, Volume 33, Issue 4, pp 391-396

Unified error analysis for Newton-type methods

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Under proper hypotheses, Rheinboldt has shown that Newtonrelated iterates \(x_{n + 1} = x_n - {\cal D}\left( {x_n } \right)^{ - 1} Fx_n \) , where some \({\cal D}\left( x \right)\) approximates the Fréchet derivative of an operatorF, converge to a rootx - ofF. Under these hypotheses, this paper establishes error bounds $$\left\| {x^* - x_n } \right\|B_n \left\| {x_n - x_{n - 1} } \right\|C_n \left\| {x_1 - x_0 } \right\|, \left\| {x_n - \xi _n } \right\|s_n ,$$ whereB n ,C n ,s n are constants, and where ξ n ; are perturbed iterates which take into account rounding errors occuring during actual computations.