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

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. Charles Hermite proved that this is effectively the case here, as soon as the x_i are pairwise different,^{[citation needed]} and provided a method for computing it, which is described below.

Simple case edit

When using divided differences to calculate the Hermite polynomial of a function f, the first step is to copy each point m times. (Here we will consider the simplest case ${\displaystyle m=1}$  for all points.) Therefore, given ${\displaystyle n+1}$  data points ${\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n}}$ , and values ${\displaystyle f(x_{0}),f(x_{1}),\ldots ,f(x_{n})}$  and ${\displaystyle f'(x_{0}),f'(x_{1}),\ldots ,f'(x_{n})}$  for a function ${\displaystyle f}$  that we want to interpolate, we create a new dataset

Now, we create a divided differences table for the points ${\displaystyle z_{0},z_{1},\ldots ,z_{2n+1}}$ . However, for some divided differences,

General case edit

In the general case, suppose a given point ${\displaystyle x_{i}}$  has k derivatives. Then the dataset ${\displaystyle z_{0},z_{1},\ldots ,z_{N}}$  contains k identical copies of ${\displaystyle x_{i}}$ . When creating the table, divided differences of ${\displaystyle j=2,3,\ldots ,k}$  identical values will be calculated as

For example,

Example edit

Consider the function ${\displaystyle f(x)=x^{8}+1}$ . Evaluating the function and its first two derivatives at ${\displaystyle x\in \{-1,0,1\}}$ , we obtain the following data:

Since we have two derivatives to work with, we construct the set ${\displaystyle \{z_{i}\}=\{-1,-1,-1,0,0,0,1,1,1\}}$ . Our divided difference table is then:

Quintic Hermite interpolation edit

The quintic Hermite interpolation based on the function (${\displaystyle f}$ ), its first (${\displaystyle f'}$ ) and second derivatives (${\displaystyle f''}$ ) at two different points (${\displaystyle x_{0}}$  and ${\displaystyle x_{1}}$ ) can be used for example to interpolate the position of an object based on its position, velocity and acceleration. The general form is given by

Call the calculated polynomial H and original function f. Evaluating a point ${\displaystyle x\in [x_{0},x_{n}]}$ , the error function is

• Cubic Hermite spline
• Newton series, also known as finite differences
• Neville's schema
• Bernstein polynomials

• Hermites Interpolating Polynomial at Mathworld