Gauss-Laguerre Quadrature Formula

Description

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

Usage

1
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.

Details

Gauss-Laguerre quadrature is used for integrating functions of the form

\int_0^{∞} f(x) x^a e^{-x} dx

over the infinite interval ]0, ∞[.

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

Value

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

Note

The basic quadrature rules are well known and can, e. g., be found in Gautschi (2004) — and explicit Matlab realizations in Trefethen (2000). These procedures have also been implemented in Matlab by Geert Van Damme, see his entries at MatlabCentral since 2010.

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

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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

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