Gauss-Laguerre Quadrature Calculator

Enter an integrand f(x) and an order n to numerically evaluate the semi-infinite integral ∫₀^∞ f(x)e^(−x)dx by Gauss-Laguerre quadrature.

Input

Enter an integrand f(x) and an order n to compute the semi-infinite integral ∫₀^∞ f(x)e^(−x)dx by Gauss-Laguerre quadrature. The weight e^(−x) is built in, so don''t include it in f(x).

Integrand f(x)

e.g. x^2, sin(x), 1/(1+x). Do not include e^(−x). See the notes for available functions, constants, and operators.

Order n

Number of nodes (1–128)

Result

Integral ∫₀^∞ f(x)e^(−x)dx

Interval [0, ∞) with weight e^(−x)

Order n

Nodes

Nodes, weights, and function values

Shows the Laguerre roots x_i, weights w_i, function values f(x_i), and contributions w_i·f(x_i).

#	Node x_i	Weight w_i	f(x_i)	w_i·f(x_i)
1	0.17027963	0.36918859	0.02899515	0.01070468
2	0.90370178	0.41878678	0.8166769	0.34201349
3	2.25108663	0.17579499	5.06739102	0.89082194
4	4.26670017	0.03334349	18.20473034	0.60700929
5	7.0459054	0.00279454	49.64478294	0.13873414
6	10.75851601	9.076509e-5	115.74566674	0.01050567
7	15.74067864	8.485747e-7	247.76896409	0.00021025
8	22.86313174	1.048001e-9	522.72279282	5.478141e-7

How it works

Gauss-Laguerre quadrature approximates integrals over the half-line with weight e^(−x), ∫₀^∞ f(x)e^(−x)dx ≈ ∑ᵢ wᵢ f(xᵢ). The weight e^(−x) is built into the rule, so do NOT include e^(−x) in your f(x) (e.g. to evaluate ∫₀^∞ x²e^(−x)dx, enter x^2 for f(x)).
The nodes xᵢ are the roots of the degree-n Laguerre polynomial Lₙ(x). This tool evaluates the polynomial with the recurrence (k+1)L_{k+1}(x) = (2k+1−x)Lₖ(x) − kL_{k−1}(x) and locates the roots by Newton's method.
Weights are computed from wᵢ = xᵢ / ((n+1)² L_{n+1}(xᵢ)²). The weights sum to ∫₀^∞ e^(−x)dx = 1.
An order-n rule is exact in theory when f(x) is a polynomial of degree at most 2n−1 (e.g. f(x)=x^k matches ∫₀^∞ x^k e^(−x)dx = k!).
For safety the integrand is evaluated by a custom expression parser instead of eval. The only variable is x; functions such as sin, cos, tan, exp, log, ln, sqrt, abs, the constants pi and e, the four basic operations, and exponentiation ^ are supported.
If f(x) grows faster than e^(−x) decays, the integral diverges and the quadrature will not converge. Results are approximate, with accuracy depending on the integrand and the chosen order n.

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