# Extrapolation

In extrapolation, the determination of an (often mathematical ) behavior through the secure area is also understood.

A statistical extrapolation is also called extrapolation. Another approach is the interpolation, within the range of safe values (possibly also established knowledge ) describes the behavior for cases that were not investigated. Usually constitutes extrapolation interpolation advance as in the case of Richardson extrapolation for numerical differentiation. Here, an interpolation is applied and then determines the value of the interpolating polynomial at the value to be calculated by some reference points. This is considered useful if the individual calculations of the function values near the limit are always more complex and therefore it appears here on the complexity as not justifiable to approach very close to the limit. To keep the extrapolation small, but is regarded as necessary to establish some criteria for the choice of the interpolation points. It turns out that the quotient of successive distances of the sampling points to the limit of a fixed number less than 1.

An application of this approach is, for example, the Romberg integration, calculation of the numerical value of the integral.

## Examples

Assuming that a vehicle travels a straight stretch of 200 meters in 0.2 minutes of 1000 m length in 1 minute. If it is assumed that the vehicle has not changed its speed, one can linearly interpolate and calculate the location at which the vehicle was located after 0.5 minutes - namely 500 meters from the starting location. Assuming that the vehicle continues to not change its speed, one can extrapolate that it will be 1500 meters after 1.5 minutes from the initial location. However, because different assumptions are required to describe the further behavior on the original course of the trip out, if necessary, extrapolation is fraught with great uncertainty. When the vehicle stops, for example, after 1000 m, the result is no longer true.

Other examples of extrapolation include, without limitation:

- Inference from the previous time to the elements already processed in the total time for all the elements
- Inference from the recent growth of a child to later body size.
- Inference from the disrupted at the surface rocks and geological structures on the same storage underground.
- Conclusion of today's physical / astronomical conditions on the Big Bang.
- Inference from the weighted past values to future values , exponential smoothing.
- Inference according to the law of Charles ( 1787) and Gay- Lussac (1808 ) to the absolute zero at -273.15 ° C (0 K ) (see Thermal equation of state of ideal gases).
- Forecasts across the population development over a very long period of time ( eg for the year 2100 ), based solely on assumptions and unsecured numbers that are inferred from the current developments are based.

## Use in the literature

The Science Fiction and Fantasy Writers of America defines extrapolation in science fiction as " spinning out " of scientific factual knowledge, so that an action to a foreseeable technical, social or other development can be built. As modern examples SF -author Joan Slonczewski calls the works of Michael Crichton (technical extrapolation), by Ursula K. Le Guin ( social scientific extrapolation) and himself (ecological extrapolation; Slonczewski is a biologist ). The use of hard data is also essential to depose SF Fantasy.