On the set of initial guesses for the secant method

Abstract

The secant method is the most used iterative method to solve an operator equation where the operator involved is nondifferentiable. A known problem that arises when applying this method is its accessibility. Then, we try to improve it by using the technique of decomposition of the operator involved, so that the operator is in turn the sum of two operators, one differentiable and the other continuous but not differentiable. For this, we use a family of Newton‐secant‐type iterative methods that arises from Newton's method and from a well‐known family of secant‐type iterative methods. We study the accessibility of the new methods in two different ways: From the convergence balls of the methods, obtained from a local study of the convergence, and from a dynamic study of the methods. Some examples related to chemistry are also presented to prove the theoretical results.