Majorizing Sequences for Newton-Like Method and Their Limit Points

Abstract

A plethora of problems from diverse disciplines of Mathematics, Mathematical Biology, Chemistry, Medicine, physics and Engineering to mention a few reduce to solving nonlinear equations or systems of equations usually in the finite dimensional Euclidean or more general spaces. The solutions of such equations are numbers or vectors of functions and can be found in closed form only in special cases. That is why researchers and practitioners develop mostly iterative methods which generate sequences approximating the solutions. The least number of iterations to be carried out in order to obtain a pre-decided error tolerance on the distances between consecutive iterates as well as the choice of initial points ensuring the convergence of the methods is very important. These two objectives can be achieved by introducing real majorizing sequences which control the behaviour of the iterates. Moreover, the closed form of the limits of the real sequences determine the radius of the ball that contains the initial points. In this paper we contribute by introducing more precise majorizing sequences and limit points.