numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...

, and

simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...

, discretization error is the

error An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'. In statistics ...

resulting from the fact that a function of a

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

variable is represented in the computer by a finite number of evaluations, for example, on a

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

. Discretization error can usually be reduced by using a more finely spaced lattice, with an increased

computational cost In computational complexity theory, a computational resource is a resource used by some computational models in the solution of computational problems. The simplest computational resources are computation time, the number of steps necessary to s ...

Examples

Discretization error is the principal source of error in methods of

finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...

s and the

pseudo-spectral method Pseudo-spectral methods, also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. They are closely rel ...

of computational physics. When we define the derivative of

\,\!f(x)

f'(x) = \lim_

f'(x)\approx\frac

, where

\,\!h

is a finitely small number, the difference between the first formula and this approximation is known as discretization error.

Related phenomena

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, the analog of discretization is sampling, and results in no loss if the conditions of the

sampling theorem Sampling may refer to: * Sampling (signal processing), converting a continuous signal into a discrete signal * Sampling (graphics), converting continuous colors into discrete color components * Sampling (music), the reuse of a sound recording in a ...

are satisfied, otherwise the resulting error is called

aliasing In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when ...

. Discretization error, which arises from finite resolution in the ''domain,'' should not be confused with

quantization error Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and ...

, which is finite resolution in the ''range'' (values), nor in

round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...

arising from

floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...

. 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.

Examples

Related phenomena

References

See also