Interpolation polynomials are commonly used for function interpolation.

Interpolation polynomials are polynomials of the form $a_0 + a_1x + a_2x^2 + \cdots$, which pass through a given set of nodes and depend only on the coordinates of these points.

Let $(n+1)$ points be given with indices from $0$ to $n$, inclusive: $(x_k, y_k),{}_{k \in \left\{0, ..., n\right\}}$.

When interpolating with polynomials, a polynomial of degree $n$ is typically used, because there is a unique polynomial of that degree that passes through $(n+1)$ points.

There are two approaches to interpolation:

1. Calculating the coefficients of the interpolation polynomial and then evaluating it at the required points:
   • Lagrange’s formula
   • Improved Lagrange’s formula
   • Newton’s formula
2. Evaluating the interpolation polynomial directly — without calculating the coefficients:
   • Lagrange’s formula
   • Improved Lagrange’s formula
   • Newton’s formula
   • Barycentric formula