= Local quadratic convergence of Newton iteration
{c}
{title2=$|e_{r+1}|\leq\frac{M}{2m}|e_r|^2$}
Let $F$ be twice <continuously differentiable> on an interval around a <zero of a function> $t_*$, with $|F'(t)|\geq m>0$ and $|F''(t)|\leq M$ there. Choose a closed interval $[t_*-\rho,t_*+\rho]$ inside that interval with $M\rho/(2m)<1$. Starting inside it makes the <Newton root-finding iteration> well defined, keeps it inside, and gives the displayed <quadratic convergence> bound for $e_r=t_r-t_*$. The nonzero <derivative> and sufficiently close start are essential hypotheses. The scalar estimate follows from the <Taylor remainder> argument in MIT's root-finding notes.https://ocw.mit.edu/courses/18-330-introduction-to-numerical-analysis-spring-2012/5b325bfa56a599794c7196de926844b0_MIT18_330S12_Chapter4.pdf[Laurent Demanet, MIT 18.330, Chapter 4, pp. 2–3, 2012]{ref}
For a short proof outline, apply <Taylor's theorem> between $t_r$ and $t_*$; subtract the defining <tangent line> equation. The linear error cancels, leaving $e_{r+1}=F''(\xi_r)e_r^2/(2F'(t_r))$. The displayed bound shrinks the error and preserves the chosen interval. If $F''(t_*)\ne0$, the error ratio tends to $|F''(t_*)/(2F'(t_*))|$; if it vanishes, convergence can be faster.
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