Q2.5.2The arithmetic-geometric mean

Write a while loop to calculate the arithmetic-geometric mean (AGM) of two positive real numbers, $$x$$ and $$y$$ , defined as the limit of the sequences:

\begin{align*} a_{n+1} &= \textstyle \frac{1}{2}(a_n + b_n)\\ b_{n+1} &= \sqrt{a_n b_n}, \end{align*}

starting with $$a_0 = x$$ , $$b_0 = y$$ . Both sequences converge to the same number, denoted $\mathrm{agm}(x,y)$ . Use your loop to determine Gauss's constant, $G = 1/\mathrm{agm}(1,\sqrt{2})$ .

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The following code calculates Gauss's constant to 14 decimal places.

>>> import math
>>> tol = 1.e-14
>>> an, bn = 1., math.sqrt(2)
>>> while abs(an - bn) > tol:
...     an, bn = (an + bn) / 2, math.sqrt(an * bn)
...
>>> print('G = {:.14f}'.format(1/an))
G = 0.83462684167407