MAPLE output for the Gauss-Legendre Method for estimating Pi.

Let's just run the loop one time and see what happens...

> restart; a:=1; b:=1/sqrt(2); t:=1/4; x:=1;

> for i from 1 to 1 do

> y:=a;

> a:=(a+b)/2;

> b:=sqrt(b*y);

> t:=t-x*(y-a)^2;

> x:=2*x;

> od;

> evalf(a); evalf(b); evalf(t); evalf(x);

> evalf(((a+b)^2)/(4*t));

This is not a bad estimate for Pi, but let's run the loop for two iterations (this time I will hide all of the output except the answer):

> restart; a:=1: b:=1/sqrt(2): t:=1/4: x:=1:

> for i from 1 to 2 do

> y:=a:

> a:=(a+b)/2:

> b:=sqrt(b*y):

> t:=t-x*(y-a)^2:

> x:=2*x:

> od:

> evalf(((a+b)^2)/(4*t));

Pretty good, we are correct to 7 places already! Let's iterate three times and check out our error.

> restart; a:=1: b:=1/sqrt(2): t:=1/4: x:=1:

> for i from 1 to 3 do

> y:=a:

> a:=(a+b)/2:

> b:=sqrt(b*y):

> t:=t-x*(y-a)^2:

> x:=2*x:

> od:

> evalf(((a+b)^2)/(4*t)); evalf(Pi-%);

Wow! Look at that precision. Let's see what iterating four times looks like.

> restart; a:=1: b:=1/sqrt(2): t:=1/4: x:=1:

> for i from 1 to 4 do

> y:=a:

> a:=(a+b)/2:

> b:=sqrt(b*y):

> t:=t-x*(y-a)^2:

> x:=2*x:

> od:

> evalf(((a+b)^2)/(4*t));

With this many digits, there is no difference from the last output, let's increase our number of digits and look at our error.

> Digits:=20:

> evalf(((a+b)^2)/(4*t)); evalf(Pi-%);

After only four iterations (and with very minimal computation time) we have Pi correct to 18 places!