Aritmetisk-geometriskt medelvärde

Det aritmetisk-geometriska medelvärdet (AGM) är ett medelvärde av två tal som fås genom att ta deras aritmetiska respektive geometriska medelvärden och i oändligheten rekursivt upprepa samma procedur med dessa. Givet två tal x och y, erhålles agm(x, y) utifrån

a_{1}=x,\,\!

b_{1}=y,\,\!

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

och

b_{n+1}={\sqrt {a_{n}b_{n}}}.

Sekvenserna a och b konvergerar mot ett gemensamt värde, vilket ger det aritmetisk-geometriska medelvärdet,

\mathrm {agm} (x,y)=\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}

Det aritmetisk-geometriska medelvärdet av två tal ligger alltid mellan talens geometriska och aritmetiska medelvärden. Om r > 0 gäller också att

\mathrm {agm} (rx,ry)=r\,\mathrm {agm} (x,y).\,\!

Andra egenskaper är,

\mathrm {agm} (x,y)=\mathrm {agm} (y,x)

\mathrm {agm} (x,x)=x

Det aritmetisk-geometriska medelvärdet utnyttjas bland annat av Gauss-Legendres algoritm som är ett mycket effektivt sätt att beräkna π numeriskt. Gauss konstant, G, kan också definieras som reciproken av det aritmetisk-geometriska medelvärdet av 1 och roten ur två,

G={\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}.

Relationen till elliptiska integraler

Anta att $0<k<1$ . Det aritmetisk-geometriska medelvärdet går att uttrycka med hjälp den fullständiga elliptiska integralen av första slaget.

\mathrm {agm} (1,k)={\frac {\pi }{2}}{\frac {1}{K(k')}}

Där $k'={\sqrt {1-k^{2}}}$ .

Det går att visa att k även kan vara mer än 1. När man räknar ut Gauss konstant stoppar man i formeln ovan så att $k={\sqrt {2}},(k'={\sqrt {-1}})$ .

K({\sqrt {-1}})={\frac {\pi }{2\mathrm {agm} ({\sqrt {2}},1)}}

Det går att på detta sätt visa att det aritmetisk-geometriska medelvärdet i väsentligt är detsamma som den fullständiga elliptiska integralen av första slaget. Genom detta kan man härleda flera egenskaper av den elliptiska integralen av första slaget.

Källor

Takebe, Takashi (2023) (på engelska). Elliptic Integrals and Elliptic Functions [Elektronisk resurs] (1). Springer. Libris 8rmwjgmm6fzzf20b. ISBN 978-3-031-30267-1

v • r Medelvärden

Aritmetiskt · Viktat aritmetiskt · Geometriskt · Harmoniskt · Kvadratiskt · Kubiskt · Aritmetiskt-geometriskt · Generaliserat · Glidande

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