We present a new convergence analysis, for the secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center-Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence analysis leads to more precise error bounds and to a better information on the location of the solution than the corresponding ones in earlier studies. Numerical examples validating the theoretical results are also provided in this study. (C) 2015 International Association for Mathematics and Computers in Simulation (IMACS).