Hermite interpolation

Summary

Hermite interpolation finds a polynomial that matches, at each node, a prescribed value and prescribed derivatives up to some order. It extends polynomial interpolation, which prescribes values only, and Taylor polynomials, which prescribe derivatives at a single point. If $N$ conditions are imposed in total, there is exactly one polynomial of degree at most $N - 1$ that satisfies them. It can be computed with the Newton form using divided differences with repeated nodes, and it underlies cubic Hermite splines and Gaussian quadrature.