Gauss–Legendre algorithm: Difference between revisions

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The values <math display="inline">\varphi=\theta={\pi\over 4}</math> can be substituted into Legendre’s identity and the approximations to K, E can be found by terms in the sequences for the arithmetic geometric mean with <math>a_0=1</math> and <math display="inline">b_0=\sin{\pi \over 4}=\frac{1}{\sqrt{2}}</math>.<ref>Adlaj, Semjon, ''An eloquent formula for the perimeter of an ellipse'', Notices of the AMS 59(8), p. 1096</ref>

=== Elementary proof with integral calculus ===

The Gauss-Legendre algorithm can be proven without elliptic modular functions. This is done here<ref>{{citation|title=Recent Calculations of π: The Gauss-Salamin Algorithm|last1=Lord|first1=Nick|DOI=10.2307/3619132|year=1992|journal=The Mathematical Gazette|volume=76|issue=476|pages=231-242|jstor=3619132}}</ref> and here<ref>{{citation|title=Easy Proof of Three Recursive π-Algorithms|last1=Milla|first1=Lorenz|arxiv=1907.04110|year=2019}}</ref> using only integral calculus.

== See also ==