Newton's method under mild differentiability conditions

Abstract

We study Newton's method for determining the solution of f(x) = 0 when f(x) is required only to be continuous and piecewise continuously differentiable in some sphere about the initial iterate, x^(0). First an existence, uniqueness and convergence theorem is obtained employing the modulus of continuity of the first derivative, f_x(x). Under the more explicit assumption of H6lder continuity several other such results are obtained, some of which extend results of Kantorovich and Akilov [1] and Ostrowski [5]. Of course, when Newton's method converges, it is now of order (1 + α), where a is the Hö1der exponent. Other results on Newton's method without second derivatives are given by Goldstein [2], Schroeder [3], Rheinboldt [6], and Antosiewicz [7], to mention a few. It seems clear that the error analysis for Newton's method given by Lancaster [4] can be extended to the present case.

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