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Task

Write a function to compute the arithmetic-geometric mean of two numbers.



The arithmetic-geometric mean of two numbers can be (usefully) denoted as 




a
g
m

(
a
,
g
)


{\displaystyle \mathrm {agm} (a,g)}

, and is equal to the limit of the sequence:

 




a

0


=
a
;


g

0


=
g


{\displaystyle a_{0}=a;\qquad g_{0}=g}


 




a

n
+
1


=



1
2



(

a

n


+

g

n


)
;


g

n
+
1


=



a

n



g

n




.


{\displaystyle a_{n+1}={\tfrac {1}{2}}(a_{n}+g_{n});\quad g_{n+1}={\sqrt {a_{n}g_{n}}}.}


Since the limit of 




a

n


−

g

n




{\displaystyle a_{n}-g_{n}}

 tends (rapidly) to zero with iterations, this is an efficient method.

Demonstrate the function by calculating:






a
g
m

(
1
,
1

/



2


)


{\displaystyle \mathrm {agm} (1,1/{\sqrt {2}})}




Also see

   mathworld.wolfram.com/Arithmetic-Geometric Mean


# The flt package (http://flt.rubyforge.org/) is useful for high-precision floating-point math.
# It lets us control 'context' of numbers, individually or collectively -- including precision
# (which adjusts the context's value of epsilon accordingly).
 
require 'flt'
include Flt
 
BinNum.Context.precision = 512  # default 53 (bits)
 
def agm(a,g)
  new_a = BinNum a
  new_g = BinNum g
  while new_a - new_g > new_a.class.Context.epsilon do
    old_g = new_g
    new_g = (new_a * new_g).sqrt
    new_a = (old_g + new_a) * 0.5
  end
  new_g
end
 
puts agm(1, 1 / BinNum(2).sqrt)