We present a new semilocal convergence analysis for the Secant and the Moser method in order to approximate a solution of an equation in a Banach space setting. Using the method of recurrent relations and weaker sufficient convergence criteria than in earlier studies such as Amat et al. (2014), Hernández and Rubio (2007), Hernández and Rubio (1999) and Hernández and Rubio (2002) we increase the convergence domain of these methods. The advantages are also obtained under less computational cost than in Amat et al. (2014), Hernández and Rubio (2007), Hernández and Rubio (1999) and Hernández and Rubio (2002). Numerical examples where the older convergence criteria are not satisfied but the new convergence criteria are satisfied are also provided in this study.
