NEW AND GENERALIZED CONVERGENCE CONDITIONS FOR THE NEWTON-KANTOROVICH METHOD

Abstract. We present new semilocal convergence theorems for New-ton methods in a Banach space. Using earlier general conditions we find more precise error estimates on the distances involved using the majorant principle. Moreover we provide a better information on the location of the solution. In the special case of Newton’s method un-der Lipschitz conditions we show that the famous Newton–Kantorovich hypothesis having gone unchallenged for a long time can be weakened under the same hypotheses/computational cost. 1