`scipy.special._spfun_stats:multigammaln`

source: /scipy/special/_spfun_stats.py :42

Signature

def multigammaln ( a , d )

Summary

Returns the log of multivariate gamma, also sometimes called the generalized gamma.

Parameters

a : ndarray: The multivariate gamma is computed for each item of a.
d : int: The dimension of the space of integration.

Returns

res : ndarray: The values of the log multivariate gamma at the given points a.

Notes

The formal definition of the multivariate gamma of dimension d for a real a is

Γ_{d} (a) = \int_{A > 0} e^{- t r (A)} ∣ A ∣^{a - (d + 1) /2} d A

with the condition $a > (d - 1) /2$ , and $A > 0$ being the set of all the positive definite matrices of dimension d. Note that a is a scalar: the integrand only is multivariate, the argument is not (the function is defined over a subset of the real set).

This can be proven to be equal to the much friendlier equation

Γ_{d} (a) = π^{d (d - 1) /4} i = 1 \prod d Γ (a - (i - 1) /2) .

Array API Standard Support

multigammaln has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ✅                     ✅                   
Dask                  ✅                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

import numpy as np
from scipy.special import multigammaln, gammaln
a = 23.5
d = 10

✓

multigammaln(a, d)

✗

Verify that the result agrees with the logarithm of the equation shown above:

d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()

✗

Aliases

scipy.special.multigammaln