# Discretization error

Discretization error

In numerical analysis, computational physics, and simulation, discretization error is error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a lattice[disambiguation needed ]. Discretization error can usually be reduced by using a more finely spaced lattice, with an increased computational cost.

## Examples

Discretization error is the principal source of error in methods of finite differences and the pseudo-spectral method of computational physics.

When we define the derivative of $\,\!f(x)$ as $f'(x) = \lim_{h\rightarrow0}{\frac{f(x+h)-f(x)}{h}}$ or $f'(x)\approx\frac{f(x+h)-f(x)}{h}$, where $\,\!h$ 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, 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.

Wikimedia Foundation. 2010.

Нужна курсовая?

### Look at other dictionaries:

• Discretization of continuous features — In statistics and machine learning, discretization refers to the process of converting or partitioning continuous attributes, features or variables to discretized or nominal attributes/features/variables/intervals. This can be useful when… …   Wikipedia

• Difference between Discretization Errors and Quantization Errors — Difference between Discretization Error and Quantization Errors = Discretization Error Real number has an important property called density property that says that between any two real number there is a another real number .and so on to… …   Wikipedia

• Quantization error — The difference between the actual analog value and quantized digital value due is called quantization error. This error is due either to rounding or truncation.Many physical quantities are actually quantized by physical entities. Examples of… …   Wikipedia

• Truncation error — or discretization error is error made by numerical algorithms that arises from taking finite number of steps in computation. It is present even with infinite precision arithmetic, because it is caused by truncation of the infinite Taylor series… …   Wikipedia

• Numerical analysis — Babylonian clay tablet BC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...[1] Numerical analysis is the …   Wikipedia

• List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra …   Wikipedia

• PID controller — A block diagram of a PID controller A proportional–integral–derivative controller (PID controller) is a generic control loop feedback mechanism (controller) widely used in industrial control systems – a PID is the most commonly used feedback… …   Wikipedia

• Floating point — In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent …   Wikipedia

• Finite difference method — In mathematics, finite difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives. Intuitive derivation Finite difference methods approximate the …   Wikipedia

• List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia