Error bound

Error limits, also called error limits, are used in error calculation , in measurement technology and in numerics . An error limit is specified with the Greek letter (epsilon) and defines an agreed or guaranteed, permitted extreme deviation from a target value. An error limit can be equated with a tolerance value. ${\ displaystyle \ epsilon}$

definition

Let be an exact value (nominal value) and an approximation of the exact value such that : ${\ displaystyle x}$ ${\ displaystyle {\ tilde {x}}}$ ${\ displaystyle {\ tilde {x}} \ approx x}$

{\ displaystyle \ Delta _ {x} = {\ tilde {x}} - x}

means absolute mistake .

{\ displaystyle \ delta _ {x} = {\ frac {\ Delta _ {x}} {x}}}

in the case of relative errors .

{\ displaystyle x \ neq 0}

If is, it is called absolute error bound. ${\ displaystyle | \ Delta _ {x} | \ leq \ epsilon}$ ${\ displaystyle \ epsilon}$

If it holds, then it is called relative error bound. ${\ displaystyle {\ frac {\ epsilon} {\ mid x \ mid}} \ leq \ rho}$ ${\ displaystyle \ rho}$

Remarks

In general, the true value is not known, only the approximate value . B. is obtained in measurement technology by a measurement .

The relative error bound is dimensionless, i.e. H. it can be assigned to unit 1 and is often expressed as a percentage. If, for example, the measured value of a measurement may only deviate by 1% from the true value, then .

{\ displaystyle \ rho = 0 {,} 01}

In the literature, the term true error also appears in some cases, which is defined (with the variables used above) as and thus has the opposite sign of the absolute error explained above . In such cases, the amount of the true error , i.e. , the absolute error . Accordingly, the relative error is the amount of the relative error explained above, i.e. H. with our variables . An advantage of the definition used above is that and if and only if (i.e. approximate value is greater than exact value). ${\ textstyle x - {\ tilde {x}}}$ ${\ textstyle \ Delta _ {x}}$ ${\ textstyle | x - {\ tilde {x}} |}$ ${\ textstyle \ delta _ {x} = | {\ frac {x - {\ tilde {x}}} {x}} |}$ ${\ textstyle \ Delta _ {x}> 0}$ ${\ textstyle \ delta _ {x}> 0}$ ${\ textstyle {\ tilde {x}}> x}$

The terms correspond to the term error limit, but are supported by standardization

in measurement technology: " Error limit" and in a more recent standard "Limit deviation"
in quality management and statistics: "Deviation limit amount".

application

measuring technology

The error limit is extremely important in measurement technology. It is not possible to make a 100% accurate measurement. A measurement is generally subject to a measurement deviation (former term: measurement error). The limit deviation indicates the measurement deviation to be tolerated with the given options.

Numerics

When calculating with floating point numbers , rounding errors inevitably occur because the number of digits (size of the mantissa ) is limited. If two floating point numbers have to be compared with one another within the framework of an algorithm or a calculation rule, the error limit should be taken into account in the comparison. In particular with numerical methods that converge to a certain value, the use of an error limit is essential, since the value will usually never exactly reach the target value due to the limited number of digits in a floating point number.

supporting documents

↑ Bronstein-Semendjajew, Taschenbuch der Mathematik, 19th ed. 1979, p. 151.

^ Gisela Engeln-Müllges , Fritz Reutter: Collection of formulas for numerical mathematics with standard FORTRAN-77 programs . 5th edition. Bibliographisches Institut, Zurich 1986, ISBN 3-411-03125-5 , p. 1 .

↑ DIN 1319-1: 1995-01 Basics of measurement technology - basic terms

↑ DIN EN 60751: 2009-05 Industrial platinum resistance thermometers and platinum temperature sensors

↑ DIN 55350-12: 1989: 03 Terms of quality assurance and statistics - feature- related terms

literature

Lothar Papula : Mathematics for engineers and natural scientists, Volume 3, vector analysis, probability calculation, mathematical statistics, error calculation and compensation calculation , Vieweg Verlag, ISBN 3-528-14937-X
Gisela Engeln-Müllges , Fritz Reutter: Numerical Algorithms. Decision-making aid for selection and use , Springer Verlag, ISBN 3-18-401539-4
Gisela Engeln-Müllges, Fritz Reutter: Collection of formulas for numerical mathematics with C programs , Springer Verlag, ISBN 3-411-03112-3
Reinhard Lerch: Electrical measurement technology. Analog, digital and computer-aided processes , Springer Verlag, ISBN 3-540-73610-7

Web links

Numerical algorithms: procedures, examples, applications Gisela Engeln-Müllges, Klaus Niederdrenk, Reinhard Wodicka on books.google.de

[1] Bronstein-Semendjajew, Taschenbuch der Mathematik, 19th ed. 1979, p. 151.

[2] Gisela Engeln-Müllges , Fritz Reutter: Collection of formulas for numerical mathematics with standard FORTRAN-77 programs . 5th edition. Bibliographisches Institut, Zurich 1986, ISBN 3-411-03125-5 , p. 1 .

[3] DIN 1319-1: 1995-01 Basics of measurement technology - basic terms

[4] DIN EN 60751: 2009-05 Industrial platinum resistance thermometers and platinum temperature sensors