Problem Statement

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: 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:

Let's start with the solution:

fun agm(a, g) = let
  fun agm'(a, g, eps) = 
    if Real.abs(a-g) < eps then
      a
    else
      agm'((a+g)/2.0, Math.sqrt(a*g), eps)
  in agm'(a, g, 1e~15)
  end;