# From Residual to Root Error

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 .

Next: Contraction Mapping Theorem Previous: Solving by Newton’s Method

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