lebedev: Lebedev quadrature on the sphere

In DirStats: Nonparametric Methods for Directional Data

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))

DirStats documentation built on June 22, 2024, 12:16 p.m.