pracma (version 1.8.8)

gaussLaguerre: Gauss-Laguerre Quadrature Formula

Description

Nodes and weights for the n-point Gauss-Laguerre quadrature formula.

Usage

gaussLaguerre(n, a = 0)

Arguments

n

Number of nodes in the interval [0, Inf[.

a

exponent of x in the integrand: must be greater or equal to 0, otherwise the integral would not converge.

Value

• List with components x, the nodes or points in[0, Inf[, and w, the weights applied at these nodes.

Details

Gauss-Laguerre quadrature is used for integrating functions of the form $$\int_0^{\infty} f(x) x^a e^{-x} dx$$ over the infinite interval $]0, \infty[$.

x and w are obtained from a tridiagonal eigenvalue problem. The value of such an integral is then sum(w*f(x)).

References

Gautschi, W. (2004). Orthogonal Polynomials: Computation and Approximation. Oxford University Press.

Trefethen, L. N. (2000). Spectral Methods in Matlab. SIAM, Society for Industrial and Applied Mathematics.

See Also

gaussLegendre, gaussHermite

Examples

cc <- gaussLaguerre(7)
# integrate exp(-x) from 0 to Inf
sum(cc$w)                     # 1
# integrate x^2 * exp(-x)     # integral x^n * exp(-x) is n!
sum(cc$w)                     # 2
# integrate sin(x) * exp(-x)
cc <- gaussLaguerre(17, 0)    # we need more nodes
sum(cc$w * sin(cc$x))         #=> 0.499999999994907 , should be 0.5