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:

MODULE Agm;
IMPORT
  Math := LRealMath,
  Out;

CONST
  epsilon = 1.0E-15;

PROCEDURE Of*(a,g: LONGREAL): LONGREAL;
VAR
  na,ng,og: LONGREAL;
BEGIN
  na := a; ng := g;
  LOOP
    og := ng;
    ng := Math.sqrt(na * ng); 
    na := (na + og) * 0.5;
    IF na - ng <= epsilon THEN EXIT END
  END;
  RETURN ng;
END Of;

BEGIN 
  Out.LongReal(Of(1,1 / Math.sqrt(2)),0,0);Out.Ln
END Agm.