# From Residual to Root Error

When computing a root of an equation $f(u)$ for a given differentiable function $f:R\rightarrow R$ by time stepping or Newton’s method,  we compute a sequence of approximate roots  $u^n$, $n=1, 2, ...$, with the property that the residual $R(u^n)\equiv \vert f(u^n)\vert$ becomes arbitarily small  as $n$ grows without bound.

The corresponding root error can be estimated as follows: Assume $\bar u$ satisfies $f(\bar u)=0$ and that $\frac{1}{\vert f^\prime (u)\vert}\leq S$ for $u\in [\bar u - \delta ,\bar u + \delta ]$ where $S$  and $\delta$ are positive numbers with $S$ referred to as a stability factor.  By the Mean Value Theorem we  then have for $u\in [\bar u - \delta ,\bar u + \delta ]$:

• $f(u) = f(u) - f(\bar u)= f^\prime (\hat u)(u -\bar u)$

for some $\hat u\in [\bar u - \delta ,\bar u + \delta ]$. We conclude that if $u^n\in [\bar u - \delta ,\bar u + \delta ]$, then

• $\vert u^n -\bar u\vert\leq S \vert f(u)\vert = S R(u^n)$

which connects the root error $\vert u^n-\bar u\vert$ to the residual $R(u^n)$ through the multiplicative stability factor $S$.

This result directly generalizes to a mapping $f:R^N\rightarrow R^N$ with $S$ the norm of the inverse of the Jacobian $f^\prime (u)$.