gaussLegendre: Gauss-Legendre Quadrature Formula

Description

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

Usage

gaussLegendre(n, a, b)

Arguments

Number of nodes in the interval [a,b].

a, b

lower and upper limit of the integral; must be finite.

Value

List with components x, the nodes or points in[a,b], and w, the weights applied at these nodes.

Details

x and w are obtained from a tridiagonal eigenvalue problem.

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.

Examples

Run this code

##  Quadrature with Gauss-Legendre nodes and weights
f <- function(x) sin(x+cos(10*exp(x))/3)
#\dontrun{ezplot(f, -1, 1, fill = TRUE)}
cc <- gaussLegendre(51, -1, 1)
Q <- sum(cc$w * f(cc$x))  #=> 0.0325036515865218 , true error: < 1e-15

# If f is not vectorized, do an explicit summation:
Q <- 0; x <- cc$x; w <- cc$w
for (i in 1:51) Q <- Q + w[i] * f(x[i])

# If f is infinite at b = 1, set  b <- b - eps  (with, e.g., eps = 1e-15)

# Use Gauss-Kronrod approach for error estimation
cc <- gaussLegendre(103, -1, 1)
abs(Q - sum(cc$w * f(cc$x)))     # rel.error < 1e-10

# Use Gauss-Hermite for vector-valued functions
f <- function(x) c(sin(pi*x), exp(x), log(1+x))
cc <- gaussLegendre(32, 0, 1)
drop(cc$w %*% matrix(f(cc$x), ncol = 3))  # c(2/pi, exp(1) - 1, 2*log(2) - 1)
# absolute error < 1e-15

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