Arithmetic–geometric mean

In mathematics is known as arithmetic- geometric mean of two positive real numbers, a certain number of which is between the arithmetic mean and the geometric mean.

Definition

Let and two non-negative real numbers. Starting from them inductively two sequences and with

Defined:

The consequences and converge to a common threshold, denoted as the geometric mean and arithmetic.

That the two limits actually exist and, moreover, are even the same, is further demonstrated below in "Important properties ".

Simple Example

Be

Then

Simple properties

Two non-negative values ​​, and the following applies:

That is, the arithmetic- geometric mean - as each averaging function - symmetric and homogeneous of degree 1 in its two variables.

Key Features

• Monotony: applies after the inequality of the arithmetic and geometric mean for two positive values ​​always start too. The result is therefore monotonically increasing and bounded by upward, so it converges to a limit. On the other hand, the sequence is monotonically decreasing and limited downward, that is, it converges to a limit. Or written differently:

If we now in the defining equation for the limit value (which is allowed because all limits exist), then we obtain, from which it follows. Thus, the two values ​​are the same and it is the arithmetic- geometric mean.

• Convergence speed: Be

Is a method with quadratic convergence.

Alternative representation

One can both sequences from each other " decouple " Let

Then we can transform the above equations to:

Historical

The arithmetic- geometric mean was from one another discovered independently by mathematicians Carl Friedrich Gauss, and previously by Adrien -Marie Legendre. They used it to calculate the arc length of ellipses, ie elliptic integrals approximation. Gauss wrote about the relationship between the arithmetic and geometric means and the elliptic integral of first kind ( arc length of a lemniscate ) the equation

In his Mathematical diary.

Method of Salamin and Brent

The following method for calculating the wave number was published by Richard P. Brent and Eugene Salamin independently in 1976. It uses essentially the findings of Gauss on the arithmetic- geometric mean. Gauss did not notice his time, however, that thus a faster algorithm can be constructed to calculate the number. Nevertheless, the method is often referred to as the method of Gauss, Brent and Salamin.

The steps of the method can be described as follows:

• Initialization: It is used as starting values

• Loop: for

One calculates

The sequence converges to the square, which means that with each iteration of the loop, the number of correctly computed digits approximately doubles. Thus, this algorithm converges much faster than many against classical methods.

Numerical example

With the starting values

Is calculated iteratively:

After three iterations we obtain the approximate value of the arithmetic- geometric mean

For the number, the approximation results

Relationship with elliptic integrals

The following applies:

The right side is a complete elliptic integral of the first kind