R: Gauss-Laguerre Quadrature Formula

gaussLaguerre {pracma} | R Documentation

## 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.

### 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

    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 * cc$x^2)            # 2
    # integrate sin(x) * exp(-x)
    cc <- gaussLaguerre(17, 0)    # we need more nodes
    sum(cc$w * sin(cc$x))         #=> 0.499999999994907 , should be 0.5

[Package pracma version 2.0.4 Index]