ndim 0.1.0

No project description provided

ndim

Useful recurrence relations for multidimensional volumes and monomial integrals.

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

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. $$ \begin{align}\nonumber I_{k_1,\dots,k_n} &= \int_{T_n} x_1^{k_1}\cdots x_n^{k_n}\\ &= \frac{\prod_{i=1}^n\Gamma(k_i)}{\Gamma\left(\sum_{i=1}^n k_i\right)}\label{simplex:closed}\\ &=\begin{cases} |T_n|&\text{if all $k_i=0$}\\ I_{k_1,\dots,k_{i_0}-1,\dots,k_n} \times \frac{k_{i_0}}{n + \sum_{i=1}^n k_i}&\text{if $k_{i_0} > 0$} \end{cases}\label{simplex:rec} \end{align} $$

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, $$ \frac{\prod_{i=1}^n\Gamma(k_i)}{\Gamma\left(\sum_{i=1}^n k_i\right)} = \exp\left(\sum_{i=1}^n\ln\Gamma(k_i) - \ln\Gamma\left(\sum_{i=1}^n k_i\right)\right), $$ 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

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

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

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

n-dimensional unit ball with Gegenbauer weight

$\lambda > -1$.

• Volume. $$ \begin{align}\nonumber |G_n^{\lambda}| &= \int_{S^n} \left(1 - \sum_{i=1}^n x_i^2\right)^\lambda\\ &= \frac{% \Gamma(1+\lambda)\sqrt{\pi}^n }{% \Gamma\left(1+\lambda + \frac{n}{2}\right) } = \begin{cases} 1&\text{for $n=0$}\\ B\left(\lambda + 1, \frac{1}{2}\right)&\text{for $n=1$}\\ |G_{n-2}^{\lambda}|\times \frac{2\pi}{2\lambda + n}&\text{otherwise} \end{cases} \end{align} $$
• Monomial integration. $$ \begin{align}\nonumber I_{k_1,\dots,k_n} &= \int_{S^n} x_1^{k_1}\cdots x_n^{k_n} \left(1 - \sum_{i=1}^n x_i^2\right)^\lambda\\ &= \frac{% \Gamma(1+\lambda)\prod_{i=1}^n \Gamma\left(\frac{k_i+1}{2}\right) }{% \Gamma\left(1+\lambda + \sum_{i=1}^n \frac{k_i+1}{2}\right) }\\ &= \begin{cases} 0&\text{if any $k_i$ is odd}\\ |G_n^{\lambda}|&\text{if all $k_i=0$}\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{2\lambda + n + \sum_{i=1}^n k_i}&\text{if $k_{i_0} > 0$} \end{cases} \end{align} $$

n-dimensional unit ball with Chebyshev-1 weight

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

• Volume. $$ \begin{align}\nonumber |G_n^{-1/2}| &= \int_{S^n} \frac{1}{\sqrt{1 - \sum_{i=1}^n x_i^2}}\\ &= \frac{% \sqrt{\pi}^{n+1} }{% \Gamma\left(\frac{n+1}{2}\right) } =\begin{cases} 1&\text{if $n=0$}\\ \pi&\text{if $n=1$}\\ |G_{n-2}^{-1/2}| \times \frac{2\pi}{n-1}&\text{otherwise} \end{cases} \end{align} $$
• Monomial integration. $$ \begin{align}\nonumber I_{k_1,\dots,k_n} &= \int_{S^n} \frac{x_1^{k_1}\cdots x_n^{k_n}}{\sqrt{1 - \sum_{i=1}^n x_i^2}}\\ &= \frac{% \sqrt{\pi} \prod_{i=1}^n \Gamma\left(\frac{k_i+1}{2}\right) }{% \Gamma\left(\frac{1}{2} + \sum_{i=1}^n \frac{k_i+1}{2}\right) }\ &= \begin{cases} 0&\text{if any $k_i$ is odd}\\ |G_n^{-1/2}|&\text{if all $k_i=0$}\\ I_{k_1,\dots,k_{i_0}-2,\dots,k_n} \times \frac{k_{i_0}-1}{n-1 + \sum_{i=1}^n k_i}&\text{if $k_{i_0} > 0$} \end{cases} \end{align} $$

n-dimensional unit ball with Chebyshev-2 weight

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

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

n-dimensional generalized Laguerre volume

$\alpha > -1$.

• Volume $$ \begin{align}\nonumber V_n &= \int_{\mathbb{R}^n} \left(\sqrt{x_1^2+\cdots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\ &= \frac{2 \sqrt{\pi}^n \Gamma(n+\alpha)}{\Gamma(\frac{n}{2})} = \begin{cases} 2\Gamma(1+\alpha)&\text{if $n=1$}\\ 2\pi\Gamma(2 + \alpha)&\text{if $n=2$}\\ V_{n-2} \times \frac{2\pi(n+\alpha-1) (n+\alpha-2)}{n-2}&\text{otherwise} \end{cases} \end{align} $$
• Monomial integration. $$ \begin{align}\nonumber I_{k_1,\dots,k_n} &= \int_{\mathbb{R}^n} x_1^{k_1}\cdots x_n^{k_n} \left(\sqrt{x_1^2+\dots+x_n^2}\right)^\alpha \exp\left(-\sqrt{x_1^2+\dots+x_n^2}\right)\\ &= \frac{% 2 \Gamma\left(\alpha + n + \sum_{i=1}^n k_i\right) \left(\prod_{i=1}^n\Gamma\left(\frac{k_i + 1}{2}\right)\right) }{% \Gamma\left(\sum_{i=1}^n\frac{k_i + 1}{2}\right) }\\ &=\begin{cases} 0&\text{if any $k_i$ is odd}\\ V_n&\text{if all $k_i=0$}\\ I_{k_1,\dots,k_{i_0}-2,\ldots,k_n} \times \frac{% (\alpha + n + p - 1) (\alpha + n + p - 2) (k_{i_0} - 1) }{% n + p - 2 }&\text{if $k_{i_0} > 0$} \end{cases} \end{align} $$ with $p=\sum_{k=1}^n k_i$.

n-dimensional Hermite (physicists')

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

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

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

License

This software is published under the GPLv3 license.