Some one-parameter families of third-order methods for solving nonlinear equations

In this paper, two one-parameter families of irrational and rational iterative methods for solving nonlinear equations are constructed. The construction of the iterative process is based on one-point approximation by the quadratic equation x2+ay+bx+cxy+d=0x2+ay+bx+cxy+d=0. Euler, Chebyshev, Halley, Super-Halley methods can be seen as the special cases of the family. And by analysis of asymptotic errors, we prove the two families are all cubically convergent. Furthermore, the rational iterative family can be extended to Banach spaces easily.