Extrapolation

Under extrapolating the determination is an (often mathematical) behavior on the secured area also understood.

Use in mathematics

A statistical extrapolation is also called a projection . Another approach is interpolation , in which the behavior is also described for cases that have not been investigated within the range of secured values ​​(possibly also secured knowledge). The extrapolation usually requires an interpolation, as in the case of the Richardson extrapolation for numerical differentiation . Here, an interpolation polynomial is placed through some support points and then the value of the interpolation polynomial is determined for the value to be calculated. This is considered sensible if the individual calculations of the function values ​​close to the limit value are becoming more and more complex and it therefore seems unreasonable in terms of complexity to approach the limit value very closely. In order to keep the extrapolation error small, it is, however, necessary to define certain criteria for the selection of the support points. This shows that the quotient of the successive distances between the support points and the limit value of a fixed number is less than 1.

One application of this procedure is, for example, the Romberg integration , for calculating the numerical value of an integral .

You can also use extrapolation to calculate the limit of sequences and series . In the case of divergent series , corresponding extrapolation methods are also called summation methods .

Application examples

Assuming a vehicle covers a straight distance of 1000 meters in 1 minute. If it is assumed that the vehicle has not changed its speed, one can linearly interpolate and calculate where the vehicle was after 0.5 minutes - namely 500 meters from the starting point. Assuming that the vehicle still does not change its speed, one can extrapolate that after 1.5 minutes it will be 1500 meters from the starting point. However, since various assumptions are required in order to describe the further behavior beyond the original course of the journey, an extrapolation may involve great uncertainty . For example, if the vehicle stops after 1000 m, the result is no longer correct.

Further examples of extrapolation include a .:

• Conclusion from the previous time for the elements already processed on the total time for all elements
• Conclusion from the previous growth of a child about the later height.
• Inference from the earth's surface at the open-minded rocks and geological structures on the same storage underground.
• Conclusion from today's physical / astronomical conditions about the Big Bang .
• Conclusion from the weighted past values ​​on future values, exponential smoothing .
• Conclusion according to the law of Charles (1787) and Gay-Lussac (1808) to the absolute zero point of −273.15 ° C (0 K) (see thermal equation of state of ideal gases ).
• Forecasts of the population development over a very long period of time (e.g. for the year 2100), which are based solely on assumptions and unsecured figures that are inferred from the current development.

Use in literature

The Science Fiction and Fantasy Writers of America defines extrapolation in science fiction as “spinning off” scientific factual knowledge so that a plot can be built around a foreseeable technical, social or other development. Science fiction author Joan Slonczewski cites the works of Michael Crichton (technical extrapolation), Ursula K. Le Guin (social science extrapolation) and herself (ecological extrapolation; Slonczewski is a biologist) as modern examples . The use of hard data is also essential to distinguish science fiction from fantasy .

literature

• C. Brezinski and M. Redivo Zaglia: Extrapolation Methods. Theory and Practice. North Holland, 1991.

Individual evidence

1. Science in Science Fiction: Making it Work ( Memento of the original dated February 22, 2009 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. , Joan Slonczewski, Science Fiction and Fantasy Writers of America, Inc, sfwa.org.