In mathematics, extrapolation is a type of interpolation. When a tabulated function is interpolated not between given values, but outside of the given range, this is called extrapolation. Extrapolation often looks sensible at first glance, but its results may sometimes be invalid or subject to substantial uncertainty. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... This article is about interpolation in mathematics. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Uncertainty is an inevitable part of the assertion of knowledge, see Bayesian probability. ...

Extrapolation also means to generalize, making use of a specific case and adjusting it to fit several cases in general.

Extrapolation in complex analysis

A problem of extrapolation may be converted into an interpolation problem by the change of variable z → 1/z. This transform exchanges the part of the complex plane inside the unit circle with the part of the complex plane outside of the unit circle. In particular, the compactification point at infinity is mapped to the origin and vice versa. Care must be taken with this transform however, since the original function may have had "features", for example poles and other singularities, at infinity that were not evident from the sampled data. This article is about interpolation in mathematics. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... The Poles are a western Slavic ethnic group primarily associated with Poland and the Polish language. ... Singularity has several different meanings: mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved gravitational singularity - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum technological singularity - a predicted point in...

Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is expanded at one of its points of convergence to produce a power series with a larger radius of convergence. In effect, a set of data from a small region is used to extrapolate a function onto a larger region. Again, analytic continuation can be thwarted by function features that were not evident from the initial data. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Convergence means approaching a definite value, as time goes on; or approaching a definite point, or a common view or opinion, or a fixed state of affairs. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

Quality of extrapolation

Typically, the quality of a particular method of extrapolation is limited by the assumptions about the function made by the method. If the method assumes the data is smooth, then a non-smooth function will be poorly extrapolated. Even for proper assumptions about the function, the extrapolation can diverge exponentially from the function. The classic example is truncated power series representations of sin(x) and related trigonometric functions. For instance, taking only data from near the x = 0, we may estimate that the function behaves as sin(x) ~ x. In the neighborhood of x = 0, this is an excellent estimate. Away from x = 0 however, the extrapolation moves arbitrarily away from the x-axis while sin(x) remains in the interval [-1,1]. I.e., the error increases without bound. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

Using the x → 1/x transform on this example indicates why the agreement becomes poorer away from x = 0 : sin(1/x) oscillates infinitely often as this x → 0, having an infinite number of zeroes there. Thus a singularity appears in sin(x) as x → . An extrapolation based on values taken in the neighborhood of x = 0 will therefore not adequately represent the non-smooth behavior at infinity. Taking more terms in the power series of sin(x) around x = 0 will produce better agreement over a larger interval near x = 0, but will still produce extrapolations that diverge away from the x-axis due to the singularity as x → .

This divergence is a specific property of extrapolation methods and is only circumvented when the functional forms assumed by the extrapolation method (inadvertently or intentionally due to additional information) accurately represent the nature of the function being extrapolated. For particular problems, this additional information may be available, but in the general case, it is impossible to satisfy all possible function behaviors with a workably small set of potential behaviors.

See also

• Richardson extrapolation