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:

procedure main(A)
    a := real(A[1]) | 1.0
    g := real(A[2]) | (1 / 2^0.5)
    epsilon := real(A[3])
    write("agm(",a,",",g,") = ",agm(a,g,epsilon))
end

procedure agm(an, gn, e)
    /e := 1e-15
    while abs(an-gn) > e do {
       ap := (an+gn)/2.0
       gn := (an*gn)^0.5
       an := ap
       }
    return an
end