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).

Next: Contraction Mapping Theorem     Previous: Solving f(u)=0 by Newton’s Method

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