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:

include ..\Utilitys.tlhy

:agm [ over over + 2 / rot rot * sqrt ] [ over over tostr swap tostr # ] while drop ;

1 1 2 sqrt / agm

pstack

" " input

include ..\Utilitys.tlhy

:agm %a %g %p !p !g !a
	$p $a $g - abs > ( [$a] [.5 $a $g + * $a $g * sqrt $p agm] ) if ;

1 .5 sqrt 1e-15 agm

pstack

" " input