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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.
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.