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.

Fitting a polynomial to given data points

We want to determine a ploynomial of degree n that interpolates some given data set

$(t_{0},y_{0}),(t_{1},y_{1}),(t_{2},y_{2}),\dots ,(t_{j},y_{j}).$

From the amount of information obtained from the data set, we see that we cannot fit a polynomial of greater degree than j, so we assume that n = j and:

$p(t_{i})=y_{i},\qquad \forall i\in \left\{0,1,\cdots j\right\}.$

If we put all these conditions in a matrix-vector combination, with the coefficients n_j as unknowns, we obtain the system:

${\begin{pmatrix}t_{0}^{n}&t_{0}^{n-1}&t_{0}^{n-2}&\ldots &t_{0}&1\\t_{1}^{n}&t_{1}^{n-1}&t_{1}^{n-2}&\ldots &t_{1}&1\\\vdots &\vdots &\vdots &&\vdots &\vdots \\t_{n}^{n}&t_{n}^{n-1}&t_{n}^{n-2}&\ldots &t_{n}&1\end{pmatrix}}{\begin{pmatrix}a_{n}\\a_{n-1}\\\vdots \\a_{0}\end{pmatrix}}={\begin{pmatrix}y_{0}\\y_{1}\\\vdots \\y_{n}\end{pmatrix}}$

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 either by hand or by machine using for example Gauss-Jordan elimination. It can be proved that given n + 1 mutually different (i.e. no two the same) t_i:s, there is only one unique nth-degree polynomial p that solves this interpolation task. This is called the unisolvence theorem.

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:

• Newton polynomials
• Lagrange polynomials
• Bernstein polynomials

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.