Multidimensional volumes and monomial integrals.

ndim computes all kinds of volumes and integrals of monomials over such volumes in a fast, numerically stable way, using recurrence relations.

Installation

Install ndim from PyPI with

pip install ndim

How to get a license

Licenses for personal and academic use can be purchased here. You'll receive a confirmation email with a license key. Install the key with

slim install <your-license-key>

on your machine and you're good to go.

For commercial use, please contact [email protected].

Use ndim

import ndim

val = ndim.nball.volume(17)
print(val)

val = ndim.nball.integrate_monomial((4, 10, 6, 0, 2), lmbda=-0.5)
print(val)

# or nsphere, enr, enr2, ncube, nsimplex

0.14098110691713894
1.0339122278806983e-07

All functions have the symbolic argument; if set to True, computations are performed symbolically.

import ndim

vol = ndim.nball.volume(17, symbolic=True)
print(vol)

512*pi**8/34459425

The formulas

A PDF version of the text can be found here.

This note gives closed formulas and recurrence expressions for many $n$-dimensional volumes and monomial integrals. The recurrence expressions are often much simpler, more instructive, and better suited for numerical computation.

n-dimensional unit cube

$$C_n = \left\{(x_1,\dots,x_n): -1 \le x_i \le 1\right\}$$

• Volume.

$$|C_n| = 2^n = \begin{cases} 1&\text{if $n=0$}\\\ |C_{n-1}| \times 2&\text{otherwise} \end{cases}$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{C_n} x_1^{k_1}\cdots x_n^{k_n}\\\ &= \prod_i \frac{1 + (-1)^{k_i}}{k_i+1} =\begin{cases} 0&\text{if any $k_i$ is odd}\\\ |C_n|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{k_{i_0}+1}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

n-dimensional unit simplex

$$T_n = \left\{(x_1,\dots,x_n):x_i \geq 0, \sum_{i=1}^n x_i \leq 1\right\}$$

• Volume.

$$|T_n| = \frac{1}{n!} = \begin{cases} 1&\text{if $n=0$}\\\ |T_{n-1}| \times \frac{1}{n}&\text{otherwise} \end{cases}$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{T_n} x_1^{k_1}\cdots x_n^{k_n}\\\ &= \frac{\prod_i\Gamma(k_i + 1)}{\Gamma\left(n + 1 + \sum_i k_i\right)}\\\ &=\begin{cases} |T_n|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-1,\dots,k_n} \times \frac{k_{i_0}}{n + \sum_i k_i}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

Remark

Note that both numerator and denominator in the closed expression will assume very large values even for polynomials of moderate degree. This can lead to difficulties when evaluating the expression on a computer; the registers will overflow. A common countermeasure is to use the log-gamma function,

$$\frac{\prod_i\Gamma(k_i)}{\Gamma\left(\sum_i k_i\right)} = \exp\left(\sum_i \ln\Gamma(k_i) - \ln\Gamma\left(\sum_i k_i\right)\right),$$

but a simpler and arguably more elegant solution is to use the recurrence. This holds true for all such expressions in this note.

n-dimensional unit sphere (surface)

$$U_n = \left\{(x_1,\dots,x_n): \sum_{i=1}^n x_i^2 = 1\right\}$$

• Volume.

$$|U_n| = \frac{n \sqrt{\pi}^n}{\Gamma(\frac{n}{2}+1)} = \begin{cases} 2&\text{if $n = 1$}\\\ 2\pi&\text{if $n = 2$}\\\ |U_{n-2}| \times \frac{2\pi}{n - 2}&\text{otherwise} \end{cases}$$

• Monomial integral.

$$\begin{align*} I_{k_1,\dots,k_n} &= \int_{U_n} x_1^{k_1}\cdots x_n^{k_n}\\\ &= \frac{2\prod_i \Gamma\left(\frac{k_i+1}{2}\right)}{\Gamma\left(\sum_i \frac{k_i+1}{2}\right)}\\\\\ &=\begin{cases} 0&\text{if any $k_i$ is odd}\\\ |U_n|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{n - 2 + \sum_i k_i}&\text{if $k_{i_0} > 0$} \end{cases} \end{align*}$$

n-dimensional unit ball

$$S_n = \left\{(x_1,\dots,x_n): \sum_{i=1}^n x_i^2 \le 1\right\}$$

• Volume.

