= Exact Newton error identity for a quadratic
{c}
{title2=$e_{k+1}=e_k^2/[2(x_k-a)]$}
For the <quadratic polynomial> $f(x)=(x-a)^2-b^2$ with $b>0$, consider the <Newton root-finding iteration> towards the <polynomial root> $x_*=a+b$. If $x_k\ne a$, its signed error $e_k=x_k-x_*$ satisfies
$$
e_{k+1}=\frac{e_k^2}{2(x_k-a)}.
$$
Indeed the iteration for $u_k=x_k-a$ is $u_{k+1}=(u_k+b^2/u_k)/2$, and subtracting $b$ yields $(u_k-b)^2/(2u_k)$. This exact identity explains <quadratic convergence> near the simple <polynomial root>. If an initial error is asymptotic to $C\varepsilon^m$ and $x_k-a\to b\ne0$ as $\varepsilon\to0$, one step changes its leading power to $2m$ and its coefficient to $C^2/(2b)$.
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