Could someone please explain to me why r should be equal to min(R,1/(2CL)) and maybe expand a little bit on line (*) and the following?:
2026-03-28 03:02:20.1774666940
Non-linear systems: proof of quadratic convergence for Newton's method
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the (*) is due to using the inequality above (7.5) now
$\|J^{-1}_F(x^*)\|\|J_F(x^{(0)})-J_F(x^{*})\|\leq CL\|x^{(0)}-x^*\|$, but $x^{(0)}\in B(x^*;r)$
So $\|x^{(0)}-x^*\|\leq r\Rightarrow CL\|x^{(0)}-x^*\|\leq CLr\leq \frac{1}{2}$
The choice of $r=min(R,1/2CL)$, is purely constructive, since we can choose $x^{(0)}$ close as we like to $x^*$ ( as long as $x^{(0)}\in B(x^*;R)$ ) to aid our proof.