When computing a root of an equation for a given differentiable function by time stepping or Newton’s method, we compute a sequence of approximate roots , , with the property that the residual becomes arbitarily small as grows without bound.
The corresponding root error can be estimated as follows: Assume satisfies and that for where and are positive numbers with referred to as a stability factor. By the Mean Value Theorem we then have for :
for some . We conclude that if , then
which connects the root error to the residual through the multiplicative stability factor .
This result directly generalizes to a mapping with the norm of the inverse of the Jacobian .