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scipy.special.

# multigammaln

scipy.special.multigammaln(a, d)

  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

  \[\Gamma_d(a) = \int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA\]

  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

  \[\Gamma_d(a) = \pi^{d(d-1)/4} \prod_{i=1}^{d} \Gamma(a - (i-1)/2).\]

  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 Support for the array API standard for more information.

  References

  R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in probability and mathematical statistics).

  Examples

      >>> import numpy as np
      >>> from scipy.special import multigammaln, gammaln
      >>> a = 23.5
      >>> d = 10
      >>> multigammaln(a, d)
      454.1488605074416

  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()
      454.1488605074416