---
tag: DZQ4EW6907
kind: definition
name: Hermite interpolation
msc: [41A05, 65D05]
defines: 9Z50M2WPTP
uses: [SS1YMS2NPC, 0V1EJCT8G6, 0P4JRMSWNH, 047SX3TD69]
sources:
  - url: https://en.wikipedia.org/wiki/Hermite_interpolation
---

## Definition

Let $m \ge 0$, let $x_0, \dots, x_m \in \R$ be pairwise distinct, and let $k_0, \dots, k_m \ge 1$ be integers with $N = k_0 + \cdots + k_m$.
For each $i$ and each $0 \le j \le k_i - 1$ let a real number $y_i^{(j)}$ be given.
The *Hermite interpolation polynomial* of these data is the polynomial $p$ with real coefficients such that:

1. $p = 0$ or $\deg p \le N - 1$;
2. $p^{(j)}(x_i) = y_i^{(j)}$ for all $0 \le i \le m$ and $0 \le j \le k_i - 1$, where $p^{(j)}$ is the $j$-th [[derivative|0V1EJCT8G6]].

## Remarks

Such a $p$ exists and is unique.
Uniqueness: the difference $q$ of two solutions is $0$ or has [[degree|0P4JRMSWNH]] at most $N - 1$.
It has a root of [[multiplicity|047SX3TD69]] at least $k_i$ at each $x_i$, so at least $N$ roots counted with multiplicity, and hence $q = 0$.
Existence follows because the conditions form a square linear system whose only solution for zero data is $0$.
If all $k_i = 1$ this is [[polynomial interpolation|SS1YMS2NPC]]; if $m = 0$ the solution is the Taylor polynomial of order $k_0 - 1$ at $x_0$.

Example: $m = 1$, nodes $0$ and $1$, $k_0 = k_1 = 2$, data $p(0) = 0$, $p'(0) = 0$, $p(1) = 1$, $p'(1) = 0$.
Then $N = 4$ and $p(x) = 3x^2 - 2x^3$: indeed $p(0) = 0$, $p'(x) = 6x - 6x^2$ gives $p'(0) = p'(1) = 0$, and $p(1) = 1$.

Non-example: $3x^2 - 2x^3 + x^2 (x - 1)^2$ satisfies clause 2 for the same data, since the added term has double roots at $0$ and $1$, but its degree is $4 > 3$, so clause 1 fails.

The conditions must cover consecutive derivative orders $0, 1, \dots, k_i - 1$ at each node; dropping this gives [[Birkhoff interpolation|FT1YKVBFC9]], where uniqueness can fail.