# gaussHermite: Gauss-Hermite Quadrature Formula In pracma: Practical Numerical Math Functions

## Description

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

## Usage

 1 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.

gaussLegendre, gaussLaguerre

## Examples

 1 2 3 4 5 6 7 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 

### Example output

[1] 1.772454
[1] 0.8862269
[1] 1.380388


pracma documentation built on Dec. 11, 2021, 9:57 a.m.