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scipy.special.

# logsumexp

scipy.special.logsumexp(a, axis=None, b=None, keepdims=False, return_sign=False)

  Compute the log of the sum of exponentials of input elements.

  Parameters:
    a array_like

      Input array.

    axis None or int or tuple of ints, optional

      Axis or axes over which the sum is taken. By default axis is None, and all elements are summed.

      Added in version 0.11.0.

    b array-like, optional

      Scaling factor for exp(a) must be of the same shape as a or broadcastable to a. These values may be negative in order to implement subtraction.

      Added in version 0.12.0.

    keepdims bool, optional

      If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array.

      Added in version 0.15.0.

    return_sign bool, optional

      If this is set to True, the result will be a pair containing sign information; if False, results that are negative will be returned as NaN. Default is False (no sign information).

      Added in version 0.16.0.

  Returns:
    res ndarray

      The result, `np.log(np.sum(np.exp(a)))` calculated in a numerically more stable way. If b is given then `np.log(np.sum(b*np.exp(a)))` is returned. If `return_sign` is True, `res` contains the log of the absolute value of the argument.

    sgn ndarray

      If `return_sign` is True, this will be an array of floating-point numbers matching res containing +1, 0, -1 (for real-valued inputs) or a complex phase (for complex inputs). This gives the sign of the argument of the logarithm in `res`. If `return_sign` is False, only one result is returned.

  See also

  `numpy.logaddexp`
  `numpy.logaddexp2`

  Notes

  NumPy has a logaddexp function which is very similar to `logsumexp`, but only handles two arguments. logaddexp.reduce is similar to this function, but may be less stable.

  The logarithm is a multivalued function: for each \(x\) there is an infinite number of \(z\) such that \(exp(z) = x\). The convention is to return the \(z\) whose imaginary part lies in \((-pi, pi]\).

  Array API Standard Support

  `logsumexp` 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.

  Examples

      >>> import numpy as np
      >>> from scipy.special import logsumexp
      >>> a = np.arange(10)
      >>> logsumexp(a)
      9.4586297444267107
      >>> np.log(np.sum(np.exp(a)))
      9.4586297444267107

  With weights

      >>> a = np.arange(10)
      >>> b = np.arange(10, 0, -1)
      >>> logsumexp(a, b=b)
      9.9170178533034665
      >>> np.log(np.sum(b*np.exp(a)))
      9.9170178533034647

  Returning a sign flag

      >>> logsumexp([1,2],b=[1,-1],return_sign=True)
      (1.5413248546129181, -1.0)

  Notice that `logsumexp` does not directly support masked arrays. To use it on a masked array, convert the mask into zero weights:

      >>> a = np.ma.array([np.log(2), 2, np.log(3)],
      ...                  mask=[False, True, False])
      >>> b = (~a.mask).astype(int)
      >>> logsumexp(a.data, b=b), np.log(5)
      1.6094379124341005, 1.6094379124341005