Almkvist Berndt 1988 begins with an investigation of why the agm is such an efficient algorithm, and proves that it converges quadratically. This is an efficient method to calculate 



π


{\displaystyle \pi }

.

With the same notations used in Arithmetic-geometric mean, we can summarize the paper by writing:





π
=



4


a
g
m

(
1
,
1

/



2



)

2




1
−

∑

n
=
1


∞



2

n
+
1


(

a

n


2


−

g

n


2


)





{\displaystyle \pi ={\frac {4\;\mathrm {agm} (1,1/{\sqrt {2}})^{2}}{1-\sum \limits _{n=1}^{\infty }2^{n+1}(a_{n}^{2}-g_{n}^{2})}}}



This allows you to make the approximation, for any large   N:





π
≈



4


a

N


2




1
−

∑

k
=
1


N



2

k
+
1


(

a

k


2


−

g

k


2


)





{\displaystyle \pi \approx {\frac {4\;a_{N}^{2}}{1-\sum \limits _{k=1}^{N}2^{k+1}(a_{k}^{2}-g_{k}^{2})}}}



The purpose of this task is to demonstrate how to use this approximation in order to compute a large number of decimals of 



π


{\displaystyle \pi }

.

DP=: 100
 
round=: DP&$: : (4 : 0)
 b %~ <.1r2+y*b=. 10x^x
)
 
sqrt=: DP&$: : (4 : 0) " 0
 assert. 0<:y
 %/ <.@%: (2 x: (2*x) round y)*10x^2*x+0>.>.10^.y
)
 
pi =: 3 : 0
 A =. N =. 1x
 'G Z HALF' =. (% sqrt 2) , 1r4 1r2
 for_I. i.18 do.
  X =. ((A + G) * HALF) , sqrt A * G
  VAR =. ({.X) - A
  Z =. Z - VAR * VAR * N
  N =. +: N
  'A G' =. X
  PI =: A * A % Z
  smoutput (0j100":PI) , 4 ": I
 end.
 PI
)