Editing Approximation error (section)

==Formal definition==
Given some true or exact value ''v'', we formally state that an approximation ''v''<sub>approx</sub> estimates or represents ''v'' where the magnitude of the '''absolute error''' is bounded by a positive value ''ε'' (i.e., ''ε''>0), if the following inequality holds: <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>

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

where the vertical bars, | |, unambiguously denote the [[absolute value]] of the difference between the true value ''v'' and its approximation ''v''<sub>approx</sub>. This mathematical operation signifies the magnitude of the error, irrespective of whether the approximation is an overestimate or an underestimate.

Similarly, we state that ''v''<sub>approx</sub> approximates the value ''v'' where the magnitude of the '''relative error''' is bounded by a positive value ''η'' (i.e., ''η''>0), provided ''v'' is not zero (''v'' ≠ 0), if the subsequent inequality is satisfied:<blockquote><math>|v-v_\text{approx}| \leq \eta\cdot |v|</math>.</blockquote>This definition ensures that ''η'' acts as an upper bound on the ratio of the absolute error to the magnitude of the true value. If ''v'' ≠ 0, then the actual '''relative error''', often also denoted by ''η'' in context (representing the calculated value rather than a bound), is precisely calculated as:
:<math> \eta = \frac{|v-v_\text{approx}|}{|v|}
    = \left| \frac{v-v_\text{approx}}{v} \right|
    = \left| 1 - \frac{v_\text{approx}}{v} \right|
</math>.
Note that the first term in the equation above implicitly defines `ε` as `|v-v_approx|` if `η` is `ε/|v|`.

The '''percent error''', often denoted as ''δ'', is a common and intuitive way of expressing the relative error, effectively scaling the relative error value to a percentage for easier interpretation and comparison across different contexts: <ref name=":0" />

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

An '''error bound''' rigorously defines an established upper limit on either the relative or the absolute magnitude of an approximation error. Such a bound thereby provides a formal guarantee on the maximum possible deviation of the approximation from the true value, which is critical in applications requiring known levels of precision.<ref>{{Cite web |title=Approximation and Error Bounds |url=http://www.math.wpi.edu/Course_Materials/MA1021B98/approx/node1.html#:~:text=Thus%20we%20introduce%20the%20term,bound%20the%20better%20the%20approximation. |access-date=2023-06-11 |website=math.wpi.edu}}</ref>