Gaussian Triangle Quadrature
Numerically integrates a function over an arbitrary triangular domain by computing the Gauss nodes and weights.
triquad(f, x, y, n = 10, tol = 1e-10, ...)
f |
the integrand as function of two variables. |
x |
x-coordinates of the three vertices of the triangle. |
y |
y-coordinates of the three vertices of the triangle. |
n |
number of nodes. |
tol |
relative tolerance to be achieved. |
... |
additional parameters to be passed to the function. |
Computes the N^2 nodes and weights for a triangle with vertices
given by 3x2 vector. The nodes are produced by collapsing the square
to a triangle.
Then f will be applied to the nodes and the result multiplied
left and right with the weights (i.e., Gaussian quadrature).
By default, the function applies Gaussian quadrature with number of
nodes n=10,21,43,87,175 until the relative error is smaller than
the tolerance.
The integral as a scalar.
A small relative tolerance is not really indicating a small absolute tolerance.
Copyright (c) 2005 Greg von Winckel Matlab code based on the publication mentioned and available from MatlabCentral (calculates nodes and weights). Translated to R (with permission) by Hans W Borchers.
Lyness, J. N., and R. Cools (1994). A Survey of Numerical Cubature over Triangles. Proceedings of the AMS Conference “Mathematics of Computation 1943–1993”, Vancouver, CA.
x <- c(-1, 1, 0); y <- c(0, 0, 1) f1 <- function(x, y) x^2 + y^2 (I <- triquad(f1, x, y)) # 0.3333333333333333 # split the unit square x1 <- c(0, 1, 1); y1 <- c(0, 0, 1) x2 <- c(0, 1, 0); y2 <- c(0, 1, 1) f2 <- function(x, y) exp(x + y) I <- triquad(f2, x1, y1) + triquad(f2, x2, y2) # 2.952492442012557 quad2d(f2, 0, 1, 0, 1) # 2.952492442012561 simpson2d(f2, 0, 1, 0, 1) # 2.952492442134769 dblquad(f2, 0, 1, 0, 1) # 2.95249244201256
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