Gauss-Legendre Quadrature Formula
Nodes and weights for the n-point Gauss-Legendre quadrature formula.
gaussLegendre(n, a, b)
n |
Number of nodes in the interval |
a, b |
lower and upper limit of the integral; must be finite. |
x and w are obtained from a tridiagonal eigenvalue problem.
List with components x, the nodes or points in[a,b], and
w, the weights applied at these nodes.
Gauss quadrature is not suitable for functions with singularities.
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.
## 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-15Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.