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%I #21 Dec 02 2020 08:15:32
%S 1,9,0,9,9,2,6,2,3,3,5,4,0,8,1,5,3,2,3,7,2,2,6,7,5,1,0,9,7,8,7,5,3,3,
%T 5,5,9,1,3,5,6,2,4,4,0,8,0,2,7,2,8,4,0,5,8,3,3,8,8,5,5,5,6,8,6,6,0,2,
%U 6,6,2,8,7,1,3,2,4,5,7,9,5,1,2,7,9,9,6,1,6,7,6,1,7,5,6,4,9,8,3,2,6
%N Decimal expansion of Arithmetic-geometric mean AGM(1, 2, 3) defined as limit of the sequence x(n+1) = P(x(n)) with x(0) = (1, 2, 3) and P(a,b,c) = ((a + b + c)/3, sqrt((ab + ac + bc)/3), (abc)^(1/3)).
%C The Arithmetic-geometric mean of two values, AGM(x,y), is the limit of the sequence defined by iterations of (x,y) -> ((x+y)/2, sqrt(xy)). This can be generalized to any number of m variables by taking the vector of the k-th roots of the normalized k-th elementary symmetric polynomials in these variables, i.e., the average of all products of k among these m variables, with k = 1 .. m. After each iteration these m components are in strictly decreasing order unless they are all equal. Once they are in this order, the first one is strictly decreasing, the last one is strictly increasing, therefore they both converge, and their limits (thus that of all components) must be the same.
%C Has this multi-variable AGM already been studied somewhere? Any references in that sense or formulas are welcome.
%C Other 3-argument generalizations of the AGM have been proposed, which all give different values whenever the three arguments are not all equal: replacing P(a,b,c) by (agm(a,b), agm(b,c), agm(a,c)) or (agm(a,agm(b,c)), cyclic...) one gets 1.9091574... resp. 1.9091504..., but these are less straightforwardly generalized to a symmetric function in more than 3 arguments. Using the average of the k-th roots rather than the root of the average (normalized elementary symmetric polynomial) yields 1.89321.... See the two StackExchange links and discussion on the math-fun list. [Edited by _M. F. Hasler_, Sep 23 2020]
%H Brad Klee, <a href="https://mailman.xmission.com/cgi-bin/mailman/private/math-fun/2020-September/035798.html">Iterated averaging of triples</a>, math-fun list (available for subscribers), Sep 18 2020.
%H User Mathlover, <a href="https://math.stackexchange.com/questions/442062/to-find-the-limit-of-three-terms-mean-iteration/3806368">To find the limit of three terms mean iteration</a>, math.StackExchange.com, Jul 12 2013.
%H Vladimir Reshetnikov, <a href="https://math.stackexchange.com/questions/1794795/arithmetic-geometric-mean-of-3-numbers">Arithmetic-geometric mean of 3 numbers</a>, math.StackExchange.com, May 22 2016.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean">Arithmetic-geometric mean</a>, created Jan 2, 2002.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial">Elementary symmetric polynomial</a>, created Jan 28, 2005.
%e 1.90992623354081532372267510978753355913562440802728405833885556866...
%o (PARI) f(k,x,S)={forvec(i=vector(k,i,[1,#x]), S+=vecprod(vecextract(x,i)),2); S/binomial(#x,k)} \\ normalized k-th elementary symmetric polynomial in x
%o AGM(x)={until(x[1]<=x[#x],x=[sqrtn(f(k,x),k)|k<-[1..#x]]);vecsum(x)/#x}
%o default(realprecision,100);digits(AGM([1,2,3])\.1^100)
%Y Cf. A332091 = AGM(1,1,2), A332092 = AGM(1,2,2).
%Y Cf. other sequences related to the AGM (of two numbers): A061979, A080504, A090852 ff, A127758 ff.
%K nonn,cons
%O 1,2
%A _M. F. Hasler_, Sep 18 2020
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