Step by Step solution about How to resolve the algorithm Arithmetic-geometric mean step by step in the Python programming language

Both code snippets calculate the arithmetic-geometric mean (AGM) of two numbers. The AGM is the limit of the sequence of arithmetic means and geometric means of two numbers. It is used in various applications, including number theory and probability.

The first snippet uses the math module to calculate the square root. It starts with the initial values a0 and g0 and iteratively updates an and gn until the difference between an and gn is smaller than the given tolerance. The final value of an is returned as the AGM.

The second snippet uses the decimal module to perform the calculations with higher precision. It also uses a different iteration strategy, where a and g are updated simultaneously. The iteration continues until the difference between a and g is smaller than the given tolerance. The final value of a is returned as the AGM.

Both snippets demonstrate how to calculate the AGM of two numbers using different approaches and precision levels. The choice of approach and precision depends on the specific application requirements.

from math import sqrt

def agm(a0, g0, tolerance=1e-10):
    """
    Calculating the arithmetic-geometric mean of two numbers a0, g0.

    tolerance     the tolerance for the converged 
                  value of the arithmetic-geometric mean
                  (default value = 1e-10)
    """
    an, gn = (a0 + g0) / 2.0, sqrt(a0 * g0)
    while abs(an - gn) > tolerance:
        an, gn = (an + gn) / 2.0, sqrt(an * gn)
    return an

print agm(1, 1 / sqrt(2))


from decimal import Decimal, getcontext

def agm(a, g, tolerance=Decimal("1e-65")):
    while True:
        a, g = (a + g) / 2, (a * g).sqrt()
        if abs(a - g) < tolerance:
            return a

getcontext().prec = 70
print agm(Decimal(1), 1 / Decimal(2).sqrt())