R: Gauss-Hermite Quadrature Formula

gaussHermite {pracma} | R Documentation

## Gauss-Hermite Quadrature Formula

### Description

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

### Usage

    gaussHermite(n)

### Arguments

`n` | Number of nodes in the interval `]-Inf, Inf[`.

### Details

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

\int_{-∞}^{∞} f(x) e^{-x^2} dx

over the infinite interval ]-∞, ∞[.

`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`]-Inf, 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`, `gaussLaguerre`

### Examples

    cc <- gaussHermite(17)
    # Integrate  exp(-x^2)  from -Inf to Inf
    sum(cc$w)                        #=> 1.77245385090552 == sqrt(pi)
    # Integrate  x^2 exp(-x^2)
    sum(cc$w * cc$x^2)               #=> 0.88622692545276 == sqrt(pi) /2
    # Integrate  cos(x) * exp(-x^2)
    sum(cc$w * cos(cc$x))            #=> 1.38038844704314 == sqrt(pi)/exp(1)^0.25

[Package pracma version 2.0.4 Index]