ndim 0.1.6

Multidimensional volumes and monomial integrals

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.

Install with

pip install ndim

and use like

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=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}
  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 + 1)}{\Gamma\left(n + 1 + \sum_{i=1}^n
  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=1}^n 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=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)}\\\\
  &=\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}
  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 $p=\sum_{i=1}^n k_i$.

n-dimensional unit ball with Gegenbauer weight

$\lambda > -1$.

• Volume.

    \begin{align}
    |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}
  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}
|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}
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}
|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}
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}
  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=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}
  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=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}
  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}

Testing

To run the meshio unit tests, check out this repository and type

tox

License

This software is published under the GPLv3 license.