EXPANDING THE APPLICABILITY OF SECANT METHOD WITH APPLICATIONS

Abstract

Abstract. We present new sufficient convergence criteria for the con-vergence of the secant-method to a locally unique solution of a nonlinearequation in a Banach space. Our idea uses Lipschitz and center–Lipschitzinstead of just Lipschitz conditions in the convergence analysis. The newconvergence criteria can always be weaker than the corresponding onesin earlier studies. Numerical examples are also provided in this study tosolve equations in cases not possible before. 1. IntroductionIn this study we are concerned with the problem of approximating a locallyunique solution x ⋆ of equation(1.1) F(x) = 0,where F is a Fr´echet–differentiable operator defined on a convex subset D of aBanach space X with values in a Banach space Y.A vast number of problems from applied science including engineering canbe solved by means of finding the solutions equations in a form like (1.1) us-ing mathematical modelling [7,11,16,19]. For example, dynamic systems aremathematically modeled by difference or differential equations, and their so-lutions usually represent the states of the systems. Except in special cases,the solutions of these equations cannot be found in closed form. This is themain reason why the most commonly used solution methods are iterative. It-eration methods are also applied for solving optimization problems. In suchcases, the iteration sequences converge to an optimal solution of the problemat hand. Since all of these methods have the same recursive structure, they canbe introduced and discussed in a general framework. The convergence analy-sis of iterative methods is usually divided into two categories: semilocal andlocal convergence analysis. In the semilocal convergence analysis one derivesconvergence criteria from the information around an initial point whereas in