Hermite interpolation

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Hermite interpolation is a method closely related to the Newton divided difference method of interpolation in numerical analysis, that allows us to consider given derivatives at data points, as well as the data points themselves. The interpolation will give a polynomial that has a degree less than or equal to the number of both data points and their derivatives, minus 1.

1 Usage
2 Example
3 Error
4 See also
5 External links

[edit] Usage

The derivatives are treated as extra points, and in the divided difference table, the points are repeated. To avoid division by zero, the values where the division by zero would take place are replaced with the derivatives, multiplied by a constant, depending on the position in the table. For example, using the notation on the Newton polynomial article, if point $x i$ is repeated $n$ times, $[x i, x i,..., x i] = f (n - 1) (x i) / (n - 1)!$ , e.g.

[x i, x i, x i, x i] = f (3) (x i) / 3!

[x i, x i, x i] = f (2) (x i) / 2!

etc.

The table is calculated in the exact same fashion as before.

[edit] Example

The example used here will be the polynomial $x 8 + 1$ . The values, first, and second derivatives at the points $x = - 1$ , $x = 0$ , and $x = 1$ will be used. This means that 9 pieces of data will be used, and so the polynomial discovered will be of degree 8.

-1      2
-1      2       -8
-1      2       -8      28
0       1       -1      7       -21
0       1       0       1       -6      15
0       1       0       0       -1      5       -10
1       2       1       1       1       1       -2      4
1       2       8       7       6       5       2       2       -1
1       2       8       28      21      15      10      4       1       1

P (x) = 2 - 8(x + 1) + 28(x + 1) 2 - 21(x + 1) 3 + 15 x (x + 1) 3 - 10 x 2 (x + 1) 3 + 4 x 3 (x + 1) 3 - 1 x 3 (x + 1) 3 (x - 1) + x 3 (x + 1) 3 (x - 1) 2

= 2 - 8 + 28 - 21 - 8 x + 56 x - 63 x + 15 x + 28 x 2 - 63 x 2 + 45 x 2 - 10 x 2 - 21 x 3 + 45 x 3 - 30 x 3 + 4 x 3 + x 3 + x 3 + 15 x 4 - 30 x 4 + 12 x 4 + 2 x 4 + x 4 - 10 x 5 + 12 x 5 - 2 x 5 + 4 x 5 - 2 x 5 - 2 x 5 - x 6 + x 6 - x 7 + x 7 + x 8

= x 8 + 1.

[edit] Error

The error of the function when used to approximate the value at a point is always going to be for some point $c$ between the furthest $x$ -value used and the $x$ -value approximated: $\frac{f^{(a)}(c)}{a!}\prod_{i=0}^n (x-x_i)^{N_i}$ where $a$ is the number of pieces of data, $n$ is the number of $x$ -values minus 1, and $N i$ is the number of pieces of data used at $x i$ . This is because the function cannot change more quickly from the estimated Hermite interpolation polynomial than its $a$ -th derivative divided by $a!$ multiplied by the distance of the point of evaluation from the known points.