  |S_n|
  = \frac{\sqrt{\pi}^n}{\Gamma(\frac{n}{2}+1)}
  = \begin{cases}
     1&\text{if $n = 0$}\\
     2&\text{if $n = 1$}\\
     |S_{n-2}| \times \frac{2\pi}{n}&\text{otherwise}
  \end{cases}
  

• Monomial integral.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{S_n} x_1^{k_1}\cdots x_n^{k_n}\\\ &= \frac{2^{n + p}}{n + p} |S_n| =\begin{cases} 0&\text{if any $k_i$ is odd}\\\ |S_n|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{n + p}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

with $p=\sum_i k_i$.

n-dimensional unit ball with Gegenbauer weight

$\lambda > -1$.

• Volume.

$$\begin{align} |G_n^{\lambda}| &= \int_{S^n} \left(1 - \sum_i x_i^2\right)^\lambda\\\ &= \frac{% \Gamma(1+\lambda)\sqrt{\pi}^n }{% \Gamma\left(1+\lambda + \frac{n}{2}\right) } = \begin{cases} 1&\text{for $n=0$}\\\ B\left(\lambda + 1, \frac{1}{2}\right)&\text{for $n=1$}\\\ |G_{n-2}^{\lambda}|\times \frac{2\pi}{2\lambda + n}&\text{otherwise} \end{cases} \end{align}$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{S^n} x_1^{k_1}\cdots x_n^{k_n} \left(1 - \sum_i x_i^2\right)^\lambda\\\ &= \frac{% \Gamma(1+\lambda)\prod_i \Gamma\left(\frac{k_i+1}{2}\right) }{% \Gamma\left(1+\lambda + \sum_i \frac{k_i+1}{2}\right) }\\\ &= \begin{cases} 0&\text{if any $k_i$ is odd}\\\ |G_n^{\lambda}|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{2\lambda + n + \sum_i k_i}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

n-dimensional unit ball with Chebyshev-1 weight

Gegenbauer with $\lambda=-\frac{1}{2}$.

• Volume.

$$\begin{align} |G_n^{-1/2}| &= \int_{S^n} \frac{1}{\sqrt{1 - \sum_i x_i^2}}\\\ &= \frac{% \sqrt{\pi}^{n+1} }{% \Gamma\left(\frac{n+1}{2}\right) } =\begin{cases} 1&\text{if $n=0$}\\\ \pi&\text{if $n=1$}\\\ |G_{n-2}^{-1/2}| \times \frac{2\pi}{n-1}&\text{otherwise} \end{cases} \end{align}$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{S^n} \frac{x_1^{k_1}\cdots x_n^{k_n}}{\sqrt{1 - \sum_i x_i^2}}\\\ &= \frac{% \sqrt{\pi} \prod_i \Gamma\left(\frac{k_i+1}{2}\right) }{% \Gamma\left(\frac{1}{2} + \sum_i \frac{k_i+1}{2}\right) }\\\ &= \begin{cases} 0&\text{if any $k_i$ is odd}\\\ |G_n^{-1/2}|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{n-1 + \sum_i k_i}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

n-dimensional unit ball with Chebyshev-2 weight

Gegenbauer with $\lambda = +\frac{1}{2}$.

• Volume.

$$\begin{align} |G_n^{+1/2}| &= \int_{S^n} \sqrt{1 - \sum_i x_i^2}\\\ &= \frac{% \sqrt{\pi}^{n+1} }{% 2\Gamma\left(\frac{n+3}{2}\right) } = \begin{cases} 1&\text{if $n=0$}\\\ \frac{\pi}{2}&\text{if $n=1$}\\\ |G_{n-2}^{+1/2}| \times \frac{2\pi}{n+1}&\text{otherwise} \end{cases} \end{align}$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{S^n} x_1^{k_1}\cdots x_n^{k_n} \sqrt{1 - \sum_i x_i^2}\\\ &= \frac{% \sqrt{\pi}\prod_i \Gamma\left(\frac{k_i+1}{2}\right) }{% 2\Gamma\left(\frac{3}{2} + \sum_i \frac{k_i+1}{2}\right) }\\\ &= \begin{cases} 0&\text{if any $k_i$ is odd}\\\ |G_n^{+1/2}|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{n + 1 + \sum_i k_i}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

n-dimensional generalized Cauchy volume

As appearing in the Cauchy distribution and Student's t-distribution.

