gauss.quad(n,kind="legendre",alpha=0,beta=0)"legendre", "chebyshev1", "chebyshev2", "hermite", "jacobi" or "laguerre"a to b of w(x)*f(x) is approximated by sum(w*f(x)) where x is the vector of nodes and w is the vector of weights. The approximation is exact if f(x) is a polynomial of order no more than 2n-1.
The possible choices for w(x), a and b are as follows:
Legendre quadrature: w(x)=1 on (-1,1).
Chebyshev quadrature of the 1st kind: w(x)=1/sqrt(1-x^2) on (-1,1).
Chebyshev quadrature of the 2nd kind: w(x)=sqrt(1-x^2) on (-1,1).
Hermite quadrature: w(x)=exp(-x^2) on (-Inf,Inf).
Jacobi quadrature: w(x)=(1-x)^alpha*(1+x)^beta on (-1,1). Note that Chebyshev quadrature is a special case of this.
Laguerre quadrature: w(x)=x^alpha*exp(-x) on (0,Inf).
The algorithm used to generated the nodes and weights is explained in Golub and Welsch (1969).gauss.quad.prob, integrate# mean of gamma distribution with alpha=6
out <- gauss.quad(10,"laguerre",alpha=5)
sum(out$weights * out$nodes) / gamma(6)Run the code above in your browser using DataLab