# Arithmetic-geometric mean process

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

Given two real numbers and , one can form the successive arithmetic and geometric means as follows: (Cf. also Arithmetic mean; Geometric mean.) The sequences and rapidly converge to a common value, denoted and called the arithmetic-geometric mean, or sometimes the arithmetic-geometric average, of and . Indeed, This so-called AGM process is useful for computing the Jacobi elliptic functions , , , , the Jacobi theta-functions (cf. also Theta-function), and the Jacobi zeta-function (see [a1], pp. 571–598, and [a4], Chap. 6; p. 663).

The number is known as the Gauss lemniscate constant, or Gauss constant, [a3]. Here, denotes the Gamma-function. The lemniscate constant which is half the total arc length of the lemniscate (cf. also Lemniscates), is closely related to the Gauss constant.

Taking , , and setting one obtains a sequence that converges quadratically to , [a2] (see Pi (number )). 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].