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In the Mathematical subfield of Numerical Analysis , interpolation is a method of constructing new data points from a Discrete Set of known data points. In Engineering and Science one often has a number of data points, as obtained by Sampling or Experiment , and tries to construct a function which closely fits those data points. This is called Curve Fitting or Regression Analysis . Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points. A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function. Suppose we know the function but it is too complex to evaluate efficiently. Then we could pick a few known data points from the complicated function, creating a Lookup Table , and try to interpolate those data points to construct a simpler function. Of course, when using the simple function to calculate new data points we usually do not receive the same result as when using the original function, but depending on the problem domain and the interpolation method used the gain in simplicity might offset the error. It should be mentioned that there is another very different kind of interpolation in mathematics, namely the " Interpolation Of Operators ". The classical results about interpolation of operators are the Riesz-Thorin Theorem and the Marcinkiewicz Theorem . There also are many other subsequent results. DEFINITION From ''inter'' meaning between and ''pole'', the points or nodes. Any means of calculating a new point between two existing data points is therefore interpolation. There are many methods for doing this, many of which involve fitting some sort of function to the data and evaluating that function at the desired point. This does not exclude other means such as statistical methods of calculating interpolated data. The simplest form of interpolation is to take the mean average of and of two adjacent points to find the mid point. This will give the same result as linear interpolation evaluated at the midpoint. Given a Sequence of ''n'' ''distinct'' numbers ''x''''k'' called nodes and for each ''x''''k'' a second number ''y''''k'', we are looking for a function ''f'' so that : A pair ''x''''k'',''y''''k'' is called a data point and ''f'' is called an '''interpolant''' for the data points. When the numbers ''y''''k'' are given by a known function ''f'', we sometimes write ''f''''k''. EXAMPLE For example, suppose we have a table like this, which gives some values of an unknown function ''f''. Interpolation provides a means of estimating the function at intermediate points, such as ''x'' = 2.5. There are many different interpolation methods, some of which are described below. Some of the concerns to take into account when choosing an appropriate is the interpolant? How many data points are needed? PIECEWISE CONSTANT INTERPOLATION .]] The simplest interpolation method is to locate the nearest data value, and assign the same value. In one dimension, there are seldom good reasons to choose this one over linear interpolation, which is almost as cheap, but in higher dimensions, in Multivariate Interpolation , this can be a favourable choice for its speed and simplicity. LINEAR INTERPOLATION See Also: Linear interpolation One of the simplest methods is Linear interpolation (sometimes known as lerp). Consider the above example of determining ''f''(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take ''f''(2.5) midway between ''f''(2) = 0.9093 and ''f''(3) = 0.1411, which yields 0.5252. Generally, linear interpolation takes two data points, say (''x''''a'',''y''''a'') and (''x''''b'',''y''''b''), and the interpolant is given by: : at the point (''x'',''y''). Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not Differentiable at the point ''x''''k''. The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by ''g'', and suppose that ''x'' lies between ''x''''a'' and ''x''''b'' and that ''g'' is twice continuously differentiable. Then the linear interpolation error is |
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