Interpolation is the task of finding a function from a certain class that matches given data , . That is,
Typically, is taken from the set of polynomials of degree .
Approximation is the task, given a function , where is a suitable space, to find a function taken from a certain class that is in some sense close to . Often we consider, for example, polynomial interpolation where is a bounded subset of and .
Different measures of closeness can be used for determining how close functions are. The most common are:
In polynomial approximation is required to be a polynomial of degree for some fixed . Often approximations can be found by interpolation: the data used is simply , for suitably chosen points .
Calculus, real analysis.
Polynomial interpolation in an interval with data , can be done with polynomials of degree provided that and all 's are distinct. The interpolant is unique (amongst all polynomials of degree ) provided in addition, . There are a number of different ways of computing the polynomial interpolant for , including solving linear systems with Van der Monde matrices, Lagrange interpolation polynomials, and Newton divided differences.
If the data for polynomial interpolation comes from a smooth function , then there is a commonly used formula for the error in the interpolant:
for some between and .