We present a convergence analysis for a Damped Secant method with modified right-hand side vector in order to approximate a locally unique solution of a nonlinear equation in a Banach spaces setting. In the special case when the method is defined on Ri, our method provides computable error estimates based on the initial data. Such estimates were not given in relevant studies such as (Herceg et al., 1996; Krejić, 2002). Numerical examples further validating the theoretical results are also presented in this study.