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 that takes the same value at given points as a given function. Instead, Hermite interpolation computes a polynomial of degree fewer than such that the polynomial and its first few derivatives have the same values at (fewer than ) 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 .
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 . For yet another method, see .
Hermite interpolating polynomials are also related to matrix generalized eigenspace projection via generalized Frobenius covariants.
Statement of the problem
In a restricted formulation, Hermite interpolation consists, given a differentiable function, of computing a polynomial function of degree as low as possible such that the two functions and their first derivatives take the same values at different points. This means that one starts from constraints, each resulting of one of the following given pairs of values:
As a polynomial of degree less than is defined by coefficients, the resulting polynomial must have a degree less than , since, otherwise, the solution could not be unique. In a more general case, there is no need for to have 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 total number of constraints .
Let us consider a polynomial of degree less than with indeterminate coefficients; that is, the coefficients of are new variables. Then, by writing the constraints that the interpolating polynomial must satisfy, one gets a system of linear equations in unknowns.
Hermite used contour integration to prove that this is effectively the case here, and to find the unique solution, provided that the are pairwise different. Hermite interpolation 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, as detailed in . Other methods are described below.
Explicit formula For a given set of distinct nodes with associated maximum derivative orders , let be the multiplicity of node , and let be the total number of conditions. The Hermite interpolating polynomial is the unique polynomial of degree at most that satisfies the interpolation conditions:
Define the complementary factor polynomial at node as:
For each node , let be the local th degree Taylor polynomial matching the target derivative conditions:
The explicit closed-form formula for the Hermite interpolating polynomial is given by:
where denotes the th degree Taylor operator of a function centered at .
When for all distinct nodes , meaning no derivative conditions are specified, this explicit closed-form formula reduces directly to the explicit formula of Lagrange polynomials.
Interpolating a function in closed form When interpolating a function that is times differentiable at each node , the interpolation conditions are determined by the function's derivatives (i.e., ). In this case, the local Taylor polynomial represents the th degree Taylor expansion of at .
Since the operator isolates terms only up to order at , substituting in place of its local Taylor expansion yields the identical Taylor expansion up to order :
The explicit closed-form Hermite interpolating polynomial for can thus be expressed concisely as:
Basis polynomials in closed form The Hermite interpolating polynomial can also be represented as a linear combination of Hermite basis polynomials :
An explicit closed-form expression for each basis polynomial can be obtained directly by choosing the local Taylor polynomial and setting for all . Substituting into the explicit formula yields:
Following the § Alternative basis construction via degree- basis polynomials, the full set of basis polynomials can be constructed entirely using only the degree- basis polynomial . For , the local Taylor polynomial simplifies to , giving the explicit closed-form expression:
Proof of the closed form The proof proceeds in three steps. First, it shows that evaluating derivatives at any node isolates only the th term in the summation. Next, it verifies that this term matches the required derivative values via local Taylor expansion. Finally, it establishes that the resulting polynomial satisfies the degree bound and uniqueness.
Vanishing of cross-terms at distinct nodes Fix an arbitrary target node and derivative order . Evaluating the th derivative of at yields:
For any node index , the factor polynomial contains as an explicit factor:
By the general Leibniz rule, evaluating the th derivative of at produces a linear combination of terms involving for . Because , each derivative evaluates to at :
Consequently, all terms in the summation vanish at .
Verification of target interpolation conditions Evaluating the remaining term at :
Since the nodes are distinct, , so the rational function is analytic at . By definition of the Taylor operator , it expands locally around as:
Multiplying by yields:
Because the big term represents an analytic function near with a zero of order at least , all of its derivatives of orders up to vanish at . Taking the th derivative () and evaluating at yields:
Thus, satisfies all interpolation conditions .
Degree bound and uniqueness For each distinct node , the degree of is at most , and . Therefore, each term in has degree at most , so .
To establish uniqueness, assume another polynomial of degree satisfies the same conditions. Because for all and , the difference satisfies for all , meaning each node is a root of multiplicity at least . This gives roots counted with multiplicity across the nodes. A nonzero polynomial of degree at most cannot have roots counted with multiplicity. Therefore, must be the zero polynomial, so .
