Gauss Laguerre

Gauss-Laguerre Quadrature is a numerical integration technique designed to evaluate semi-infinite definite integrals featuring exponential decay weight functions. Unlike Gauss-Legendre quadrature which targets finite intervals, Gauss-Laguerre quadrature naturally handles integration domains from zero to infinity by weighting samples according to the Laguerre polynomials. Mathematical Foundation To evaluate a semi-infinite integral with an exponential weight function over the interval $[0, \infty)$:

The nodes $x_i$ are selected as the roots of the n-th degree Laguerre polynomial $L_n(x)$. The weights $w_i$ are calculated using the derivative of the Laguerre polynomial at each node:

An n-point Gauss-Laguerre rule integrates polynomials $f(x)$ of degree up to 2n - 1 exactly when weighted by $e^{-x}$. General Domain & Weight Transformation For integrals lacking an explicit $e^{-x}$ weight or evaluated over $[0, \infty)$ with a scaling parameter $\alpha > 0$:

Applied Examples

.. Admonition:: Example 1 : Example 1: Standard Exponential-Weighted Polynomial Integration

Evaluate

var I = Integrators.GaussLag(x => Pow(x, 3));
Console.WriteLine($"I = {I:F4}");

I = 6.0000

Exact Analytical Result:

.. Admonition:: Example 2 : Example 2: Quantum Physics Atomic Orbital Integral

Evaluate

var I = GaussLag(x => x * x);
Console.WriteLine($"I = {I:F4}");

I = 2.0000

Exact Analytical Result:

Commonly used in radial wave-function integrals for hydrogen-like atom probability densities.

.. Admonition:: Example 3 : Example 3: Unweighted Semi-Infinite Exponential Decay

Evaluate

var I = GaussLag(x => Exp(-x));
Console.WriteLine($"I = {I:F4}");

I = 0.5000

Exact Analytical Result:

Computed by expressing the integrand as $e^{-x} f(x)$ where $f(x) = e^{-x}$.

.. Admonition:: Example 4 : Example 4: Trigonometric Oscillatory Decay

Evaluate :

var I = GaussLag(x => Cos(x));
Console.WriteLine($"I = {I:F4}");

I = 0.5000

Exact Analytical Result:

Demonstrates high-precision convergence on decaying trigonometric response functions.