# extrapolation

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## extrapolation

[ik‚strap·ə′lā·shən]
(mathematics)
Estimating a function at a point which is larger than (or smaller than) all the points at which the value of the function is known.

## Extrapolation

in mathematics and statistics, the approximate determination of the values of a function f(x) at points x lying outside the interval [x0, xn] on the basis of the function’s values at the points x0 < x1 <... < xn In parabolic extrapolation, which is the most widely encountered type, the value of f(x) at x is approximated by the value of a polynomial Pn(x ) of degree n that assumes at the n + 1 points xi the specified values yi = f (xi). Interpolation formulas are used for parabolic extrapolation.

## extrapolation

(mathematics, algorithm)
A mathematical procedure which estimates values of a function for certain desired inputs given values for known inputs.

If the desired input is outside the range of the known values this is called extrapolation, if it is inside then it is called interpolation.

The method works by fitting a "curve" (i.e. a function) to two or more given points and then applying this function to the required input. Example uses are calculating trigonometric functions from tables and audio waveform sythesis.

The simplest form of interpolation is where a function, f(x), is estimated by drawing a straight line ("linear interpolation") between the nearest given points on either side of the required input value:

f(x) ~ f(x1) + (f(x2) - f(x1))(x-x1)/(x2 - x1)

There are many variations using more than two points or higher degree polynomial functions. The technique can also be extended to functions of more than one input.
References in periodicals archive ?
Caption: Figure 5: Reconstruction results of Shepp-Logan phantom using the proposed extrapolation method. From left to right: reference from nontruncated data, proposed correction method with medium truncation, and proposed correction method with severe truncation.
In Section 2, a brief review of popular vector extrapolation methods is provided.
Due to the uncertainty within this extrapolation method and the limited information available from the measured data, the estimation of the cumulative probability might not be accurate in the absolute sense.
(i) the extrapolation method for period and cohort effects is linear trend-preserving:
When using the non-exercise prediction equation, individuals who were more physically active had higher cardiorespiratory fitness levels, which is consistent with other studies among persons with diabetes; (25) however, when using estimates from the heart rate extrapolation method, no significant findings emerged.
From the specialized literature results that the sports forecasting problems centres round the using of the various forecasting methods (extrapolation method, modeling method and examination method).
Based on comparisons with field-collected adults, the authors used an extrapolation method and a formula-based estimate to hypothesize that males of the species mature at either the sixth or seventh instar and females at either the seventh or eighth.
Corrosion rates were determined by the Tafel extrapolation method. The corrosion rates of the Ti-alloys were comparable but lower than that of 316L stainless steel.
The assumptions underlying the single-deflation method were judged more likely to hold under a wider variety of macroeconomic conditions than those underlying the extrapolation method. Because of data limitations, however, those conclusions were based on only 3 years of testing with experimental accelerated estimates.
However, unlike in the leading political/economic circles where ideology may prevail, the caveats due to extrapolation method are well understood in Science.
The pattern of the time steps, however, must be equally spaced when Richardson extrapolation method is used to derive the two-point or three-point Geske and Johnson  formula.
The resolution of this issue therefore depends on the applicable state laws, as well as the applicability of a mutually agreeable extrapolation method.

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