Preprint A434/2006
Kantorovich's Majorants Principle for Newton's Method

B. F. Svaiter | Ferreira, Orizon

**Keywords: **
Kantorovich Theorem | Newton method

We prove Kantorovich's theorem on Newton's method using a convergence analysis which makes clear, with respect to Newton's Method, the relationship of the majorant function and the non-linear operator under consideration. This approach enable us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results.