In mathematics, the arithmeticâÂÂgeometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmeticâÂÂgeometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some mathematical constants, in particular, computing.
The AGM is defined as the limit of the interdependent sequences and . Assuming , we write:These two sequences converge to the same number, the arithmeticâÂÂgeometric mean of and ; it is denoted by , or sometimes by or .
The arithmeticâÂÂgeometric mean can be extended to complex numbers and, when the branches of the square root are allowed to be taken inconsistently, it is a multivalued function.
To find the arithmeticâÂÂgeometric mean of and , iterate as follows:The first five iterations give the following values:
The number of digits in which and agree (underlined) approximately doubles with each iteration. The arithmeticâÂÂgeometric mean of 24 and 6 is the common limit of these two sequences, which is approximately .
The first algorithm based on this sequence pair appeared in the works of Joseph-Louis Lagrange. Its properties were further analyzed by Carl Friedrich Gauss.
Both the geometric mean and arithmetic mean of two positive numbers and are between the two numbers. (They are strictly between when .) The geometric mean of two positive numbers is never greater than the arithmetic mean. So the geometric means are an increasing sequence ; the arithmetic means are a decreasing sequence ; and for any . These are strict inequalities if .
is thus a number between and ; it is also between the geometric and arithmetic mean of and .
If then .
There is an integral-form expression for :where is the complete elliptic integral of the first kind:Since the arithmeticâÂÂgeometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in elliptic filter design.
The arithmeticâÂÂgeometric mean is connected to the Jacobi theta function bywhich upon setting gives
The reciprocal of the arithmeticâÂÂgeometric mean of 1 and the square root of 2 is Gauss's constant.In 1799, Gauss proved thatwhere is the lemniscate constant.
In 1941, (and hence ) was proved transcendental by Theodor Schneider. The set is algebraically independent over , but the set (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over . In fact,The geometricâÂÂharmonic mean GH can be calculated using analogous sequences of geometric and harmonic means, and in fact . The arithmeticâÂÂharmonic mean is equivalent to the geometric mean.
The arithmeticâÂÂgeometric mean can be used to compute â among others â logarithms, complete and incomplete elliptic integrals of the first and second kind, and Jacobi elliptic functions.
The inequality of arithmetic and geometric means implies thatand thusthat is, the sequence is nondecreasing and bounded above by the larger of and . By the monotone convergence theorem, the sequence is convergent, so there exists a such that:However, we can also see that: and so:
This proof is given by Gauss. Let
Changing the variable of integration to , where
This yields
gives
Thus, we have
The last equality comes from observing that .
Finally, we obtain the desired result
According to the GaussâÂÂLegendre algorithm,
where
with and , which can be computed without loss of precision using
Taking and yields the AGM
where is a complete elliptic integral of the first kind:
That is to say that this quarter period may be efficiently computed through the AGM,
Using this property of the AGM along with the ascending transformations of John Landen, Richard P. Brent suggested the first AGM algorithms for the fast evaluation of elementary transcendental functions (, , ). Subsequently, many authors went on to study the use of the AGM algorithms.