Generalizing the arithmetic geometric mean - a hapless computer experiment. (English) Zbl 0707.26005

As the title shows, this paper contains more of a chat about the author’s trials and tribulations around Gauss’s medium arithmetico-geometricum and similar iterative processes of Meissel and of Borchardt (which start with 3 or 4 numbers, respectively) than solid results. He surveys, however, several of those too and states some problems and conjectures, one of which, as he points out at the end of the paper, has been solved “in a truly marvelous paper” by J. M. Borwein and P. B. Borwein [Math. Comput. 53, No.187, 311-326 (1989; Zbl 0675.30023)].
{Reviewer’s remark: The paper is carelessly typeset: abstract is twice printed, “link” in the first line of p. 236 should be “line”, etc. There are also some unusual notations, like “\(=^{fed}\)” for “=:”.}
Reviewer: J.Aczél


26A18 Iteration of real functions in one variable
37B99 Topological dynamics
39B12 Iteration theory, iterative and composite equations
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)


Zbl 0675.30023
Full Text: DOI EuDML