Gauss-Laguerre Quadrature Nodes and Weights

Computes generalized Gauss-Laguerre quadrature nodes and weights via eigendecomposition of the Jacobi matrix. These are used to approximate integrals of the form \(\int_0^\infty f(x) x^\beta e^{-x} dx\).

Usage

gauss_laguerre_nodes(M, alpha_param = 0)

Arguments

M: Integer; number of quadrature nodes (at least 1, typically 40-120).
alpha_param: Numeric; Laguerre parameter (must be > -1). For standard Laguerre polynomials, use 0. For integrating against Gamma(a, b), use alpha_param = a - 1.

Value

A list with components:

nodes: Numeric vector of quadrature nodes \(x_m\).
weights: Numeric vector of quadrature weights \(w_m\).
weights_log: Numeric vector of \(\log(w_m)\) for numerical stability.

Details

The algorithm constructs a tridiagonal Jacobi matrix \(J\) of size \(M \times M\):

Diagonal: \(a_k = 2k + 1 + \alpha\) for \(k = 0, \ldots, M-1\)
Off-diagonal: \(b_k = \sqrt{k(k + \alpha)}\) for \(k = 1, \ldots, M-1\)

Eigendecomposition \(J = V D V^T\) yields:

Nodes = eigenvalues (diagonal of \(D\))
Weights = \(\Gamma(\alpha + 1) \cdot V[1,:]^2\)

The generalized Laguerre polynomials \(L_n^{(\alpha)}(x)\) are orthogonal with respect to the weight function \(w(x) = x^\alpha e^{-x}\) on \([0, \infty)\).

References

Golub, G. H., & Welsch, J. H. (1969). Calculation of Gauss Quadrature Rules. Mathematics of Computation, 23(106), 221-230.

Examples

# Standard Laguerre (alpha_param = 0)
quad <- gauss_laguerre_nodes(40)
sum(quad$weights)  # Should equal Gamma(1) = 1
#> [1] 1

# For Gamma(2.5, b) integration, use alpha_param = 1.5
quad <- gauss_laguerre_nodes(80, alpha_param = 1.5)
sum(quad$weights)  # Should equal Gamma(2.5)
#> [1] 1.32934