Lagrange Polynomial - Barycentric Interpolation

Barycentric Interpolation

Using

we can rewrite the Lagrange basis polynomials as

or, by defining the barycentric weights

we can simply write

which is commonly referred to as the first form of the barycentric interpolation formula.

The advantage of this representation is that the interpolation polynomial may now be evaluated as

which, if the weights have been pre-computed, requires only operations (evaluating and the weights ) as opposed to for evaluating the Lagrange basis polynomials individually.

The barycentric interpolation formula can also easily be updated to incorporate a new node by dividing each of the, by and constructing the new as above.

We can further simplify the first form by first considering the barycentric interpolation of the constant function :

Dividing by does not modify the interpolation, yet yields

which is referred to as the second form or true form of the barycentric interpolation formula. This second form has the advantage that need not be evaluated for each evaluation of .