Gauss-Laguerre Quadrature Nodes and Weights
Enter the order n to compute the nodes (zeros of the Laguerre polynomial Lₙ) and weights of Gauss-Laguerre quadrature in a table. Useful for evaluating ∫₀^∞ e^(-x) f(x) dx numerically.
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Input
Enter an order n to compute the nodes (zeros of the Laguerre polynomial Lₙ) and weights of Gauss-Laguerre quadrature, for integrating ∫₀^∞ e^(-x) f(x) dx with weight function e^(-x).
Integer from 1 to 128. Equals the number of nodes.
Result
Order n
8
8 nodes
Sum of weights
1
Smallest node
0.1702796323
Largest node
22.8631317369
Nodes and weights
Nodes xᵢ are zeros of the Laguerre polynomial Lₙ(x); weights are wᵢ = xᵢ / ((n+1)²(L_n+1(xᵢ))²). The ''Weight × eˣ'' column is wᵢ·e^(xᵢ).
| i | Node xᵢ | Weight wᵢ | Weight × eˣ |
|---|---|---|---|
| 1 | 0.1702796323 | 0.3691885893 | 0.4377234105 |
| 2 | 0.9037017768 | 0.4187867808 | 1.0338693477 |
| 3 | 2.2510866299 | 0.1757949866 | 1.6697097657 |
| 4 | 4.2667001703 | 0.0333434923 | 2.3769247018 |
| 5 | 7.0459054024 | 0.0027945362 | 3.2085409133 |
| 6 | 10.7585160102 | 9.07650877e-5 | 4.2685755108 |
| 7 | 15.7406786413 | 8.48574672e-7 | 5.8180833687 |
| 8 | 22.8631317369 | 1.04800117e-9 | 8.9062262153 |
How it works
- Gauss-Laguerre quadrature approximates the semi-infinite integral ∫₀^∞ e^(-x) f(x) dx ≈ Σ wᵢ f(xᵢ) with the weight function e^(-x).
- The nodes xᵢ are the zeros of the order-n Laguerre polynomial Lₙ(x). This tool evaluates the polynomial via its three-term recurrence and locates each zero with Newton's method (deflating previously found roots).
- Weights use the standard formula wᵢ = xᵢ / ((n+1)² (L_{n+1}(xᵢ))²). The weights sum to 1 in exact arithmetic.
- The 'Weight × eˣ' column is wᵢ·e^(xᵢ), the effective weight when integrating ∫₀^∞ g(x) dx without folding the e^(-x) weight into the integrand.
- The rule is exact when the integrand f(x) is a polynomial of degree at most 2n−1. It is especially effective for exponentially decaying integrands.
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