# gaussLaguerre: Gauss-Laguerre Quadrature Formula In pracma: Practical Numerical Math Functions

### 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.

gaussLegendre, gaussHermite
 1 2 3 4 5 6 7 8 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