| lebedev {DirStats} | R Documentation |
Lebedev quadrature on the sphere
Description
Nodes and weights for Lebedev quadrature on the sphere
S^2. The rule has 5810 points and is exact up to polynomials
of order 131.
Usage
lebedev
Format
A data frame with 5810 rows and two variables:
- xyz
nodes for quadrature, a matrix with three columns.
- w
weights for quadrature, a vector.
Details
The approximation to the integral of f has the form
\int_{S^2} f(x, y, z) \,\mathrm{d}x \,\mathrm{d}y \,\mathrm{d}z =
4 \pi \sum_{i = 1}^N w_i f(x_i, y_i, z_i)
where N = 5810. The nodes (in spherical coordinates) and weights
are processed from
lebedev_131.txt.
Source
https://people.sc.fsu.edu/~jburkardt/datasets/sphere_lebedev_rule/sphere_lebedev_rule.html
References
Lebedev, V. I. and Laikov, D. N. (1999). A quadrature formula for the sphere of the 131st algebraic order of accuracy. Doklady Mathematics, 59(3):477–481.
Examples
# Load data
data("lebedev")
# Integrate x_1 * x_2^2 (zero integral)
f_1 <- function(x) x[, 1] * x[, 2]^2
4 * pi * sum(lebedev$w * f_1(lebedev$xyz))