New convergence conditions for Secant-type higher-order methods in Banach spaces

Derivative-free iterative methods are valuable for finding numerical solutions in cases where explicit derivative information is unavailable or the computation of derivatives is prohibitively expensive. It is essential to investigate the convergence properties of such methods to ensure their effectiveness. Previous studies made assumptions about the existence of high-order derivatives, despite the fact that methods are not relying on derivatives. These assumptions restrict their applicability. To address this limitation, our study extends the analysis by providing error estimates. As a result, the methods’ applicability appears to encompass a wider range of problems. The findings presented in this paper provide a non Taylor series convergence analysis of derivative-free numerical algorithms. Numerical examples complete this paper.