Residual of a root equation ($r=f(\hat{x})$)

The residual of a root equation $f(x)=0$ at a proposed solution $\hat{x}$ is the value $f(\hat{x})$. It measures violation of the equation, while the forward error $|\hat{x}-x_{\ast }|$ measures distance from a chosen exact root. Rescaling $f$ rescales the residual of a root equation without moving its roots, so a small residual needs a scale or a derivative bound before it certifies a small forward error.