Hermite Polynomials

Hermite polynomials are a special case of osculating polynomials. These polynomials are capable of replicating both the function values and their derivatives at a given set of nodes.

Definition

For a set of \(n+1\) points, the Hermite polynomial \(H_{2n+1}(x)\) is the unique polynomial of least degree (\(2n+1\)) such that:

• \(P(x_i) = f(x_i)\) for all \(i = 0, \dots, n\).
• \(P^\prime(x_i) = f^\prime(x_i)\) for all \(i = 0, \dots, n\).

Construction using Lagrange Polynomials

It can be expressed as:

\[ P_{2n +1}(x) = \sum\limits_{j = 0}^nf(x_j)H_{n, j}(x) + \sum\limits_{j = 0}^nf^\prime(x_j)\hat{H}_{n, j}(x) \]

Where: - \(H_{n, j}(x) = [1-2(x-x_j)L^\prime_{n, j}(x_j)] L^2_{n, j}(x)\) - \(\hat{H}_{n, j}(x) = (x - x_j)L^2_{n, j}(x)\)

\(L_{n,j}\) represents the basic Lagrange polynomials.

Construction using Divided Differences

Newton's method can be extended. We define a new sequence \(z_i\) such that each node is repeated: $$ z_{2i} = z_{2i+1} = x_i $$

To resolve the indeterminacy at repeated points, we use the derivative: $$ f[z_{2i}, z_{2i+1}] = f^\prime(x_i) $$

Then, the polynomial is constructed normally: $$ P_{2n + 1}(x) = f[z_0] + \sum_{k = 1}^{2n+1}\left( f[z_0, z_1, \dots, z_k] \prod_{i = 0}^{k-1}(x-z_{i}) \right) $$

Step-by-Step Example

Given points \(x_0 = -1, x_1= 0, x_2 = 2\): - \(f(x_0) = 5, f(x_1) = 2, f(x_2) = -4\) - \(f^\prime(x_0) = 12, f^\prime(x_1) = -7, f^\prime(x_2) = -51\)

1. Lagrange:
2. \(L_0 = (x^2 - 2x)/3, L^\prime_0(-1) = -4/3\)
3. \(L_1 = (x^2 -x-2)/(-2), L^\prime_1(0) = 1/2\)
4. \(L_2 = (x^2 + x)/6, L^\prime_2(2) = 5/6\)
5. Hermite Components:
6. \(H_0 = [1 - 2(x+1)(-4/3)]L_0^2\)
7. \(H_1 = [1 - 2(x)(1/2)]L_1^2\)
8. \(H_2 = [1 - 2(x-2)(5/6)]L_2^2\)
9. Result: The final \(P_5(x)\) polynomial sums these contributions multiplied by \(f(x_j)\) and \(f^\prime(x_j)\).