 Local quadratic convergence of Newton iteration

Let $F$ be twice continuously differentiable on an interval around a zero of a function $t_{\ast }$, with $|F^{\prime }(t)|\ge m>0$ and $|F^{\prime \prime }(t)|\le M$ there. Choose a closed interval $[t_{\ast }-\rho ,t_{\ast }+\rho ]$ 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 $e_r=t_r-t_{\ast }$. 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.^[ref]

For a short proof outline, apply Taylor's theorem between $t_r$ and $t_{\ast }$; subtract the defining tangent line equation. The linear error cancels, leaving $e_{r+1}=F^{\prime \prime }({\xi }_r)e_r^2/(2F^{\prime }(t_r))$. The displayed bound shrinks the error and preserves the chosen interval. If $F^{\prime \prime }(t_{\ast })\ne 0$, the error ratio tends to $|F^{\prime \prime }(t_{\ast })/(2F^{\prime }(t_{\ast }))|$; if it vanishes, convergence can be faster.