standardSimplex_Dirichlet-class: An S4 class to represent the function...

In multIntTestFunc: Provides Test Functions for Multivariate Integration

An S4 class to represent the function \prod_{i=1}^{n}x_i^{v_i-1}(1 - x_1 - \ldots - x_n)^{v_{n+1}-1} on T_n

Description

Implementation of the function

f \colon T_n \to (0,\infty),\, \vec{x} \mapsto f(\vec{x}) = \prod_{i=1}^{n}x_i^{v_i-1}(1 - x_1 - \ldots - x_n)^{v_{n+1}-1},

where n \in \{1,2,3,\ldots\} is the dimension of the integration domain T_n = \{\vec{x} \in \R^n : x_i\geq 0, \Vert \vec{x} \Vert_1 \leq 1\} and v_i>0, i=1,\ldots,n+1, are constants. The integral is known to be

\int_{T_n} f(\vec{x}) d\vec{x} = \frac{\prod_{i=1}^{n+1}\Gamma(v_i)}{\Gamma(\sum_{i=1}^{n+1}v_i)},

where v_i>0 for i=1,\ldots,n+1.

Details

The instance needs to be created with two parameters representing the dimension n and the vector of positive parameters.

Slots

dim

An integer that captures the dimension

v

A vector of dimension n+1 with positive entries representing the constants

Author(s)

Klaus Herrmann

Examples

n <- as.integer(3)
f <- new("standardSimplex_Dirichlet",dim=n,v=c(1,2,3,4))

multIntTestFunc documentation built on Sept. 11, 2024, 5:18 p.m.