# scipy.special.agm

scipy.special.agm(a, b, out=None) = <ufunc 'agm'>

  Compute the arithmetic-geometric mean of a and b.

  Start with a_0 = a and b_0 = b and iteratively compute:

      a_{n+1} = (a_n + b_n)/2
      b_{n+1} = sqrt(a_n*b_n)

  a_n and b_n converge to the same limit as n increases; their common limit is agm(a, b).

  Parameters:
    a, b array_like

      Real values only. If the values are both negative, the result is negative. If one value is negative and the other is positive, nan is returned.

    out ndarray, optional

      Optional output array for the function values

  Returns:
    scalar or ndarray

      The arithmetic-geometric mean of a and b.

  Examples

      >>> import numpy as np
      >>> from scipy.special import agm
      >>> a, b = 24.0, 6.0
      >>> agm(a, b)
      13.458171481725614

  Compare that result to the iteration:

      >>> while a != b:
      ...     a, b = (a + b)/2, np.sqrt(a*b)
      ...     print("a = %19.16f  b=%19.16f" % (a, b))
      ...
      a = 15.0000000000000000  b=12.0000000000000000
      a = 13.5000000000000000  b=13.4164078649987388
      a = 13.4582039324993694  b=13.4581390309909850
      a = 13.4581714817451772  b=13.4581714817060547
      a = 13.4581714817256159  b=13.4581714817256159

  When array-like arguments are given, broadcasting applies:

      >>> a = np.array([[1.5], [3], [6]])  # a has shape (3, 1).
      >>> b = np.array([6, 12, 24, 48])    # b has shape (4,).
      >>> agm(a, b)
      array([[  3.36454287,   5.42363427,   9.05798751,  15.53650756],
             [  4.37037309,   6.72908574,  10.84726853,  18.11597502],
             [  6.        ,   8.74074619,  13.45817148,  21.69453707]])