Gauss-Legendre Quadrature Formula

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

gaussLegendre(n, a, b)

Arguments

n

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

a, b

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

Details

x and w are obtained from a tridiagonal eigenvalue problem.

Value

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

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.

Note

Gauss quadrature is not suitable for functions with singularities.

Examples

##  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
#> [1] 9.327597e-11

# 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)
#> [1] 0.6366198 1.7182818 0.3862944
# absolute error < 1e-15