Arithmetic-geometric mean

Plot of the arithmetic-geometric mean (in dark blue) ${\displaystyle \operatorname {agm} (1,x)}$

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

definition

Let and be two nonnegative real numbers . Based on them, inductive two sequences and with ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a_{n}}$ ${\displaystyle b_{n}}$

${\displaystyle a_{0}=a}$

(1a)

${\displaystyle b_{0}=b}$

(1b)

Are defined:

${\displaystyle a_{n+1}={\frac {a_{n}+b_{n}}{2}}}$

(2)

( arithmetic mean )

${\displaystyle b_{n+1}={\sqrt {a_{n}b_{n}}}}$

(3)

( geometric mean )

The sequences and converge to a common limit , which is referred to as the arithmetic-geometric mean of and . ${\displaystyle a_{n}}$ ${\displaystyle b_{n}}$ ${\displaystyle M(a,b)}$ ${\displaystyle a}$ ${\displaystyle b}$

The fact that the two limit values ​​actually exist and are even the same is shown below in “Important properties”.

Simple example

Be

${\displaystyle a_{0}=4}$ and ${\displaystyle b_{0}=9}$

(4a, b)

Then

${\displaystyle a_{1}={\frac {4+9}{2}}=6{,}5}$ and ${\displaystyle b_{1}={\sqrt {4\cdot 9}}=6}$

(5)

${\displaystyle a_{2}=6{,}25\,}$ and ${\displaystyle b_{2}\approx 6{,}245\,}$

(6)

${\displaystyle a_{3}\approx b_{3}\approx M(a,b)\approx 6{,}2475\,}$

(7)

Simple properties

For two non-negative values and the following applies: ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle M(a,b)=M(b,a)\,}$

(10)

${\displaystyle M(ta,tb)=t\cdot M(a,b)}$ For ${\displaystyle t\geq 0}$

(11)

That is, the arithmetic-geometric mean is - like every mean value function - symmetrical and homogeneous of degree 1 in its two variables and . ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \min\{a,b\}\leq {\sqrt {ab}}\leq M(a,b)\leq {\frac {a+b}{2}}\leq \max\{a,b\}}$

(12)

Equality applies to exactly ${\displaystyle a=b}$

${\displaystyle M(a,b)=M\left({\frac {a+b}{2}},{\sqrt {ab}}\right)}$

(13)

Important properties

• Monotony : For two positive starting values, the inequality of the arithmetic and geometric mean alwaysapplies. The sequenceis thus monotonically increasing and boundedby upwards, therefore it converges to a limit value. On the other hand, the sequence ismonotonically falling and bounded downwards, that is to say it converges to a limit value. Or written differently: ${\displaystyle 0<b_{0}<a_{0}}$ ${\displaystyle b_{n}<a_{n}}$ ${\displaystyle b_{0},b_{1},b_{2},\dotsc }$ ${\displaystyle a_{0}}$ ${\displaystyle \beta }$ ${\displaystyle a_{0},a_{1},a_{2},\dotsc }$ ${\displaystyle \alpha }$

${\displaystyle b_{0}<b_{1}<\dotsb <b_{n}<b_{n+1}<\dotsb <\beta \leq \alpha <\dotsb <a_{n+1}<a_{n}<\dotsb <a_{1}<a_{0}.}$

(14)

If you now go over to the limit value in the definition equation (this is allowed because all limit values ​​exist), you get what follows. Thus the two limit values ​​are the same and it is the arithmetic-geometric mean. ${\displaystyle a_{n+1}=(a_{n}+b_{n})/2}$ ${\displaystyle \alpha =(\alpha +\beta )/2}$ ${\displaystyle \alpha =\beta }$ ${\displaystyle \alpha =\beta =M(a,b)}$

• Convergence speed : Let

${\displaystyle c_{n}:={\sqrt {{a}_{n}^{2}-{b}_{n}^{2}}}}$

(15)

Because of the estimate

${\displaystyle c_{n+1}={\frac {1}{2}}(a_{n}-b_{n})={\frac {{c}_{n}^{2}}{4{a}_{n+1}}}\leq {\frac {{c}_{n}^{2}}{M(a,b)}}}$

(16)

there is a method with quadratic convergence .

Alternative representation

One can also "decouple" both sequences from one another: Be

${\displaystyle a_{0}=a}$, , And . ${\displaystyle b_{0}=b}$ ${\displaystyle a_{1}={\frac {a_{0}+b_{0}}{2}}}$ ${\displaystyle b_{1}={\sqrt {a_{0}b_{0}}}}$

(21)

Then you can transform the above equations to:

${\displaystyle a_{n}={\frac {a_{n-1}\;+{\sqrt {(2a_{n-1}-a_{n-2})\cdot a_{n-2}}}}{2}}}$

(22)

${\displaystyle b_{n}={\sqrt {\frac {b_{n-1}\cdot (b_{n-1}^{2}+b_{n-2}^{2})}{2b_{n-2}}}}}$

(23)

Historical

The arithmetic-geometric mean was discovered independently by the mathematicians Carl Friedrich Gauß and previously by Adrien-Marie Legendre . They used it to approximate the arc length of ellipses, i.e. elliptical integrals. Gauss, for example, noted the equation for the relationship between the arithmetic-geometric mean and the elliptic integral of genus 1 (arc length of a lemniscate )

${\displaystyle {\frac {\pi }{2M(1,{\sqrt {2}})}}=\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}}$

(24)

in his math diary.

