Arithmetic-geometric mean process

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