On a theorem of S. Smale about Newton's method for analytic mappings
Applied Mathematics Letters
The theory of Smale's point estimation and its applications
Proceedings of the international meeting on Linear/nonlinear iterative methods and verification of solution
Convergence of Newton's method and inverse function theorem in Banach space
Mathematics of Computation
Finding zeros of analytic functions: &agr; theory for secant type methods
Journal of Complexity
On the Newton-Kantorovich hypothesis for solving equations
Journal of Computational and Applied Mathematics
Kantorovich-type convergence criterion for inexact Newton methods
Applied Numerical Mathematics
Extending the Newton-Kantorovich hypothesis for solving equations
Journal of Computational and Applied Mathematics
Improved local convergence of Newton's method under weak majorant condition
Journal of Computational and Applied Mathematics
Weaker conditions for the convergence of Newton's method
Journal of Complexity
| Hi-index | 7.29 |
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 @a-theory by introducing the notion of the center @c"0-condition. In particular, in the semilocal convergence case we show that if the center @c"0-condition is smaller than the @c-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.