 Exact Newton error identity for a quadratic

For the quadratic polynomial $f(x)=(x-a{)}^2-b^2$ with $b>0$, consider the Newton root-finding iteration towards the polynomial root $x_{\ast }=a+b$. If $x_k\ne a$, its signed error $e_k=x_k-x_{\ast }$ satisfiesIndeed 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 $C{\epsilon }^m$ and $x_k-a\to b\ne 0$ as $\epsilon \to 0$, one step changes its leading power to $2m$ and its coefficient to $C^2/(2b)$.