Expanding the applicability of Newton's method using Smale's α-theory

We present a tighter convergence analysis than earlier studies such as in Cianciaruso (2007), Guo (2007), Shen and Li (2010), Smale (1986, 1987), Wang and Zhao (1995), Wang (1999), Wang and Han (1990) of Newton's method using Smale's α-theory by introducing the notion of the center γ0-condition. In particular, in the semilocal convergence case we show that if the center γ0-condition is smaller than the γ-condition, then the new majorizing sequence is tighter than the old majorizing sequence. The new convergence criteria are weaker than the older convergence criteria. Furthermore, in the local convergence case, we obtain a larger radius of convergence and tighter error estimates on the distances involved. These improvements are obtained under the same computational cost. Numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations.