Nicolas of Cues
(1401  1464)
Algorithm to remember
Slices of life
Nicolas de Cues was born in 1401 at Kues, Nikolaus
Krebs is a German theologian known by the name gallicized of Nicolas of Cues. His
main action was to support the popes and the principle of the infallibility against
councils. While touching mathematics, he also left an important theological and philosophical
work.
He dies in 1464, leaving behind him an approximation method of Pi in the lineage of
the sequences from archimedian origin.
As Leibniz and Descartes, his
theological and scientific thought are often connected. he considers as well that
the search for the solution of the quadrature of the circle is comparable in search
of the truth!
About
As I have yet wrote in the history of the geometrical
period of Pi.
This formula poses me problem. it is indeed often given as the "official"
version of Archimedes' formula (Le fascinant nombre
Pi ), and nevertheless I found it on
an exercice of mathematics entitled "Cues's method ". And in Le Petit Archimède, this formula is considered as the one of Gregory... Then,
I renew my appeal, if somebody knows the real paternity of this sequence, I wish
he clarifies it to me!
Let us note however that this sequence uses the perimeter of a polygone having a
fixed value. We try then to estimate the radius of inscribed and circumscribed circles
what is still more close to the Descartes's method
of isoperimeters. But after all, I decided to choose De Cues to make known a little
this remarkable person!
Proof
Although this algorithm much looks like the arithmetico
 geometrical mean (see Salamin), convergence is
(regrettably!) in no way comparable and more close to a linear convergence than to
a quadratic convergence!
As for Archimede,a polygone in 2^{n} sides is considered with perimeter equal to 1 and we note a_{n} and b_{n} the respective radius of the inscribed and circumscribed circles.
So we have :
From which, because the perimeter of the polygone is 1, one deducts from it : 2^{n}A_{1}A_{2}=1 and
according to previously, one has so :
Let us calculate now :
Let us show that these two series are adjacent :
Because tan > sin.
As well, so b_{n+1}b_{n} is of the sign of because cos
< 1 for n
> 0.
From which (b_{n}) is decreasing...
Furthermore, tan(x) ~ x and sin(x) ~ x in 0 so it is immediate that :
and we also have :
so b_{n}>a_{n} then a_{n} and b_{n} are adjacent.
Finally, so we have :
Let us estimate the convergence speed now :
so :
That gives us a linear convergence (if we apply
the log).
Let us verify all this by some attempts...
Trials
Nevertheless it is not necessary to be discouraged
of this lineat convergence because sequence is all the same easy to calculate and
to accelerate...

a_{n}= 
b_{n}= 
n=5 
3,1517
(1) 
3,136
(1) 
n=10 
3,14160
(3) 
3,141587
(4) 
n=20 
3,141592653599
(10) 
3,14159265358606
(11) 
n=50 
28 decimals
exact 
29 decimals
exact 
A convergence in about 3n / 5 is obtained here
Acceleration of the convergence
Aitken and the Delta2 is here particularly effective. Let us see it!

Delta2(a_{n})= 
Delta2(b_{n})= 
n=5 
3,1422
(2) 
3,14163
(3) 
n=10 
3,14159265418
(8) 
3,14159265362
(9) 
n=20 
20 décimales
justes 
22 décimales
justes 
n=50 
56 décimales
exactes 
58 décimales
exactes 
It is rather incredible, but the Delta2 doubles the speed of convergence! A convergence of the
order 1.2n is obtained indeed
Then let us continue and let us iterate the same process (Delta2 applied twice !)

Delta2^{2}(a_{n})= 
Delta2^{2}(b_{n})= 
n=5 
non available (div by zero) 
3,1415953
(5) 
n=10 
12 decimals
exact 
14 decimals
exact 
n=20 
32 decimals
exact 
32 decimals
exact 
n=50 
87 decimals
exact 
87 decimals
exact 
Factor of 50 % is gained again! One has then a convergence 1.75n that is very honorable
for a series of this visible simplicity!
We will so be able to remember this series among archimedian's as one of the fastest.
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