Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate (i.e. estimate) the value of that function for an intermediate value of the independent variable.

A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complex to evaluate efficiently. A few known data points from the original function can be used to create an interpolation based on a simpler function. Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in precision.

In the examples below if we consider x as a topological space and the function f forms a different kind of Banach spaces then the problem is treated as "interpolation of operators". The classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other subsequent results.

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 algorithm are: How accurate is the method? How expensive is it? How smooth is the interpolant? How many data points are needed?

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