Steffensen’s method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen’s method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen’s method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.
Argyros, I.K.: A new convergence theorem for Steffensen’s method on Banach spaces and applications. Southwest J. Pure Appl. Math. 1, 23–29 (1997)zbMATHGoogle Scholar
Ezquerro, J.A., Grau-Sánchez, M., Hernández, M.A.: Solving non-differentiable equations by a new one-point iterative method with memory. J. Complexity 28, 48–58 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
Ezquerro, J.A., Grau-Sánchez, M., Hernández, M.A., Noguera, M.: Semilocal convergence of secant-like methots for differentiable and non-differentiable operator equations. J. Math. Anal. Appl. 398, 100–112 (2013)CrossRefzbMATHMathSciNetGoogle Scholar