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.
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.
In the restricted formulation studied in,[2] 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.
(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.)
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,[1] Charles Hermite used contour integration to prove that this is effectively the case here, and to find the unique solution, provided that the xi 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.
[3] The general methods of linear algebra, and specific methods for confluent Vandermonde matrices are often used for computing the interpolation polynomial.
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
These conditions implies that the Taylor polynomial of f of degree
The Chinese remainder theorem for polynomials implies that there is exactly one solution of degree less than
arithmetic operations, or even faster with fast polynomial multiplication.
This approach does not works in positive characteristic, because of the denominators of the coefficients of the Taylor polynomial.
The approach through divided differences, below, works in every characteristic.
When using divided differences to calculate the Hermite polynomial of a function f, the first step is to copy each point m times.
Now, we create a divided differences table for the points
In this case, the divided difference is replaced by
In the general case, suppose a given point
When creating the table, divided differences of
A fast algorithm for the fully general case is given in.
[4] A slower but more numerically stable algorithm is described in.
, we obtain the following data: Since we have two derivatives to work with, we construct the set
The quintic Hermite interpolation based on the function (
Call the calculated polynomial H and original function f. Consider first the real-valued case.