Talk:Arithmetic–geometric mean

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What is the arithmetic-geometric mean of 1 and 2? AxelBoldt

To answer this, I added a Scheme implementation of the algorithm. (agmean 1 2 .000000001) => 1.4567910310469068 -- Damian Yerrick

Here is an alternative implementation in Scheme that is conceptually similar but is simpler (I checked this in Guile and it works):

(define (agm a b epsilon)
  ;; Determine if two numbers are already very close together
  (define (ratio-diff a b) (abs (/ (- a b) b)))
  ;; Actually do the computation
  (define (loop a b)
    ;; If they're already really close together, just return the arithmetic mean
    (if (< (ratio-diff a b) epsilon)
        (/ (+ a b) 2)
        (loop (sqrt (* a b)) (/ (+ a b) 2)))) ; otherwise, do another step
  ;; Error checking

  (if (or (not (real? a))
          (not (real? b))
          (<= a 0)
          (<= b 0))
    (error 'agm "~s and ~s must both be positive real numbers" a b)
    (loop a b)))

What's it used for?[edit]

It's easy enough to find a use for an arithmetic mean, and a use for a geometric mean, but when do you use an arithmetic-geometric mean? — Daniel 20:52, 1 July 2007 (UTC)

I am personally not familiar with any application of the agm as is (and I guess I would be if there was anything very important). It seems to me like more of a mathematical trivia (there is nothing bad with that, of course). However, it is equivalent to a certain elliptic integral, which itself is useful in evaluating some integrals. -- Meni Rosenfeld (talk) 09:11, 2 July 2007 (UTC)
It is a remarkable computing method. The Springer OEM article is considerably more helpful than ours, and some of its most important sources, like Abramowitz & Stegun, are available online. --KSmrqT 10:49, 2 July 2007 (UTC)
It converges quadratically and inspired the fastest known method for computing the digits of π. I've added the Borwein – Borwein book to the list of references. Arcfrk 06:16, 22 July 2007 (UTC)

I've added an engineering application. Tijfo098 (talk) 03:16, 19 September 2012 (UTC)


Doesn't the AGM property hold for any (finite?) set of numbers? Not just two, as represented in the article?Akshayaj 20:20, 2 July 2007 (UTC)

What do you mean by "the AGM property"? If you mean the inequality of arithmetic and geometric means (which AFAIK is only loosely related to the topic of this article), then yes, the arithmetic mean of any set of positive numbers is at least their geometric mean (and equal iff all numbers are the same). -- Meni Rosenfeld (talk) 21:11, 2 July 2007 (UTC)
So, does this mean there ought to be some generalization of the AGM to more than 2 arguments? Would one simply take a_1 and g_1 as the normal means of an arbitrary list of real numbers and then proceed as for two arguments? Nosuchforever (talk) 21:06, 2 April 2012 (UTC)

Source code listings?[edit]

I found source code implementations in three languages on this page. Having three variations does not give encyclopedic value, and I personally hardly find much value in having an implementation at all on the page (after all, the defining equation is in there already, and is for practical purposes identical to the source code). Therefore, I was bold and removed all three of them (instead of choosing one to keep).

The source codes in question are stored in the previous version:

Remember, Wikipedia is not a source code repository, WP:NOT. --Berland (talk) 13:20, 14 May 2009 (UTC)

New long proof section[edit]

The new long proof section by Shlav seems to be not very useful, as encyclopedic content goes. Do we really need it? If we do, can we at least cite a source so we can verify it? Dicklyon (talk) 04:03, 4 June 2012 (UTC)

There are many mathematical proofs inside the Wikipedia so I think there is no problem to keep this one. I'll add reference to the source of the proof. Shlav(talk)

I vote for removing the proof -- while I think it's important to have proofs widely available, I think formal proofs are not appropriate for wikipedia in general, and this one in particular doesn't add much value. (talk) 13:37, 17 August 2012 (UTC)

I agree that the lengthy proof has minimal encyclopedic value and should be removed, it would be appropriate for a monograph rather than Wikipedia.—Emil J. 13:56, 20 November 2012 (UTC)

I agree, this proof is overdone. What people need are more identities and here's a couple I derived but since I am not of the class that gets recognition, someone else will need to independently come up with them (or they exist already but there are so few good sources so that is why I made a "blog" page that goes into some of this) so they may get posted on wiki for massive consumption:

