On computational order of convergence of some multi-precision solvers of nonlinear systems of equations

Report d'un treball de recerca on es presenten noves tècniques de càlcul de l'ordre de convergència amb una aritmètica adaptativa.In this paper the local order of convergence used in iterative methods to solve nonlinear systems of equations is revisited, where shorter alternative analytic proofs of the order based on developments of multilineal functions are shown. Most important, an adaptive multi-precision arithmetics is used hereof, where in each step the length of the mantissa is defined independently of the knowledge of the root. Furthermore, generalizations of the one dimensional case to m-dimensions of three approximations of computational order of convergence are defined. Examples illustrating the previous results are given.Preprin

Multi-step derivative-free preconditioned Newton method for solving systems of nonlinear equations

Preconditioning of systems of nonlinear equations modifies the associated Jacobian and provides rapid convergence. The preconditioners are introduced in a way that they do not affect the convergence order of parent iterative method. The multi-step derivative-free iterative method consists of a base method and multi-step part. In the base method, the Jacobian of the system of nonlinear equation is approximated by finite difference operator and preconditioners add an extra term to modify it. The inversion of modified finite difference operator is avoided by computing LU factors. Once we have LU factors, we repeatedly use them to solve lower and upper triangular systems in the multi-step part to enhance the convergence order. The convergence order of m-step Newton iterative method is m + 1. The claimed convergence orders are verified by computing the computational order of convergence and numerical simulations clearly show that the good selection of preconditioning provides numerical stability, accuracy and rapid convergence.Peer ReviewedPostprint (author's final draft

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung and Traub's conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basin of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basin of attraction.Comment: arXiv admin note: substantial text overlap with arXiv:1508.0174