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''arithmetic-geometric mean method, AGM process, AGM method, Lagrange arithmetic-geometric mean algorithm''
 
''arithmetic-geometric mean method, AGM process, AGM method, Lagrange arithmetic-geometric mean algorithm''
  
Given two real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302802.png" />, one can form the successive arithmetic and geometric means as follows:
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Given two real numbers $a = a_0$ and $b = b _ { 0 }$, one can form the successive arithmetic and geometric means as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302803.png" /></td> </tr></table>
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\begin{equation*} a _ { n  + 1} = \frac { 1 } { 2 } ( a _ { n } + b _ { n } ) ,\, b _ { n  + 1} = \sqrt { a _ { n } b _ { n } }. \end{equation*}
  
(Cf. also [[Arithmetic mean|Arithmetic mean]]; [[Geometric mean|Geometric mean]].) The sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302805.png" /> rapidly converge to a common value, denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302806.png" /> and called the arithmetic-geometric mean, or sometimes the arithmetic-geometric average, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302808.png" />. Indeed,
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(Cf. also [[Arithmetic mean|Arithmetic mean]]; [[Geometric mean|Geometric mean]].) The sequences $a_0 , a _ { 1 } , \dots$ and $b _ { 0 } , b _ { 1 } , \dots$ rapidly converge to a common value, denoted $\operatorname { agm } ( a , b )$ and called the arithmetic-geometric mean, or sometimes the arithmetic-geometric average, of $a$ and $b$. Indeed,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a1302809.png" /></td> </tr></table>
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\begin{equation*} | a _ { n  + 1} - b _ { n  + 1} | &lt; \frac { 1 } { 2 } | a _ { n } - b _ { n } |. \end{equation*}
  
This so-called AGM process is useful for computing the [[Jacobi elliptic functions|Jacobi elliptic functions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028013.png" />, the Jacobi theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028014.png" /> (cf. also [[Theta-function|Theta-function]]), and the Jacobi zeta-function (see [[#References|[a1]]], pp. 571–598, and [[#References|[a4]]], Chap. 6; p. 663).
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This so-called AGM process is useful for computing the [[Jacobi elliptic functions|Jacobi elliptic functions]] $\operatorname { sn } ( u | k )$, $\operatorname { sn } ( u | k )$, $\operatorname { cn } ( u | k )$, $\operatorname { dn } ( u | k )$, the Jacobi theta-functions $\theta _ { i } ( v )$ (cf. also [[Theta-function|Theta-function]]), and the Jacobi zeta-function (see [[#References|[a1]]], pp. 571–598, and [[#References|[a4]]], Chap. 6; p. 663).
  
 
The number
 
The number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028015.png" /></td> </tr></table>
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\begin{equation*} \operatorname { agm } ( 1 , \sqrt { 2 } ) ^ { - 1 } = ( 2 \pi ) ^ { - 3 / 2 } \Gamma \left( \frac { 1 } { 4 } \right) ^ { 2 } = 0.83462684\dots \end{equation*}
  
is known as the Gauss lemniscate constant, or Gauss constant, [[#References|[a3]]]. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028016.png" /> denotes the [[Gamma-function|Gamma-function]]. The lemniscate constant
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is known as the Gauss lemniscate constant, or Gauss constant, [[#References|[a3]]]. Here, $\Gamma$ denotes the [[Gamma-function|Gamma-function]]. The lemniscate constant
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028017.png" /></td> </tr></table>
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\begin{equation*} L = \frac { 1 } { 2 } ( 2 \pi ) ^ { - 1 / 2 } \Gamma \left( \frac { 1 } { 4 } \right) ^ { 2 }, \end{equation*}
  
which is half the total arc length of the lemniscate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028018.png" /> (cf. also [[Lemniscates|Lemniscates]]), is closely related to the Gauss constant.
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which is half the total arc length of the lemniscate $r ^ { 2 } = \operatorname { cos } ( 2 \phi )$ (cf. also [[Lemniscates|Lemniscates]]), is closely related to the Gauss constant.
  
Taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028021.png" /> and setting
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Taking $a = 1$, $b = ( \sqrt { 2 } ) ^ { - 1 }$, $s _ { 0 } = 1 / 2$ and setting
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028022.png" /></td> </tr></table>
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\begin{equation*} c _ { k } = a _ { k } ^ { 2 } - b _ { k } ^ { 2 } ,\, s _ { k } = s _ { k - 1 } - 2 ^ { k } c _ { k } ,\, p _ { k } = 2 s _ { k } ^ { - 1 } a _ { k } ^ { 2 }, \end{equation*}
  
one obtains a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028023.png" /> that converges quadratically to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028025.png" />, [[#References|[a2]]] (see [[Pi(number)|Pi (number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130280/a13028026.png" />)]]). This means that each iteration roughly doubles the number of correct digits. This algorithm is variously known as the Brent–Salamin algorithm, the Gauss–Salamin algorithm, or Salamin–Brent algorithm. There are also corresponding cubic, quartic, etc. algorithms, [[#References|[a2]]].
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one obtains a sequence $p_0 , p _ { 1 } , \dots$ that converges quadratically to $\pi$, [[#References|[a2]]] (see [[Pi(number)|Pi (number $\pi$)]]). This means that each iteration roughly doubles the number of correct digits. This algorithm is variously known as the Brent–Salamin algorithm, the Gauss–Salamin algorithm, or Salamin–Brent algorithm. There are also corresponding cubic, quartic, etc. algorithms, [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  "Handbook of mathematical functions"  M. Abramowitz (ed.)  J.A. Stegun (ed.) , Nat. Bureau Standards  (1964)  ((Dover reprint 1965))</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.H. Bailey,  J.M. Borwein,  P.B. Borwein,  S. Plouffe,  "The quest for pi"  ''Math. Intelligencer'' , '''19''' :  1  (1997)  pp. 50–57</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Finch,  "Favorite mathematical constants"  ''WEB: www.mathsoft.com/asolve/constant/gauss/gauss.html''  (2000)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.H. Press,  B.P. Flannery,  S.A. Teukolsky,  W.T. Vetterling,  "Numerical recipes" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.P. Brent,  "Fast multiple-precision evaluation of elementary functions"  ''J. Assoc. Comput. Mach.'' , '''23'''  (1976)  pp. 242–251</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Salamin,  "Computation of pi using arithmetic-geometric mean"  ''Math. Comput.'' , '''30'''  (1976)  pp. 565–570</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  "Handbook of mathematical functions"  M. Abramowitz (ed.)  J.A. Stegun (ed.) , Nat. Bureau Standards  (1964)  ((Dover reprint 1965))</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  D.H. Bailey,  J.M. Borwein,  P.B. Borwein,  S. Plouffe,  "The quest for pi"  ''Math. Intelligencer'' , '''19''' :  1  (1997)  pp. 50–57</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. Finch,  "Favorite mathematical constants"  ''WEB: www.mathsoft.com/asolve/constant/gauss/gauss.html''  (2000)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  W.H. Press,  B.P. Flannery,  S.A. Teukolsky,  W.T. Vetterling,  "Numerical recipes" , Cambridge Univ. Press  (1986)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R.P. Brent,  "Fast multiple-precision evaluation of elementary functions"  ''J. Assoc. Comput. Mach.'' , '''23'''  (1976)  pp. 242–251</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E. Salamin,  "Computation of pi using arithmetic-geometric mean"  ''Math. Comput.'' , '''30'''  (1976)  pp. 565–570</td></tr></table>

Latest revision as of 16:57, 1 July 2020

arithmetic-geometric mean method, AGM process, AGM method, Lagrange arithmetic-geometric mean algorithm

