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Some observations on algorithms of the Gauss-Borchardt type. (English) Zbl 0746.39006

The author continues in the informal style of his 1989 paper [Int. J. Math. Math. Sci. 12, No. 2, 235-245 (1989; Zbl 0707.26005)]: He heaps lavish praise on the papers of J. M. Borwein and P. M. Borwein [Math. Comput. 53, No. 187, 311-326 (1989; Zbl 0675.30023) and Proc. Am. Math. Soc. 323, 691-701 (1991; Zbl 0725.33014)] and is somewhat vague in stating his own results: “we need not be very explicit about the assumptions made”, “ the following considerations are somewhat formal” (meaning that they consist of formal calculations “without being very explicit about the assumptions made”), “with appropriate assumptions and with appropriate interpretation of the summation”, “we assume that we are in the situation that there is an asymptotic formula”, etc.
To put it in another way, the author stops writing his paper(s) where others start: after making some calculations and guessing what kind of results to expect but before doing the housecleaning job of formulating assumptions and thus results exactly. So it is somewhat difficult to report on solidly proved results in his review.
We just state that the paper is about algorithms of the form \[ a_{n+1}=(a_ n+(\lambda-1)b_ n)/\lambda, \qquad b_{n+1}=(2(a_ nb_ n)^{1\over2}+(\lambda-2)b_ n)/\lambda \qquad (n=0,1,2,\dots) \] (\(\lambda=2\) gives Gauss’s medium arithmetico-geometrium algorithm, \(\lambda=4\) a truncated Borchardt algorithm) and some of their generalizations, their convergence, the functional equations attached to them, existence and uniqueness of the latter, fixed points, asymptotic formulas, etc. In the first four sections \(a_ 0\) and \(b_ 0\) are real, in Section 5 they are supposed to be complex and the convergences of the double values of square roots are considered.
Nothing said at the beginning of this review is meant to indicate that the contents of this paper are not interesting: they are, and they would deserve a more careful and exact exposition.

MSC:

39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
39B32 Functional equations for complex functions
26A18 Iteration of real functions in one variable
47H10 Fixed-point theorems
37B99 Topological dynamics
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
65D15 Algorithms for approximation of functions
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
33C75 Elliptic integrals as hypergeometric functions
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References:

[1] Borwein, Pi and the AGM–A Study in Analytic Number Theory and Computational Complexity (1987)
[2] Arazy, Proc. of the Nineteenth Nordic Congress of Mathematicians, Reykjavík, 1984 pp 191– (1985)
[3] DOI: 10.1155/S016117128900027X · Zbl 0707.26005
[4] Nishiwada, Proc. Japan Acad. Ser. A64 pp 322– (1988)
[5] DOI: 10.2307/2008364 · Zbl 0675.30023
[6] Peetre, Colloquia Mathematica Societatis Janos Bolyai 49 pp 711– (1986)
[7] Cox, Enseign. Math. 30 pp 275– (1984)
[8] DOI: 10.2307/2001551 · Zbl 0725.33014
[9] Gauss, Bestimmung der Anziehung eines elliptischen Ringes. Nachlass zur Theorie des arithmetisch-geometrischen Mittels 225 (1927)
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