An improved convergence theorem for the Newton method under relaxed continuity assumptions
For researchers in numerical analysis, this provides an incremental improvement to convergence theory for Newton's method.
The paper proposes an improved convergence condition for Newton's method under continuous derivative, yielding weaker conditions and better solution bounds compared to prior work by Argyros. In the Lipschitz case, the new condition improves the Kantorovich condition.
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.