Convergence of Traub's iteration under $\omega$ continuity condition in Banach spaces

We provide the local and semi-local convergence analysis of the most celebrated cubically convergent Traub's iterative method (TM) for obtaining solutions of Banach space valued nonlinear operator equations. The significance of our work is that the convergence study only needs the $\omega$ continuity condition on the first-order Fréchet derivative and avoids the use of higher order derivatives, which do not occur in this scheme. Also, the proposed local analysis extends the domain of convergence and applicability of this scheme. Using basins of attraction technique the complex dynamics of the scheme are also explored when it is applied on various complex polynomials. Finally, convergence radii for benchmark numerical problems are computed applying our analytical results. From these numerical tests, it is confirmed that the proposed analysis provides a larger convergence domain in comparison with the earlier work.