Approximation error: Difference between revisions

Revision as of 16:43, 18 December 2023

Revision as of 13:12, 28 January 2024

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==Formal definition==

:<math>\epsilonGiven =some |v-v_\text{approx}|\value ''v'', we say that ''v''<sub>approx</mathsub> approximates ''v'' with '''absolute error''' ''ε''>0 if <ref>{{Cite web |last=Weisstein |first=Eric W. |title=Absolute Error |url=https://mathworld.wolfram.com/ |access-date=2023-06-11 |website=mathworld.wolfram.com |language=en}}</ref><ref name=":0">{{Cite web |title=Absolute and Relative Error {{!}} Calculus II |url=https://courses.lumenlearning.com/calculus2/chapter/absolute-and-relative-error/ |access-date=2023-06-11 |website=courses.lumenlearning.com}}</ref>

Given some value ''v'' and its approximation ''v''_approx, the '''absolute error''' is

:<math>|v-v_\text{approx}| \leq \varepsilon</math>

:<math>\epsilon = |v-v_\text{approx}|\ ,</math> <ref>{{Cite web |last=Weisstein |first=Eric W. |title=Absolute Error |url=https://mathworld.wolfram.com/ |access-date=2023-06-11 |website=mathworld.wolfram.com |language=en}}</ref><ref name=":0">{{Cite web |title=Absolute and Relative Error {{!}} Calculus II |url=https://courses.lumenlearning.com/calculus2/chapter/absolute-and-relative-error/ |access-date=2023-06-11 |website=courses.lumenlearning.com}}</ref>

where the vertical bars denote the [[absolute value]].

If <math>v \ne 0,</math> the '''relative error''' is

We say that ''v''_approx approximates ''v'' with '''relative error''' ''η''>0 if<blockquote><math>|v-v_\text{approx}| \leq \eta\cdot v</math>.</blockquote>If ''v'' ≠ 0, then

:<math> \eta = \frac{\epsilon}{|v|}

= \left| \frac{v-v_\text{approx}}{v} \right|

= \left| 1 - \frac{v_\text{approx}}{v} \right|,

</math>.

and theThe '''percent error''' (an expression of the relative error) is <ref name=":0" />

:<math>\delta = 100\%\times\eta = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|.</math>