nonlinear systems in engineering

A new family of iterative methods with a strong converging order of twenty to solve nonlinear equations and systems is presented in this study. A simple strategy of blending some existing methods is used to develop the proposed family. The theoretical order of convergence is derived by employing Taylor’s series. The performance of the iterative methods in the proposed family is examined by applying the methods on real-world engineering problems. A nonlinear equation modeled by NASA for launching “Wind” satellite and some other complex applied systems, such as combustion problem, tank-reactor problem, kinematic synthesis mechanism, neurophysiology application and one boundary-value problem, have been solved to check the performance of the proposed family against other methods under similar test conditions. All the numerical results show that the proposed family converges very fast in complex and difficult problems as compared to other well-known methods. The methods in the proposed family have an efficiency improvement of 11.99% over the classical Newton method for scalar nonlinear equations