Secant-type methods and nondiscrete induction
For solving the equation \(f(x)= 0\), where \(f\) is a Fréchet-differentiable operator defined on a subset \(D\) of a Banach space \(X\) with values in a Banach space \(Y\), the authors consider secant-type methods based on the concept of nondiscrete mathematical induction. Using Lipschitz and center-Lipschitz conditions on the Fréchet-differential of the involved operator instead of the only Lipschitz condition, new sufficient convergence conditions are derived. The results obtained in this paper extend the applicability of the methods under discussion by providing a new semilocal convergence analysis with the advantages: (1) weaker sufficient convergence conditions; (2) tighter error bounds on the distances involved. The presented results of numerical experiments confirm the theoretical considerations.
scientific article; zbMATH DE number 6110505
scientific article