On an improved local convergence analysis for the Secant method
The authors are concerned with the problem of approximating the locally unique solution of a nonlinear operator equation in Banach spaces. The operator F involving in the operator equation is supposed Fréchet-differentiable on an open convex subset of the given Banach space \(X\). The classical secant method has been generalized for equations in Banach spaces and there has been an extensive literature on the local and semilocal convergence of the secant method under various Lipschitz-type conditions. In the case of the local convergence of the secant method, the idea is to find a ball (the convergence ball), as large as possible, such that the convergence of the sequence of iterations to the solution of the equation can be achieved for initial guesses chosen from the convergence ball. Under a special Hölder continuous-type condition for the divided differences, the radius of convergence given by \textit{H. Ren} and \textit{Q. Wu} [J. Comput. Appl. Math. 194, No.~2, 284--293 (2006; Zbl 1101.65057)], is enlarged and the local convergence of the secant method is proved. One numerical example is performed to illustrate the performance of the obtained result.
scientific article; zbMATH DE number 5615009
scientific article