Direct algorithm for polynomial interpolation

Given the data points for , we want to find a degree polynomial that ``passes through'' all the data points. This polynomial is

To determine the coefficients we write the interpolating conditions which require the value of the polynomial in be equal to

These relation can be formulated as an linear system:

with the system matrix (it has a special form and is called a Vandermonde matrix)

The vector of measurements and the vector of polynomial coefficients are