= Scale-aware stopping criterion for Newton iteration
{title2=$|x_{n+1}-x_n|\le C\epsilon\max(1,|x_n|)$}
Near a well-conditioned simple root, a numerically stable <Newton root-finding iteration> cannot usually resolve steps far below the local spacing of <floating-point numbers>. That spacing has relative scale comparable to <machine epsilon> $\epsilon$, motivating the displayed position-based criterion with a modest constant $C$. It combines an absolute scale near zero with a relative scale for large inputs. A residual-only test is changed by rescaling the equation, whereas the root and exact Newton step are unchanged. A tiny step alone is not a universal accuracy certificate: derivative errors, stagnation and cycles require separate consideration.
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