GENERALIZING THE ARITHMETIC GEOMETRIC MEAN - A HAPLESS COMPUTER EXPERIMENT

JAAK PEETRE

《International Journal of Mathematics and Mathematical Sciences》 1989卷 (1989/12) Pp. 235-245

https://doi.org/10.1155/S016117128900027X

摘要

The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ＂hapless computer experiment＂ in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ＂fluctuations＂ are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt. The paper discusses the asymptotic behavior of generalizations of the Gauss's arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The ＂hapless computer experiment＂ in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general ＂fluctuations＂ are present. However, no very conclusive results are obtained so the paper ends in a conjecture concerning the special rôle of the algorithms of Gauss and Borchardt.

關鍵字

Arithmetic-geometric mean ； iteration ； algorithm ； asymptotic formula