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Hermite interpolation

Hermite interpolation is an application of the Chinese remainder theorem for univariate polynomials, which may involve moduli of arbitrary degrees (Lagrange interpolation involves only moduli of degree one).

The problem consists of finding a polynomial of the least possible degree, such that the polynomial and its first derivatives take given values at some fixed points.

More precisely, let x 1, …, x k {\displaystyle x_{1},\ldots,x_{k}} be k {\displaystyle k} elements of the ground field K, {\displaystyle K,} and, for i = 1, …, k, {\displaystyle i=1,\ldots,k,} let a i, 0, a i, 1, …, a i, r i − 1 {\displaystyle a_{i,0},a_{i,1},\ldots,a_{i,r_{i}-1}} be the values of the first r i {\displaystyle r_{i}} derivatives of the sought polynomial at x i {\displaystyle x_{i}} (including the 0th derivative, which is the value of the polynomial itself). The problem is to find a polynomial P (X) {\displaystyle P(X)} such that its j th derivative takes the value a i, j {\displaystyle a_{i,j}} at x i, {\displaystyle x_{i},} for i = 1, …, k {\displaystyle i=1,\ldots,k} and j = 0, …, r j. {\displaystyle j=0,\ldots,r_{j}.}

Consider the polynomial

This is the Taylor polynomial of order r i − 1 {\displaystyle r_{i}-1} at x i {\displaystyle x_{i}}, of the unknown polynomial P (X). {\displaystyle P(X).} Therefore, we must have

Conversely, any polynomial P (X) {\displaystyle P(X)} that satisfies these k {\displaystyle k} congruences, in particular verifies, for any i = 1, …, k {\displaystyle i=1,\ldots,k}

therefore P i (X) {\displaystyle P_{i}(X)} is its Taylor polynomial of order r i − 1 {\displaystyle r_{i}-1} at x i {\displaystyle x_{i}}, that is, P (X) {\displaystyle P(X)} solves the initial Hermite interpolation problem.The Chinese remainder theorem asserts that there exists exactly one polynomial of degree less than the sum of the r i, {\displaystyle r_{i},} which satisfies these k {\displaystyle k} congruences.

There are several ways for computing the solution P (X). {\displaystyle P(X).} One may use the method described at the beginning of § Over univariate polynomial rings and Euclidean domains. One may also use the constructions given in § Existence (constructive proof) or § Existence (direct proof).