Method of Salamin and Brent

The following procedure for calculating the circle number was published in 1976 independently by Richard P. Brent and Eugene Salamin . It essentially uses Gauss' knowledge of the arithmetic-geometric mean. At the time, Gauss did not notice that a faster algorithm for calculating the number could be constructed with it. Nevertheless, the method is often referred to as the Gauss, Brent and Salamin method. ${\displaystyle \pi }$ ${\displaystyle \pi }$

The steps of the procedure can be described as follows:

• Initialization : One uses as start values

${\displaystyle a_{0}=1\qquad b_{0}={\frac {1}{\sqrt {2}}}\qquad s_{0}={\frac {1}{2}}\qquad }$

(31)

• Loop : For

${\displaystyle n=1,2,...}$

one calculates

${\displaystyle a_{n}={\frac {a_{n-1}+b_{n-1}}{2}}\,}$

(32)

${\displaystyle b_{n}={\sqrt {a_{n-1}b_{n-1}}}\,}$

(33)

${\displaystyle c_{n}=a_{n}^{2}-b_{n}^{2}\,}$

(34)

${\displaystyle s_{n}=s_{n-1}-2^{n}c_{n}\,}$

(35)

${\displaystyle p_{n}={\frac {2a_{n}^{2}}{s_{n}}}\,}$

(36)

The sequence of converges to the square of the square , which means that the number of correctly calculated digits roughly doubles each time the loop is run through. This algorithm converges much faster than many classic methods. ${\displaystyle (p_{n})}$ ${\displaystyle \pi }$ ${\displaystyle \pi }$

Numerical example

With the starting values

${\displaystyle a_{0}=1\qquad b_{0}={\frac {1}{\sqrt {2}}}\approx 0{,}707106781186547\qquad s_{0}={\frac {1}{2}}=0{,}5\qquad }$

(37)

one computes iteratively:

index ${\displaystyle n}$	${\displaystyle a_{n}}$	${\displaystyle b_{n}}$	${\displaystyle c_{n}}$	${\displaystyle s_{n}}$	${\displaystyle p_{n}}$
${\displaystyle n=0}$	1	0.70710 67811 86547		0.5
${\displaystyle n=1}$	0.8 5355 33905 93274	0.8 4089 64152 53715	0.02144 66094 06726	0.45710 67811 86547	3.1 8767 26427 12110
${\displaystyle n=2}$	0.8472 2 49029 23494	0.8472 0 12667 46891	0.00004 00497 56187	0.45694 65821 61801	3.141 68 02932 97660
${\displaystyle n=3}$	0.84721 3084 8 35193	0.84721 3084 7 52765	0.00000 00001 39667	0.45694 65810 44462	3.14159 2653 8 95460

After three iterations, the approximate value for the arithmetic-geometric mean is obtained ${\displaystyle M(1,1/{\sqrt {2}})\approx a_{3}\approx 0{,}847213084.}$

The approximation is obtained for the number ${\displaystyle \pi }$ ${\displaystyle \pi \approx p_{3}\approx 3{,}141592653.\,}$

Relationship to elliptic integrals

The following applies:

${\displaystyle {\frac {\pi /4}{M(a,b)}}=\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {(1-t^{2})((a+b)^{2}-(a-b)^{2}t^{2})}}}}$

(41)

The right hand side is a complete elliptical integral of the first kind.

literature

• Jonathan M. Borwein , Peter B. Borwein : Pi and the AGM, A Study in Analytic Number Theory and Computational Complexity . John Wiley, New York 1987, ISBN 0-471-31515-X .
• Louis Vessot King: On the direct numerical calculation of elliptic functions and integrals . Cambridge University Press, 1924, archive.org

Web links

• David H. Bailey et al. The Quest for Pi (PDF; 119 kB)
• Eric W. Weisstein : Arithmetic-Geometric Mean . In: MathWorld (English).

Individual evidence

1. ↑ Cf. Carl Friedrich Gauß: Mathematisches Tagebuch 1796–1814. With a historical introduction by Kurt-R. Beer man. Reviewed and annotated by Hans Wußing and Olaf Neumann. 5th edition. Harri Deutsch, Frankfurt am Main 2005. (Ostwalds Klassiker der exacten Wissenschaften, Volume 256.), No. 98 (Braunschweig, May 30, 1798): “ Terminum medium arithmetico-geometricum inter 1 et esse usque ad figuram undecimam comprobavimus, qua re demonstrata prorsus novus campus in analysi certo aperietur. "" We have shown up to the eleventh digit that the value of the arithmetic-geometric mean is between 1 and ; this proof will certainly open up a completely new field in analysis for us. ”This is the lemniscatic constant introduced by Gauss . ${\displaystyle {\sqrt {2}}}$ ${\displaystyle ={\frac {\pi }{\varpi }}}$ ${\displaystyle {\sqrt {2}}={\frac {\pi }{\varpi }}}$ ${\displaystyle \varpi :=2\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}}$