= Residual of a root equation
{title2=$r=f(\widehat x)$}
The <residual of a root equation> $f(x)=0$ at a proposed solution $\widehat x$ is the value $f(\widehat x)$. It measures violation of the equation, while the forward error $|\widehat x-x_*|$ 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.
Back to article page
