 Thresholded stopping of a capped-tolerance multiplier search

Thresholded stopping of a capped-tolerance multiplier search ends a numerical allocation when a feasible physical budget is sufficiently well used. With the decreasing budget function $Q(\lambda )$ in tolerance-budget residual bound for multiplier bisection, an exact upper bracket endpoint $u$ satisfies $Q(u)\le E$. A residual requirement is $E-Q(u)\le \zeta$, where $\zeta >0$ has the same physical units as the permitted deterministic bias $E$. A small change in $u$ alone does not establish that condition.

Numerical evaluation also needs a bound. Suppose the reported value $\overline{Q}$ has validated absolute error $|Q-\overline{Q}|\le {ϵ}_Q$. Both feasibility and the residual requirement are certified ifEquivalently, the acceptable reported value lies inThis window is empty if $\zeta <2{ϵ}_Q$. Iterating the multiplier more accurately cannot by itself overcome an evaluation-error bound larger than the desired physical certificate.

For an original numerical certificate, let $E=10\mathrm{nV}$, $\zeta =0.10\mathrm{nV}$ and ${ϵ}_Q=0.02\mathrm{nV}$. Values $9.92\le \overline{Q}\le 9.98\mathrm{nV}$ are certifiable. A reported $9.95\mathrm{nV}$ places the actual budget in $[9.93,9.97]\mathrm{nV}$, so it is feasible and leaves at most $0.07\mathrm{nV}$ unused. A reported $9.99\mathrm{nV}$ is closer to ten but could represent $10.01\mathrm{nV}$ and therefore does not certify feasibility.

The evaluation allowance must itself be justified, for example by validated arithmetic or an explicit forward-error analysis. Uncertain physical coefficients also need their stipulated worst-case bounds before the numerical sum can certify the physical error. NIST's pulse-radiometer discussion motivates the offset and leakage terms; it does not provide the hypothetical numerical tolerances above.^[ref]