pracma (version 1.8.8)

gaussHermite: 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[.

Value

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

Details

Gauss-Hermite quadrature is used for integrating functions of the form $$\int_{-\infty}^{\infty} f(x) e^{-x^2} dx$$ over the infinite interval $]-\infty, \infty[$.

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

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