Given two real numbers $a = a_0$ and $b = b _ { 0 }$, one can form the successive arithmetic and geometric means as follows:

\begin{equation*} a _ { n + 1} = \frac { 1 } { 2 } ( a _ { n } + b _ { n } ) ,\, b _ { n + 1} = \sqrt { a _ { n } b _ { n } }. \end{equation*}

(Cf. also Arithmetic mean; Geometric mean.) The sequences $a_0 , a _ { 1 } , \dots$ and $b _ { 0 } , b _ { 1 } , \dots$ rapidly converge to a common value, denoted $\operatorname { agm } ( a , b )$ and called the arithmetic-geometric mean, or sometimes the arithmetic-geometric average, of $a$ and $b$. Indeed,

\begin{equation*} | a _ { n + 1} - b _ { n + 1} | < \frac { 1 } { 2 } | a _ { n } - b _ { n } |. \end{equation*}

This so-called AGM process is useful for computing the Jacobi elliptic functions $\operatorname { sn } ( u | k )$, $\operatorname { sn } ( u | k )$, $\operatorname { cn } ( u | k )$, $\operatorname { dn } ( u | k )$, the Jacobi theta-functions $\theta _ { i } ( v )$ (cf. also Theta-function), and the Jacobi zeta-function (see [a1], pp. 571–598, and [a4], Chap. 6; p. 663).

The number

\begin{equation*} \operatorname { agm } ( 1 , \sqrt { 2 } ) ^ { - 1 } = ( 2 \pi ) ^ { - 3 / 2 } \Gamma \left( \frac { 1 } { 4 } \right) ^ { 2 } = 0.83462684\dots \end{equation*}

is known as the Gauss lemniscate constant, or Gauss constant, [a3]. Here, $\Gamma$ denotes the Gamma-function. The lemniscate constant

\begin{equation*} L = \frac { 1 } { 2 } ( 2 \pi ) ^ { - 1 / 2 } \Gamma \left( \frac { 1 } { 4 } \right) ^ { 2 }, \end{equation*}

which is half the total arc length of the lemniscate $r ^ { 2 } = \operatorname { cos } ( 2 \phi )$ (cf. also Lemniscates), is closely related to the Gauss constant.

Taking $a = 1$, $b = ( \sqrt { 2 } ) ^ { - 1 }$, $s _ { 0 } = 1 / 2$ and setting

\begin{equation*} c _ { k } = a _ { k } ^ { 2 } - b _ { k } ^ { 2 } ,\, s _ { k } = s _ { k - 1 } - 2 ^ { k } c _ { k } ,\, p _ { k } = 2 s _ { k } ^ { - 1 } a _ { k } ^ { 2 }, \end{equation*}

one obtains a sequence $p_0 , p _ { 1 } , \dots$ that converges quadratically to $\pi$, [a2] (see Pi (number $\pi$)). This means that each iteration roughly doubles the number of correct digits. This algorithm is variously known as the Brent–Salamin algorithm, the Gauss–Salamin algorithm, or Salamin–Brent algorithm. There are also corresponding cubic, quartic, etc. algorithms, [a2].

References

[a1] "Handbook of mathematical functions" M. Abramowitz (ed.) J.A. Stegun (ed.) , Nat. Bureau Standards (1964) ((Dover reprint 1965))
[a2] D.H. Bailey, J.M. Borwein, P.B. Borwein, S. Plouffe, "The quest for pi" Math. Intelligencer , 19 : 1 (1997) pp. 50–57
[a3] S. Finch, "Favorite mathematical constants" WEB: www.mathsoft.com/asolve/constant/gauss/gauss.html (2000)
[a4] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, "Numerical recipes" , Cambridge Univ. Press (1986)
[a5] R.P. Brent, "Fast multiple-precision evaluation of elementary functions" J. Assoc. Comput. Mach. , 23 (1976) pp. 242–251
[a6] E. Salamin, "Computation of pi using arithmetic-geometric mean" Math. Comput. , 30 (1976) pp. 565–570
How to Cite This Entry:
Arithmetic-geometric mean process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic-geometric_mean_process&oldid=50214
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article