When is the Lagrange interpolation polynomial exact?

Question

Find the Lagrange interpolation polynomial for data points $x_k=k$ and $f(k)=k^2$, where $k=0,1,2,3$. Also, find the Lagrange interpolation polynomial for the same data points but with $g(k)= k^4$.

I would like to say that in both cases the Lagrange polynomial (L.p.) is just the given function, as the number of data points is greater than or equal to the degrees of $f$ and $g$ and the L.p. is unique.

Is this true? More generally when is the Lagrange interpolation polynomial exact? Many thanks!

Answer 1

Given the collocation points at $x_k=k$ where $k=0,1,2,3$ with $\mathcal{N}=4$ your Lagrange polynomial $\mathcal{L}(k)$ is of order $\mathcal{P}=3$. This means:

1. For the first function $f_1(k)=k^2$ the Lagrange interpolation polynomial $\mathcal{L_{P=3}}(k)$ is exact.

2. For the second function $f_2(k)=k^4$ the Lagrange interpolation polynomial $\mathcal{L_{P=3}}(k)$ is not exact and only an approximation.

You mentioned:

"I would like to say that in both cases the Lagrange polynomial (L.p.) is just the given function."

You have to distinguish between the Langrange interpolation problem itself and the representation of the given polynomial.

• In both cases the given functions are not in a Langrange representation $$\mathcal{L(k)}=\sum_{i=0}^{\mathcal{P}} a_i \phi_i(k).$$ In the given plots you can see that the i'th basis function $\phi_i$ intersects with $f_1, f_2$ at the i'th collocation point.

• The functions $f_1, f_2$ are in a Monomial representation $$\mathcal{T(k)}=\sum_{i=0}^{\mathcal{P}} b_i \psi_i(k),$$ where $b_2=1,~\psi_2=k^2$ and $b_4=1,~\psi_4=k^4$

Regards