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Arithmetic–geometric mean

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In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers x and y is defined as follows:

Contents

First, compute the arithmetic and geometric means of x and y, calling them a1 and g1 respectively (the latter is the principal square root of the product xy):

a 1 = 1 2 ( x + y ) g 1 = x y

Then, use iteration, with a1 taking the place of x and g1 taking the place of y. In this way, two sequences (an) and (gn) are defined:

a n + 1 = 1 2 ( a n + g n ) g n + 1 = a n g n

These two sequences converge to the same number, which is the arithmetic–geometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y).

This can be used for algorithmic purposes as in the AGM method, which makes it possible to construct fast algorithms for calculating exponential and trigonometric functions, as well as some mathematical constants, in particular, to quickly compute π .

Example

To find the arithmetic–geometric mean of a0 = 24 and g0 = 6, first calculate their arithmetic and geometric means, thus:

a 1 = 1 2 ( 24 + 6 ) = 15 g 1 = 24 × 6 = 12

and then iterate as follows:

a 2 = 1 2 ( 15 + 12 ) = 13.5 g 2 = 15 × 12 = 13.41640786500 … …

The first five iterations give the following values:

As can be seen, the number of digits in agreement (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.

History

The first algorithm based on this sequence pair appeared in the works of Lagrange. Its properties were further analyzed by Gauss.

Properties

The geometric mean of two positive numbers is never bigger than the arithmetic mean (see inequality of arithmetic and geometric means); as a consequence, (gn) is an increasing sequence, (an) is a decreasing sequence, and gn ≤ M(x, y) ≤ an. These are strict inequalities if x ≠ y.

M(x, y) is thus a number between the geometric and arithmetic mean of x and y; it is also between x and y.

If r ≥ 0, then M(rx,ry) = r M(x,y).

There is an integral-form expression for M(x,y):

M ( x , y ) = π 2 / ∫ 0 π 2 d θ x 2 cos 2 ⁡ θ + y 2 sin 2 ⁡ θ = π 4 ⋅ x + y K ( x − y x + y )

where K(k) is the complete elliptic integral of the first kind:

K ( k ) = ∫ 0 π 2 d θ 1 − k 2 sin 2 ⁡ ( θ )

Indeed, since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals via this formula. In engineering, it is used for instance in elliptic filter design.

The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is called Gauss's constant, after Carl Friedrich Gauss.

1 M ( 1 , 2 ) = G = 0.8346268 …

The geometric–harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. The arithmetic–harmonic mean can be similarly defined, but takes the same value as 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

From the inequality of arithmetic and geometric means we can conclude that:

g n ⩽ a n

and thus

g n + 1 = g n ⋅ a n ⩾ g n ⋅ g n = g n

that is, the sequence gn is nondecreasing.

Furthermore, it is easy to see that it is also bounded above by the larger of x and y (which follows from the fact that both the arithmetic and geometric means of two numbers lie between them). Thus, by the monotone convergence theorem, the sequence is convergent, so there exists a g such that:

lim n → ∞ g n = g

However, we can also see that:

a n = g n + 1 2 g n

and so:

lim n → ∞ a n = lim n → ∞ g n + 1 2 g n = g 2 g = g

Q.E.D.

Proof of the integral-form expression

This proof is given by Gauss. Let

I ( x , y ) = ∫ 0 π / 2 d θ x 2 cos 2 ⁡ θ + y 2 sin 2 ⁡ θ ,

Changing the variable of integration to θ ′ , where

sin ⁡ θ = 2 x sin ⁡ θ ′ ( x + y ) + ( x − y ) sin 2 ⁡ θ ′ ,

gives

I ( x , y ) = ∫ 0 π / 2 d θ ′ ( 1 2 ( x + y ) ) 2 cos 2 ⁡ θ ′ + ( x y ) 2 sin 2 ⁡ θ ′ = I ( 1 2 ( x + y ) , x y ) .

Thus, we have

I ( x , y ) = I ( a 1 , g 1 ) = I ( a 2 , g 2 ) = ⋯ = I ( M ( x , y ) , M ( x , y ) ) = π / ( 2 M ( x , y ) ) .

The last equality comes from observing that I ( z , z ) = π / ( 2 z ) .

Finally, we obtain the desired result

M ( x , y ) = π / ( 2 I ( x , y ) ) .

The AGM method

Gauss noticed that the sequences

a 0 b 0 a 1 = a 0 + b 0 2 , b 1 = a 0 b 0 a 2 = a 1 + b 1 2 , b 2 = a 1 b 1     ⋮     ⋮ a N + 1 = a N + b N 2 , b N + 1 = a N b N

as

N → + ∞ ,

have the same limit:

lim N → ∞ a N = lim N → ∞ b N = M ( a , b ) ,

the arithmetic–geometric mean, agm.

It is possible to use this fact to construct fast algorithms for calculating elementary transcendental functions and some classical constants, in particular, the constant π.

The number π

For example, according to the Gauss–Salamin formula:

π = 4 ( M ( 1 ; 1 2 ) ) 2 1 − ∑ j = 1 ∞ 2 j + 1 c j 2 ,

where

c j = 1 2 ( a j − 1 − b j − 1 )

which can be computed without loss of precision using

c j = c j − 1 2 4 a j .

Complete elliptic integral K(sinα)

Taking a 0 = 1 , b 0 = cos ⁡ α , yields the agm,

lim N → ∞ a N = π 2 K ( sin ⁡ α )   ,

where K(k) is a complete elliptic integral of the first kind,

K ( k ) = ∫ 0 π / 2 ( 1 − k 2 sin 2 ⁡ θ ) − 1 / 2 d θ   .

That is to say that this quarter period may be efficiently computed through the agm,

K ( k ) = π 2   M ( 1 , 1 − k 2 )   .

Other applications

Using this property of the AGM along with the ascending transformations of Landen, Richard Brent suggested the first AGM algorithms for the fast evaluation of elementary transcendental functions (ex, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms.

References

Arithmetic–geometric mean Wikipedia
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