int {DirStats}

Integration routines

Description

Several quadrature rules for integration of functions on S^1, S^2, and S^q, q \ge 3.

Usage

int_cir(f, N = 500, na.rm = TRUE, f_vect = TRUE, ...)

int_sph(f, na.rm = TRUE, f_vect = TRUE, ...)

int_hypsph(f, q, M = 1e+05, na.rm = TRUE, f_vect = TRUE, ...)

Arguments

f

function to be integrated on S^q. Must be vectorized and accept matrix inputs of size c(nx, q + 1).

N

Defaults to 5e2.

na.rm

ignore possible NAs arising from the evaluation of f? Defaults to TRUE.

f_vect

can f be called in a vectorized form, with matrix input? Defaults to TRUE.

...

further arguments passed to f.

q

dimension of S^q, q\ge 1.

M

number of Monte Carlo replicates. Defaults to 1e5.

Details

int_cir is an extension of equation (4.1.11) in Press et al. (1997), a periodic trapezoidal rule. int_sph employs the Lebedev quadrature on S^2. int_hypsph implements a Monte Carlo integration on S^q.

Value

A scalar approximating the integral.

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.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery B. P. (1997). Numerical Recipes in Fortran 77: The Art of Scientific Computing. Volume 1. Cambridge University Press, Cambridge. Second edition.

Examples

# S^1, trapezoidal rule
f <- function(x) rotasym::d_vMF(x = x, mu = c(0, 1), kappa = 2)
int_cir(f = f)

# S^2, Lebedev rule
f <- function(x) rotasym::d_vMF(x = x, mu = c(0, 0, 1), kappa = 2)
int_sph(f = f)

# S^2, Monte Carlo
f <- function(x) rotasym::d_vMF(x = x, mu = c(0, 0, 1), kappa = 2)
int_hypsph(f = f, q = 2)