Andrei Dubin

Andrei Dubin

In the framework of the majorization technique, an improved condition is proposed for the semilocal convergence of the Newton method under the mild assumption that the derivative of the involved operator F(x) is continuous. Our starting point is the Argyros representation of the optimal upper bound for the distance between the adjacent members of the Newton sequence. The major novel element of our proposal is the optimally reconstructed 'first integral' approximation to the recurrence relation defining the scalar majorizing sequence. Compared to the previous results of Argyros, it enables one to obtain a weaker convergence condition that leads to a better bound on the location of the solution of the equation F(x)=0 and allows for a wider choice of initial guesses. In the simplest case of the Lipschitz continuous derivative operator, an explicit restriction is found which guarantees that the new convergence condition improves the famous Kantorovich condition.

Andrei Dubin

The extended modification of the Newton method is considered when the inverse of the derivative (of the operator F(x) in the equation F(x)=0) is replaced by an invertible bounded x-independent operator B. The continuity assumption is relaxed to the requirement that F(x) is continuously Frechet-differentiable. The Kantorovich majorization technique is adapted to formulate and prove the corresponding generalization of the Kantorovich theorem originally stated for the standard modified Newton method (MNM) when the derivative is Lipschitz continuous. In the MNM case, the generalized theorem is shown to extend the existing one due to Argyros. For a generic B and a Holder continuous derivative, the proposed theorem leads to a weaker condition of the semilocal convergence, larger uniqueness domain and finer error bounds compared to the previous results of Ahues and Argyros.