Polynomial interpolation
Polynomial interpolation is the act of fitting a polynomial to a given function with defined values in certain discrete data points. This "function" may actually be any discrete data (such as obtained by sampling), but it is generally assumed that such data may be described by a function. Polynomial interpolation is an area of inquiry in numerical analysis.
Polynomial interpolation relies on Weierstrass' theorem which states that for any function  that is continuous on the interval  there exists a sequence of polynomials such that if:
then
holds, where  is the degree of the polynomial.  is the set of all n:th degree polynomials, and also form a linear space with the dimension  . The monomials  form a basis for this of this space.
Fitting a Polynomial to Given Data Points
We want to determine the constants  so that the resulting polynomial of degree interpolates some given data set  . From the amount of information obtained from the data set, we see that we cannot fit a polynomial of greater degree than  , so we assume that  and:
If we put all these conditions in a matrix-vector combination, with the coefficients  as unknowns, we obtain the system:
Where the leftmost matrix is commonly referred to as a vandermonde matrix, so named after the mathematician Alexandre-Théophile Vandermonde. This equation may be solved both by hand and by machine using for example Gauss-Jordan elimination. It can be proved that given mutually different (i.e. no two the same)  :s, there is only one unique polynomial  in of maximum degree that solves this interpolation task. This is called the Unisolvence theorem. (It can be proven by assuming the opposite.)
Non-Vandermonde Solutions
Solving the vandermonde matrix is (mostly) a costly operation (as counted in clock cycles of a computer trying to do the job). Therefore, several other clever ways of constructing the unique polynomial have been devised:
The Error of Polynomial Interpolation
- To be written
Disadvantages of Polynomial Interpolation
When the interpolation polynomial reach a certain degree, it will tend to oscillate wildly in the undetermined areas. This is called Runge's phenomenon. Even though these problems can be partially avoided by using for example Chebyshev polynomials, the solution that is mostly preferred in practice is to use several polynomials of a lower degree, connected in chains. These are called splines.
Referenced By
Interpolation | List of mathematical topics (P-R) | List of numerical analysis topics | List of polynomial topics | Simpson's rule
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