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:
Table of contents |
2 Non-Vandermonde Solutions 3 The Error of Polynomial Interpolation 4 Disadvantages of Polynomial Interpolation |
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:
Fitting a Polynomial to Given Data Points
If we put all these conditions in a matrix-vector combination, with the coefficients as unknowns, we obtain the system:
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:
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.Non-Vandermonde Solutions
The Error of Polynomial Interpolation
Disadvantages of Polynomial Interpolation