Ploynomial interpolation relies on Weierstrass' theorem[?] which states that for any function <math>f</math> that is continuous on the interval <math>[a,b]</math> there exists a sequence of polynomials such that if:
then
holds, where <math>n</math> is the degree of the polynomial. <math>P_n</math> is the set of all n:th degree polynomials, and also form a linear space[?] with the dimension <math>n+1</math>. The monomials <math>1, t, t^2, t^3 \cdots t^n</math> form a basis for this of this space.

Fitting a Polynomial to Given Data Points
We want to determine the constants <math>a_0, a_1, a_2 \cdots a_n</math> so that the resulting polynomial of degree <math>n</math> interpolates some given data set <math>(t_0,y_0), (t_1,y_1), (t_2,y_2) \cdots (t_j,y_j)</math>. From the amount of information obtained from the data set, we see that we cannot fit a polynomial of greater degree than <math>j</math>, so we assume that <math>n = j</math> and:
If we put all these conditions in a matrixvector combination, with the coefficients <math>a_j</math> as unknowns, we obtain the system:
Where the horrible leftmost matrix is commonly referred to as a vandermonde matrix, so named after the mathematician AlexandreThéophile Vandermonde[?]. This equation may be solved both by hand and by machine using for example GaussJordan elimination. It can be proved that given <math>n+1</math> mutually different (i.e. no two the same) <math>t_i</math>:s, there is only one unique polynomial <math>p</math> in <math>P_n</math> of maximum degree <math>n</math> that solves this interpolation task. This is called the Unisolvence theorem. (It can be proven by assuming the opposite.)
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
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[?].
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