Arithmetic–geometric mean
In mathematics, the arithmetic–geometric mean of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric 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 arithmetic–geometric mean of and ; it is denoted by, or sometimes by or.
The arithmetic–geometric 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.
Example
To find the arithmetic–geometric mean of and, iterate as follows:The first five iterations give the following values:| 0 | 24 | 6 |
| 1 | 5 | 2 |
| 2 | .5 | .416 407 864 998 738 178 455 042... |
| 3 | 203 932 499 369 089 227 521... | 139 030 990 984 877 207 090... |
| 4 | 45 176 983 217 305... | 06 053 858 316 334... |
| 5 | 20... | 06... |
The number of digits in which and agree approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately.
History
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.Properties
Both the geometric mean and arithmetic mean of two positive numbers and are between the two numbers. 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 arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in elliptic filter design.
The arithmetic–geometric mean is connected to the Jacobi theta function bywhich upon setting gives
Related concepts
The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is Gauss's constant.In 1799, Gauss proved thatwhere is the lemniscate constant.In 1941, was proved transcendental by Theodor Schneider. The set is algebraically independent over, but the set is not algebraically independent over. In fact,The geometric–harmonic mean GH can be calculated using analogous sequences of geometric and harmonic means, and in fact.
The arithmetic–harmonic mean is equivalent to the geometric mean.
The arithmetic–geometric mean can be used to compute – among others – logarithms, complete and incomplete elliptic [integrals of the first and second kind], and Jacobi elliptic functions.
Proof of existence
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:
Q.E.D.
Proof of the integral-form expression
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
Applications
The number ''π''
According to the Gauss–Legendre algorithm,where
with and, which can be computed without loss of precision using
Complete elliptic integral ''K''(sin''α'')
Taking and yields the AGMwhere is a complete elliptic integral of the first kind:
That is to say that this quarter period may be efficiently computed through the AGM,