Gauss-Hermite Quadrature Nodes & Weights
Enter the order n to list the Gauss-Hermite quadrature nodes (roots of the Hermite polynomial Hₙ) and weights for the weight function e^(-x²) in a table.
Input
Enter the order n to compute the nodes and weights of Gauss-Hermite quadrature (weight function e^(-x²)). Use them in the form ∫ f(x) e^(-x²) dx ≈ Σ wᵢ f(xᵢ), with e^(-x²) excluded from f(x).
An integer from 1 to 128, equal to the number of nodes.
Result
Order n
5
The roots of the Hermite polynomial Hₙ are used as nodes
Number of nodes
5
Sum of weights
1.7724538509
Weight function
e^(-x²)
Nodes and weights
Nodes xᵢ are in ascending order. The weights sum to the theoretical value √π ≈ 1.7724539.
| # | Node xᵢ | Weight wᵢ |
|---|---|---|
| 1 | -0.9585724646 | 0.3936193232 |
| 2 | -2.0201828705 | 0.0199532421 |
| 3 | 0 | 0.9453087205 |
| 4 | 2.0201828705 | 0.0199532421 |
| 5 | 0.9585724646 | 0.3936193232 |
How it works
- Gauss-Hermite quadrature approximates integrals over the whole real line with the weight e^(-x²): ∫_{-∞}^{∞} f(x) e^(-x²) dx ≈ Σ wᵢ f(xᵢ).
- The nodes xᵢ are the roots of the (physicists') Hermite polynomial Hₙ(x) of order n, found here by Newton's method.
- The weights are wᵢ = 2^(n-1)·n!·√π / (n²·Hₙ₋₁(xᵢ)²), computed in the log domain to avoid factorial overflow.
- Nodes and weights are symmetric about the origin. The weights sum to √π ≈ 1.7724539, which serves as an accuracy check.
- An order-n rule integrates polynomials f(x) of degree up to 2n−1 exactly.
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Gauss-Hermite Quadrature Nodes & Weights