In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than n such that the polynomial and its first few derivatives have the same values at m (fewer than n) given points as the given function and its first few derivatives at those points. The number of pieces of information, function values and derivative values, must add up to ${\displaystyle n}$.

Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both can be derived from the calculation of divided differences. However, there are other methods for computing a Hermite interpolating polynomial. One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see Chinese remainder theorem § Hermite interpolation. For yet another method, see, which uses contour integration.

In the restricted formulation studied in, Hermite interpolation consists of computing a polynomial of degree as low as possible that matches an unknown function both in observed value, and the observed value of its first m derivatives. This means that n(m + 1) values must be known. $${\displaystyle {\begin{matrix}(x_{0},y_{0}),&(x_{1},y_{1}),&\ldots ,&(x_{n-1},y_{n-1}),\\[1ex](x_{0},y_{0}'),&(x_{1},y_{1}'),&\ldots ,&(x_{n-1},y_{n-1}'),\\[1ex]\vdots &\vdots &&\vdots \\[1.2ex](x_{0},y_{0}^{(m)}),&(x_{1},y_{1}^{(m)}),&\ldots ,&(x_{n-1},y_{n-1}^{(m)})\end{matrix}}}$$ The resulting polynomial has a degree less than n(m + 1). (In a more general case, there is no need for m to be a fixed value; that is, some points may have more known derivatives than others. In this case the resulting polynomial has a degree less than the number of data points.)

Let us consider a polynomial P(x) of degree less than n(m + 1) with indeterminate coefficients; that is, the coefficients of P(x) are n(m + 1) new variables. Then, by writing the constraints that the interpolating polynomial must satisfy, one gets a system of n(m + 1) linear equations in n(m + 1) unknowns.

In general, such a system has exactly one solution. In, Charles Hermite used contour integration to prove that this is effectively the case here, and to find the unique solution, provided that the x_i are pairwise different. The Hermite interpolation problem is a problem of linear algebra that has the coefficients of the interpolation polynomial as unknown variables and a confluent Vandermonde matrix as its matrix. The general methods of linear algebra, and specific methods for confluent Vandermonde matrices are often used for computing the interpolation polynomial. Another method is described below.

Let k be a positive integer, ⁠${\displaystyle m_{1},\ldots ,m_{k}}$⁠ be nonnegative integers, and values ⁠${\displaystyle x_{1},\ldots ,x_{k}}$⁠ that are real numbers or belong to any other field of characteristic zero. Hermite interpolation problem consists of finding a polynomial f such that

for ⁠${\displaystyle i=1,\ldots ,k}$⁠, where the ⁠${\displaystyle y_{i,j}}$⁠ are given values in the same field as the ⁠${\displaystyle x_{i}}$⁠.

These conditions implies that the Taylor polynomial of f of degree ⁠${\displaystyle m_{i}}$⁠ at ⁠${\displaystyle x_{i}}$⁠ is