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:

#!/usr/bin/awk -f
BEGIN {
	printf "%.16g\n", agm(1.0,sqrt(0.5))
}
function agm(a,g) {
	while (1) {
		a0=a
		a=(a0+g)/2
		g=sqrt(a0*g)
		if (abs(a0-a) < abs(a)*1e-15) break
	}
	return a
}
function abs(x) {
	return (x<0 ? -x : x)
}