The construction of rational iterative methods for solving nonlinear equations

In this paper, based on the idea of constructing iterative methods by Sharma [J.R. Sharma, A family of third-order methods to solve nonlinear equations, Appl. Math. Comput. 184 (2007) 210–215], Jiang and Han [Dongdong Jiang, Danfu Han, Some one-parameter families of third-order methods for solving nonlinear equations, Appl. Math. Comput. 195 (2008) 392–396], we propose some rational iterative methods by linearizing the quadratic curve equation. By analysis of asymptotic errors, we prove the new families are all cubically convergent. As an example, we present a new concrete iterative family with better convergent performance than classic third-order methods through the adjustment of parameter λnλn automatically.