An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.
Approximations may be used because incomplete information prevents use of exact representations. Many problems in physics are either too complex to solve analytically, or impossible to solve using the available analytical tools. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.
For instance, physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical behaviours — e.g. gravity — are much easier to calculate for a sphere than for other shapes.
It is difficult to exactly analyze the motion of several planets orbiting a star, for example, due to the complex interactions of the planets' gravitational effects on each other, so an approximate solution is effected by performing iterations. In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained. The use of perturbations to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions.
As another example, in order to accelerate the convergence rate of evolutionary algorithms, fitness approximation—that leads to build model of the fitness function to choose smart search steps—is a good solution.
The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.
Read more about Approximation: Science, Mathematics, Unicode
Other articles related to "approximation":
... In algebraic topology, in the cellular approximation theorem, a map between CW-complexes can always be taken to be of a specific type ... The content of the cellular approximation theorem is then that any continuous map f X → Y between CW-complexes X and Y is homotopic to a cellular map ...
... The model solid approximation is a method used for determining the extrema of energy bands in semiconductors ... potential in a semiconductor crystal fluctuates on an atomic scale, the model solid approximation averages these fluctuations out to obtain a constant energy level for each material ...
... Sparse approximation (also referred to as sparse decomposition) is the problem of estimating a sparse multi-dimensional vector, satisfying a linear ... Sparse approximation techniques have found wide use in applications such as image processing, audio processing, biology, and document analysis ...
... for the difficulty of understanding its approximation hardness" ... The best polynomial time approximation algorithm known for this case achieves only a very weak approximation ratio ... there is a big gap between this inapproximability result and the known approximation algorithms for this problem ...
... the integral can be evaluated in the stationary phase approximation ... In this approximation the integral is over the path in which the action is a minimum ... Therefore, this approximation recovers the classical limit of mechanics ...