We will sample at different points and construct a quadratic polynomial which interpolates these samples.

The position and the number of the points are not random. For example, if we try to construct a polynomial of degree 1 (a plane), which interpolates locally a function

(3.1) |

The matrix above is called the ``Vandermonde Matrix''.

We can say even more: What happens if these three points are on the same line? There is a simple infinity of planes which passes through three aligned points. The determinant of the Vandermonde Matrix (called here after the ``Vandermonde determinant'') will be null. The interpolation problem is not solvable. We will say that ''the problem is NOT poised''.

In opposition to the univariate polynomial interpolation (where we can take a random number of point, at random different places), the multivariate polynomial interpolation imposes a precise number of interpolation points at precise places.

In fact, if we want to interpolate by a polynomial of degree a function

If we already have a polynomial of degree and want to use information contained in new points, we will need a block of exactly new points. The new interpolating polynomial will have a degree of . This is called