Unified error analysis for Newton-type methods

Summary

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.