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Concept
Approximation error
Summary
The approximation error in a data value is the discrepancy between an exact value and some approximation to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value).
An approximation error can occur for a variety of reasons, among them a computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm).
In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates the extent to which errors in the input of the algorithm will lead to large errors of the output; numerically stable algorithms to not yield a significant error in output when the input is malformed and vice versa.
Given some value v and its approximation vapprox, the absolute error is
where the vertical bars denote the absolute value.
If the relative error is
and the percent error (an expression of the relative error) is
An error bound is an upper limit on the relative or absolute size of an approximation error.
These definitions can be extended to the case when and are n-dimensional vectors, by replacing the absolute value with an n-norm.
As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1/50 = 0.002 = 0.2%. As a practical example, when measuring a 6 mL beaker, the value read was 5 mL. The correct reading being 6 mL, this means the percent error in that particular situation is, rounded, 16.7%.
The relative error is often used to compare approximations of numbers of widely differing size; for example, approximating the number 1,000 with an absolute error of 3 is, in most applications, much worse than approximating the number 1,000,000 with an absolute error of 3; in the first case the relative error is 0.003 while in the second it is only 0.000003.
There are two features of relative error that should be kept in mind.
Official source
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