The Secant method for nondifferentiable operators

Convergence Conditions for the Secant Method

We provide new sufficient convergence conditions for the convergence of the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, Lipschitz–type and center–Lipschitz–type instead of just Lipschitz–type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise ...

Associativity of the Secant Method

1. INTRODUCTION. This paper has its genesis in a problem the author first came upon while in college. Although the areas covered here are well travelled and nothing here is guaranteed original, it covers a pleasant nexus of many mathematical strands. Furthermore, we show the value of good notation and of reading an old master for solving a problem. Consider iterates of the function m (x) = 1 I ...

A secant method for nonsmooth optimization

The notion of a secant for locally Lipschitz continuous functions is introduced and a new algorithm to locally minimize nonsmooth, nonconvex functions based on secants is developed. We demonstrate that the secants can be used to design an algorithm to find descent directions of locally Lipschitz continuous functions. This algorithm is applied to design a minimization method, called a secant met...

A modified secant method for unconstrained minimization

Newton - Secant method for solving operator equations ∗

where F is a Fréchet-differentiable operator defined on an open subset D of a Banach space X with values in a Banach space Y . Finding roots of Eq.(1) is a classical problem arising in many areas of applied mathematics and engineering. In this study we are concerned with the problem of approximating a locally unique solution α of Eq.(1). Some of the well known methods for this purpose are the f...