Modern era
Superfast algorithms based on the theory of modular equations and the arithmetico-geometric mean
(XXth century)
Salamin/Brent :
*1976*
:
a derived algorithm
I haven't tested the following algorithms yet
Quadratic convergence
Cubic convergence
This sequence converges towards the multiple of *Pi* closest to *f*_{0}.
*f*_{n}=f_{n-1}+sin(f_{n-1})
J and P.Borwein :
1) *1984* : quadratic convergence (based on the arithmetico-geometric mean)
2) *1987*
: quadratic convergence (also based on the *AGM*)
3) quadratic convergence (based on modular equations as the following equations)
4) quadratic convergence :
5) cubic convergence :
6) quartic convergence :
7) quintic convergence:
8) septic convergence :
9) nonic convergence :
10) "hexadecimalic" (!) convergence (order *16*)
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