Editing Discretization error

{{Short description|Error from taking a finite number of steps in a computation to approximate an infinite process}}
{{refimprove|date=December 2009}}
In [[numerical analysis]], [[computational physics]], and [[simulation]], '''discretization error'''  is the [[error]] resulting from the fact that a [[function (mathematics)|function]] of a [[continuum (set theory)|continuous]] variable is represented in the computer by a finite number of evaluations, for example, on a [[lattice model (physics)|lattice]].  Discretization error can usually be reduced by using a more finely spaced lattice, with an increased [[Computational complexity theory|computational cost]].

==Examples==
Discretization error is the principal source of error in methods of [[finite difference]]s and the [[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]], the analog of discretization is [[Sampling (signal processing)|sampling]], and results in no loss if the conditions of the [[sampling theorem]] are satisfied, otherwise the resulting error is called [[aliasing]].

Discretization error, which arises from finite resolution in the ''domain,'' should not be confused with [[quantization error]], which is finite resolution in the ''range'' (values), nor in [[round-off error]] arising from [[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>

==References==
{{Reflist}}

==See also==
* [[Discretization]]
* [[Linear multistep method]]
* [[Quantization error]]

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