# 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 [*x*_{0}, *x*_{n}] on the basis of the function’s values at the points x_{0} < *x*_{1} <... < *x*_{n} 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 P_{n}(*x* ) of degree *n* that assumes at the *n* + 1 points *x*_{i} the specified values *y*_{i} = *f* (*x*_{i}). 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.

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.