Gauss–Legendre algorithm: Difference between revisions

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{{Short description|Quickly converging computation of π}}

The '''Gauss–Legendre algorithm''' is an [[algorithm]] to compute the digits of [[Pi|π{{pi}}]]. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π {{pi}}. However, theit drawbackhas issome thatdrawbacks (for example, it is [[Random-access_memory|computer memory]]-intensive) and therefore sometimesall record-breaking calculations for many years have used other methods, almost always the [[Machin-likeChudnovsky formulasalgorithm]]. areFor useddetails, insteadsee [[chronology of computation of π|Chronology of computation of {{pi}}]].

The method is based on the individual work of [[Carl Friedrich Gauss]] (1777–1855) and [[Adrien-Marie Legendre]] (1752–1833) combined with modern algorithms for multiplication and [[square root]]s. It repeatedly replaces two numbers by their [[arithmetic mean|arithmetic]] and [[geometric mean]], in order to approximate their [[arithmetic-geometric mean]].

The version presented below is also known as the '''Gauss–Euler''', '''Brent–Salamin''' (or '''Salamin–Brent''') '''algorithm''';<ref>[[Richard Brent (scientist)|Brent, Richard]], ''Old and New Algorithms for pi'', Letters to the Editor, Notices of the AMS 60(1), p. 7</ref> it was independently discovered in 1975 by [[Richard Brent (scientist)|Richard Brent]] and [[Eugene Salamin (mathematician)|Eugene Salamin]]. It was used to compute the first 206,158,430,000 decimal digits of π{{pi}} on September 18 to 20, 1999, and the results were checked with [[Borwein's algorithm]].

== Algorithm ==

# Initial value setting:<br /><math display="block">a_0 = 1\qquad b_0 = \frac{1}{\sqrt{2}}\qquad p_0 = 1\qquad t_0 = \frac{1}{4}\qquad p_0 = 1.\!</math>

# Repeat the following instructions until the difference ofbetween <math>a_n\!a_{n+1}</math> and <math>b_n\!b_{n+1}</math> is within the desired accuracy:<br /><math display="block"> \begin{align}
a_{n+1} & = \frac{a_n + b_n}{2}, \\

\\
b_{n+1} & = \sqrt{a_n b_n}, \\

t_{n+1} & = t_n - p_n(a_{n}-a_{n+1})^2, \\

p_{n+1} & = 2p_n., \\

\\

t_{n+1} & = t_n - p_n(a_{n+1}-a_{n})^2. \\

\end{align}

</math>

# π{{pi}} is then approximated as:<br /><math display="block">\pi \approx \frac{(a_{n+1}+b_{n+1})^2}{4t_{n+1}}.\!</math>

The first three iterations give (approximations given up to and including the first incorrect digit):

:<math>3.140\dots\!</math>

:<math>3.14159264\dots\!</math>

:<math>3.1415926535897932382\dots\!</math>

:<math>3.14159265358979323846264338327950288419711\dots</math>

:<math>3.141592653589793238462643383279502884197169399375105820974944592307816406286208998625\dots</math>

The algorithm has second-order[[quadratic convergent natureconvergence]], which essentially means that the number of correct digits doubles with each step[[iteration]] of the algorithm.

== Mathematical background ==

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The [[arithmetic–geometric mean]] of two numbers, a₀ and b₀, is found by calculating the limit of the sequences

:<math>\begin{align} a_{n+1} & = \frac{a_n+b_n}{2}, \\[6pt]

b_{n+1} & = \sqrt{a_n b_n},

\end{align}

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which both converge to the same limit.<br />

If <math>a_0=1\!</math> and <math>b_0=\cos\varphi\!</math> then the limit is <math display="inline">{\pi \over 2K(\sin\varphi)}\!</math> where <math>K(k)\!</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]]

:<math>K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}.\!</math>

If <math>c_0 = \sin\varphi\!</math>, <math>c_{i+1} = a_i - a_{i+1}\!</math>., then

:<math>\sum_{i=0}^\infty 2^{i-1} c_i^2 = 1 - {E(\sin\varphi)\over K(\sin\varphi)}\!</math>

where <math>E(k)\!</math> is the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]:

:<math>E(k) = \int_0^{\pi/2}\sqrt {1-k^2 \sin^2\theta}\,; d\theta.\!</math>

Gauss knew of boththese of thesetwo results.<ref name="brent">{{Citation

| last=Brent

| first=Richard

| author-link=Richard Brent (scientist)

| publication-date=

| date=

| year=1975

| title=Multiple-precision zero-finding methods and the complexity of elementary function evaluation

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| oclc=

| accessdate=8 September 2007

| archive-date=23 July 2008

}}</ref>

| archive-url=https://web.archive.org/web/20080723170157/http://wwwmaths.anu.edu.au/~brent/pub/pub028.html

| url-status=dead

}}</ref>

<ref name="salamin1">[[Eugene Salamin (mathematician)|Salamin, Eugene]], ''Computation of pi'', Charles Stark Draper Laboratory ISS memo 74–19, 30 January 1974, Cambridge, Massachusetts</ref>

<ref name="salamin2">{{Citation

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| issn=0025-5718

| doi=10.2307/2005327

| jstor=2005327

| oclc=

| accessdate=

Line 97 ⟶ 106:

=== Legendre’s identity ===

Legendre proved the following identity:

For <math>\varphi\!</math> and <math>\theta\!</math> such that <math>\varphi+\theta={1 \over 2}\pi\!</math> Legendre proved the identity:

:<math display="block">K(\sincos \varphitheta) E(\sin \theta ) + K(\sin \theta ) E(\sincos \varphitheta) - K(\sincos \varphitheta) K(\sin \theta) = {1\pi \over 2},</math>
for all <math>\pi.\!theta</math>.<ref name="brent" />

=== Elementary proof with integral calculus ===

=== Gauss–Euler method ===

The Gauss-Legendre algorithm can be proven to give results converging to <math>\pi</math> using only integral calculus. 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|s2cid=125865215 }}</ref> and here.<ref>{{citation|title=Easy Proof of Three Recursive π-Algorithms|last1=Milla|first1=Lorenz|arxiv=1907.04110|year=2019}}</ref>

The values <math>\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>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>

== See also ==

* [[Numerical approximations of π|Numerical approximations of {{pi}}]]

== References ==