The theory of Smale's point estimation and its applications

Abstract

The main result of this paper is that we exact Smale's point estimation theory, i.e., without assuming γ k = ‖P′(z) −1P (k)(z) k! ‖ (k ⩾ 2) being bounded by γ, the point estimation convergence theorem of the Ne method is set up by making use of the majorizing method. The proof of the theorem is simple and precise, while the required point estimation conditions are weaker than all those of known point estimation convergence theorems. Another result of this paper is an application of the above new theory to the Durand-Kerner method. We compare the point estimation conditions for the Durand-Kerner method with other known point estimation conditions. Numerical results show that our results have evident advantages.