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# scipy.special.kl_div

scipy.special.kl_div(x, y, out=None) = <ufunc 'kl_div'>

  Elementwise function for computing Kullback-Leibler divergence.

  \[\begin{split}\mathrm{kl\_div}(x, y) = \begin{cases} x \log(x / y) - x + y & x > 0, y > 0 \\ y & x = 0, y \ge 0 \\ \infty & \text{otherwise} \end{cases}\end{split}\]
  Parameters:
    x, y array_like

      Real arguments

    out ndarray, optional

      Optional output array for the function results

  Returns:
    scalar or ndarray

      Values of the Kullback-Liebler divergence.

  See also

  `entr`, `rel_entr`, `scipy.stats.entropy`

  Notes

  Added in version 0.15.0.

  This function is non-negative and is jointly convex in x and y.

  The origin of this function is in convex programming; see [1] for details. This is why the function contains the extra \(-x + y\) terms over what might be expected from the Kullback-Leibler divergence. For a version of the function without the extra terms, see `rel_entr`.

  Array API Standard Support

  `kl_div` 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

  [1]

  Boyd, Stephen and Lieven Vandenberghe. Convex optimization. Cambridge University Press, 2004. DOI:10.1017/CBO9780511804441.