scipy.special.agm

scipy.special.agm(a, b) = <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.

Returns:

float

The arithmetic-geometric mean of a and b.

Examples

>>> 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]])