On the applicability of two Newton methods for solving equations in Banach space
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In this study we examine the applicability of Newton’s method and the modified Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space. We assume that the Newton-Kantorovich hypothesis for Newton’s method is violated, but the corresponding condition for the modified Newton method holds. Under these conditions there is no guarantee that Newton’s method starting from the same initial guess as the modified Newton’s method converges. Hence, it seems that we must always use the modified Newton method under these conditions. However, we provide a numerical example to demonstrate that in practice this may not be a good decision.
AMS Mathematics Subject Classification65B05 47H17 49D15
Key words and phrasesBanach space Newton-Kantorovich hypothesis Newton’s method modified Newton’s method
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