Content deleted Content added
←Replaced page with 'HEY~~~~' |
←Undid revision 107365941 by 24.147.196.238 (talk) |
||
Line 1:
In the [[mathematics|mathematical]] sub-field of [[numerical analysis]] the '''approximation error''' in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because
#the [[measurement]] of the [[data]] is not precise (due to the instruments), or
#approximations are used instead of the real data (e.g., 3.14 instead of π).
One commonly distinguishes between the '''relative error''' and the '''absolute error'''.
The [[numerical stability]] of an [[algorithm]] in numerical analysis indicates how the error is propagated by the algorithm.
==Definitions==
Given some value ''a'' and an approximation ''b'' of ''a'', the '''absolute error''' is
:<math>\epsilon = |a - b|\,</math>
the '''relative error''' is
:<math>\eta = \frac{|a - b|}{|a|} </math>
if <math>|a|\ne 0</math> and the '''percent error''' is
:<math>\delta = \frac{|a-b|}{|a|}\times{}100\%</math>
where the vertical bars denote the [[absolute value]], ''a'' represents the true value, and ''b'' represents the approximation to ''a''.
==Linear Algebra Definition==
Where <math>\left\| a\right\|_p</math> is the [[Norm_%28mathematics%29#p-norm|p-norm]] of vertex ''a'', and <math>\hat{x}</math> is an approximation of vertex ''x'' where <math>x\in\mathbb{R}^n</math>.
*<math>\epsilon=\left\|\hat{x}-x\right\|_2</math>
*<math>\eta=\frac{\left\|\hat{x}-x\right\|_2}{\left\| x\right\|_2}</math>
*<math>\left\lceil -\operatorname{log}_{10}\left(\frac{\left\|\hat{x}-x\right\|_\infty}{\left\| x\right\|_\infty}\right)\right\rceil</math> is the number of [[Significant figures]] of the largest magnitude entry of x.<ref name="GOLUB_MAT_COMP2.2.3">{{cite book|last=Golub|first=Gene|authorlink=Gene_H._Golub|coauthors=Charles F. Van Loan|title=Matrix Computations - Third Edition|publisher=The Johns Hopkins University Press|date=1996|___location=Baltimore|pages=53|id=ISBN 0-8018-5413-X}}
</ref>
==Propagation of errors==
When calculating using approximate values it is important to be able to calculate the errors involved.
For measured values '''X & Y''' with absolute errors '''<math>\epsilon x\,</math> & <math>\epsilon y\,</math>''' and relative errors '''<math>\eta x\,</math> & <math>\eta y\,</math>''' respectively, we can use:
* For <math>Z = X + Y\,</math>:
:<math>\epsilon z = \epsilon x + \epsilon y\,</math>
* For <math>Z = X - Y\,</math>::
:<math>\epsilon z = |\epsilon x - \epsilon y|\,</math>
* For <math>Z = X \times Y\,</math>:
:<math>\eta z = \eta x + \eta y\,</math>
* For <math>Z = X / Y\,</math>:
:<math>\eta z = |\eta x - \eta y|\,</math>
==References==
<!--<nowiki>
See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref> and </ref> tags, and the template below.
</nowiki>-->
{{FootnotesSmall|resize=100%}}
[[Category:Numerical analysis]]
[[cs:Chyba aproximace]]
[[de:Rundungsfehler]]
[[es:Error de aproximación]]
[[fr:Erreur d'approximation]]
[[nl:Benaderingsfout]]
[[pl:Błąd przybliżenia]]
[[sv:Absolutfel]]
[[zh:逼近误差]]
| |||