• Volume. $2 \lambda > n$.

$$\begin{align} |Y_n^{\lambda}| &= \int_{\mathbb{R}^n} \left(1 + \sum_i x_i^2\right)^{-\lambda}\\\ &= |U_{n-1}| \frac{1}{2} B(\lambda - \frac{n}{2}, \frac{n}{2})\\\ &= \begin{cases} 1&\text{for $n=0$}\\\ B\left(\lambda - \frac{1}{2}, \frac{1}{2}\right)&\text{for $n=1$}\\\ |Y_{n-2}^{\lambda}|\times \frac{2\pi}{2\lambda - n}&\text{otherwise} \end{cases} \end{align}$$

• Monomial integration. $2 \lambda > n + \sum_i k_i$.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \left(1 + \sum_i x_i^2\right)^{-\lambda}\\\ &= \frac{\Gamma(\frac{n+\sum k_i}{2}) \Gamma(\lambda - \frac{n - \sum k_i}{2})}{2 \Gamma(\lambda)} \times \frac{2\prod_i \Gamma(\tfrac{k_i+1}{2})}{\Gamma(\sum_i \tfrac{k_i+1}{2})}\\\ &= \begin{cases} 0&\text{if any $k_i$ is odd}\\\ |Y_n^{\lambda}|&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{2\lambda - \left(n + \sum_i k_i\right)}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

n-dimensional generalized Laguerre volume

$\alpha > -1$.

• Volume

$$\begin{align} V_n &= \int_{\mathbb{R}^n} \left(\sqrt{x_1^2+\cdots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\\ &= \frac{2 \sqrt{\pi}^n \Gamma(n+\alpha)}{\Gamma(\frac{n}{2})} = \begin{cases} 2\Gamma(1+\alpha)&\text{if $n=1$}\\\ 2\pi\Gamma(2 + \alpha)&\text{if $n=2$}\\\ V_{n-2} \times \frac{2\pi(n+\alpha-1) (n+\alpha-2)}{n-2}&\text{otherwise} \end{cases} \end{align}$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \left(\sqrt{x_1^2+\dots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\\ &= \frac{% 2 \Gamma\left(\alpha + n + \sum_i k_i\right) \left(\prod_i \Gamma\left(\frac{k_i + 1}{2}\right)\right) }{% \Gamma\left(\sum_i \frac{k_i + 1}{2}\right) }\\\ &=\begin{cases} 0&\text{if any $k_i$ is odd}\\\ V_n&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\ldots,k_n} \times \frac{% (\alpha + n + p - 1) (\alpha + n + p - 2) (k_{i_0} - 1) }{% n + p - 2 }&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

with $p=\sum_i k_i$.

n-dimensional Hermite (physicists')

• Volume.

$$\begin{align} V_n &= \int_{\mathbb{R}^n} \exp\left(-(x_1^2+\cdots+x_n^2)\right)\\\ &= \sqrt{\pi}^n = \begin{cases} 1&\text{if $n=0$}\\\ \sqrt{\pi}&\text{if $n=1$}\\\ V_{n-2} \times \pi&\text{otherwise} \end{cases} \end{align}$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \exp(-(x_1^2+\cdots+x_n^2))\\\ &= \prod_i \frac{(-1)^{k_i} + 1}{2} \times \Gamma\left(\frac{k_i+1}{2}\right)\\\ &=\begin{cases} 0&\text{if any $k_i$ is odd}\\\ V_n&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0} - 1}{2}&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$

n-dimensional Hermite (probabilists')

• Volume.

$$V_n = \frac{1}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} \exp\left(-\frac{1}{2}(x_1^2+\cdots+x_n^2)\right) = 1$$

• Monomial integration.

$$\begin{align} I_{k_1,\dots,k_n} &= \frac{1}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \exp\left(-\frac{1}{2}(x_1^2+\cdots+x_n^2)\right)\\\ &= \prod_i \frac{(-1)^{k_i} + 1}{2} \times \frac{2^{\frac{k_i+1}{2}}}{\sqrt{2\pi}} \Gamma\left(\frac{k_i+1}{2}\right)\\\ &=\begin{cases} 0&\text{if any $k_i$ is odd}\\\ V_n&\text{if all $k_i=0$}\\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times (k_{i_0} - 1)&\text{if $k_{i_0} > 0$} \end{cases} \end{align}$$