Using Chinese remainder theorem
Let be a positive integer representing the number of distinct nodes, be nonnegative integers specifying the maximum derivative order at each node, be the multiplicity of node , and the nodes be real numbers or elements of any other field of characteristic zero. The Hermite interpolation problem consists of finding a polynomial such that
for , where the are given values in the same field as the .
These conditions implies that the Taylor polynomial of of degree at is
In other words, the desired polynomial is congruent to this polynomial modulo .
The Chinese remainder theorem for polynomials implies that there is exactly one solution of degree less than .
Moreover, this solution can be computed with arithmetic operations, or even faster with fast polynomial multiplication.
This approach does not work in positive characteristic, because of the denominators of the coefficients of the Taylor polynomial. The approach through divided differences, below, works in every characteristic.
Explicit construction via Chinese remainder theorem For general Hermite interpolation, an explicit solution satisfying the system of simultaneous congruences can be constructed directly by following the constructive proof of the Chinese remainder theorem.
The target polynomial must satisfy the system of modular equivalences:
where is the modulus polynomial at node .
The solution is constructed iteratively by building a sequence of polynomials such that satisfies the first congruences.
1. Base Step: Set the initial partial solution for :
2. Inductive Step: Define for . For , since and share no roots and thus are coprime. By Bézout's identity, there exist polynomials and satisfying:
The polynomials and can be solved together using the polynomial extended Euclidean algorithm.
Define the th partial solution as:
This construction satisfies all conditions up to node : Modulo , the term containing vanishes while , giving . Modulo for any , and , giving .
After inductive steps, the final interpolating polynomial is obtained by reducing modulo the total degree product polynomial :
Basis polynomials via Chinese remainder theorem Following § Alternative basis construction via degree- basis polynomials, a full basis can be constructed by only solving the degree- basis polynomials , which can be obtained in a simplified manner.
For each node , the degree- basis polynomial must satisfy: , .
To satisfy the zero-remainder conditions at all other nodes , must contain the product factor . Thus, for some polynomial .
Since and share no roots and thus are coprime, Bézout's identity guarantees the existence of polynomials and satisfying:
Taking this relation modulo , the term vanishes, showing that is a modular multiplicative inverse of :
Since the polynomial is not required for the construction, can be computed directly using half extended Euclidean algorithm that tracks only one set of Bézout coefficients.
Because the degree of the modular inverse obtained via the extended Euclidean algorithm is strictly less than , and , the product degree is bounded by:
Thus, , meaning the product already satisfies the degree constraint without requiring any further modular reduction, yielding directly the unique degree- basis polynomial .
For any individual simple node with no derivative conditions (), . By the polynomial remainder theorem, the modular condition simplifies to the point evaluation . In this case, reduces to a scalar constant that can be evaluated directly in closed form:
Using divided differences
For a given set of distinct nodes with associated maximum derivative orders , let be the multiplicity of node , and let be the total number of conditions. An efficient way to construct the Hermite interpolating polynomial is by extending Newton's divided difference method. Since Newton interpolation and Lagrange interpolation construct the exact same unique polynomial for a distinct set of nodes, Newton's method can be generalized to the confluent case where distinct nodes coalesce via confluent divided differences. To interpolate a function or a given set of derivative conditions (for and ), the original set of nodes is transformed into a dataset by repeating each node according to its multiplicity times:
When constructing the divided difference table for this dataset, the standard finite difference quotient for identical confluent nodes would result in an undefined expression; these entries are directly replaced by the corresponding scaled derivative values (or ). The interpolating polynomial is then represented in Newton form:
Once the target function and derivative values are known, this generalized construction requires only basic algebraic operations (addition, subtraction, multiplication, and division) to compute the divided difference table and construct the polynomial.
Through Cauchy's integral formula for confluent divided differences, constructing the Hermite interpolating polynomial via divided differences can also be viewed as a contour integration approach, providing an alternative formulation to the integral representation described in .
Simple case with only first derivatives