# Discretization error

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{{Short description|Quantization error in numerical analysis}}
{{refimprove|date=December 2009}}
In [numerical analysis](/source/numerical_analysis), [computational physics](/source/computational_physics), and [simulation](/source/simulation), '''discretization error'''  is the [error](/source/error) resulting from the fact that a [function](/source/function_(mathematics)) of a [continuous](/source/continuum_(set_theory)) variable is represented in the computer by a finite number of evaluations, for example, on a [lattice](/source/lattice_model_(physics)).  Discretization error can usually be reduced by using a more finely spaced lattice, with an increased [computational cost](/source/Computational_complexity_theory).

==Examples==
Discretization error is the principal source of error in methods of [finite difference](/source/finite_difference)s and the [pseudo-spectral method](/source/pseudo-spectral_method) of computational physics.

When we define the derivative of <math>\,\!f(x)</math> as <math>f'(x) = \lim_{h\rightarrow0}{\frac{f(x+h)-f(x)}{h}}</math> or <math>f'(x)\approx\frac{f(x+h)-f(x)}{h}</math>, where <math>\,\!h</math> is a finitely small number, the difference between the first formula and this approximation is known as discretization error.

==Related phenomena==
In [signal processing](/source/signal_processing), the analog of discretization is [sampling](/source/Sampling_(signal_processing)), and results in no loss if the conditions of the [sampling theorem](/source/sampling_theorem) are satisfied, otherwise the resulting error is called [aliasing](/source/aliasing).

Discretization error, which arises from finite resolution in the ''domain,'' should not be confused with [quantization error](/source/quantization_error), which is finite resolution in the ''range'' (values), nor in [round-off error](/source/round-off_error) arising from [floating-point arithmetic](/source/floating-point_arithmetic).  Discretization error would occur even if it were possible to represent the values exactly and use exact arithmetic – it is the error from representing a function by its values at a discrete set of points, not an error in these values.<ref>{{cite book | first = Nicholas | last=Higham | title=Accuracy and Stability of Numerical Algorithms |edition = 2 | doi = 10.1137/1.9780898718027 | publisher = SIAM | year=2002 | pages=5 | isbn = 978-0-89871-521-7 | series = Other Titles in Applied Mathematics | url=http://eprints.maths.manchester.ac.uk/238/4/asna2_cover.pdf }}</ref>

==See also==
* [Discretization](/source/Discretization)
* [Linear multistep method](/source/Linear_multistep_method)
* [Quantization error](/source/Quantization_error)

==References==
{{Reflist}}

{{DEFAULTSORT:Discretization Error}}
Category:Numerical analysis

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Adapted from the Wikipedia article [Discretization error](https://en.wikipedia.org/wiki/Discretization_error) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Discretization_error?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.