AGM(1,sqrt(1-x))=AGM(1-sqrt(x),1+sqrt(x)) AGM(1,z)=AGM(2/(1+z),2z/(1+z))/(2/(1+z)) Numbertruth (talk) 08:22, 22 December 2012 (UTC)

Numbertruth's identities for the AGM are easily derived. Applying one round of AGM to AGM(1-sqrt(x), 1+sqrt(x)) gives AGM(1, sqrt(1-x)). The second is just an application of the scaling rule AGM(rx, ry) = r AGM(x, y). I recommend against including these in the article. cffk (talk) 12:02, 17 July 2013 (UTC)

I replaced the long proof with a much shorter (and clearer!) one. cffk (talk) 18:37, 15 July 2013 (UTC)

AGM method[edit]

Does anyone think there needs a separate article for the "AGM method"? It overlaps substantially with the contents of this one. Arguably, that one is slightly better written. Tijfo098 (talk) 03:21, 19 September 2012 (UTC)

I recommend combining these articles. cffk (talk) 19:59, 15 July 2013 (UTC)
I think they should be combined too. Bubba73 You talkin' to me? 02:53, 11 October 2013 (UTC)
I think they should be merged but possibly with "AGM method" as the title. The AGM (not method) article does not provide any motivation, context or history and as far as I know the AGM method is the reason why the AGM was defined, but I could be wrong on that. MATThematical (talk) 06:41, 6 April 2014 (UTC)

Edit Kerfuffle over minor improvement in Sec 8.1.1[edit]

Please discuss the addition of the extraneous formula π, instead of huffy reverts and score settling in the comment boxes of the edits. It is presumed that ‎PseudoScientist and 2A00:1370:8128:73E:C811:B591:263B:B6C6: are the same, or related users, attempting to thereby bypass the WP:3RR rule. Very bad karma. The edit would not be unwarranted provided the eponymy nonsense is taken out, and only survives in the reference. Internet-linking the author's personal page, however, is against WP policy, and rather fulsome. Discuss first. I'm sure this could be resolved. Cuzkatzimhut (talk) 22:17, 16 July 2016 (UTC)

@Cuzkatzimhut:: you may want to review the talk page for Ellipse -- something similar has been going on there. Just FYI. LaurentianShield (talk) 22:48, 16 July 2016 (UTC)

Indeed, User:2a00:1370:8128:73e:c811:b591:263b:b6c6 appears to be concerted (WP:SOCK?) with PseudoScientist, and involved in serial tendentious reverts and personal taunts on qualifications. I hope he can count RRRs. I for one, would be sympathetic to the insertion here, if it were not for the bizarre & crude (self?) promotional slant involved. One may have to launch a WP:COI case. I urge him to state his case at length here, instead of misusing the term "vandalize" in the edit fields. The point is not so much the content of the snippets inserted, which would make decent footnote asides, but the crude and aggrandizing framing them, instead of a NPOV reference. If he were persuasive here, and attenuated the trolling, I might undertake to insert a neutral footnote with the remark. Cuzkatzimhut (talk) 23:58, 16 July 2016 (UTC)

I guess that 2a00: and User:PesudoScientist didn't notice that the Adlaj paper is already referenced and written about in the last paragraph of the related concepts section. However important this might be, we certainly should not be repeating this twice in the same article. - MrOllie (talk) 01:46, 17 July 2016 (UTC)

I fear @PseudoScientist: doesn't get it. I asked him and his avatar to discuss matters here first, instead of specious snipes and reverts on the edit boxes, but evidently he chose the deaf ear approach. Ritual reverts cannot allow WP to become a promotional vehicle for marginal results, in my professional judgment, passing themselves as central (through indelicate priority disputes, cf. Brent's letter), all the while building personal adulation shrines.Cuzkatzimhut (talk) 02:58, 17 July 2016 (UTC)

I have asked for sock-puppet investigation of PseudoScientist and IP and FeelUs: Wikipedia:Sockpuppet_investigations/PseudoScientist. Dicklyon (talk) 04:28, 17 July 2016 (UTC)

calculation of K[edit]

the formula for K must be wrong because K(1)=infinity and this requires agm()=0, that is possible only if an argument is zero. I suspect that cos(k) should be sqr(1-k)

There is also a confusion arising from using two different definitions of K [I regret that this could cause friction between two different contributors for the nomenclature to be deleted]

pietro (I am writing from a site where I cannot run any numerical program)

2001:760:2C00:8001:192E:BA47:5CBE:D663 (talk) 15:13, 5 October 2016 (UTC)

Yes, if y=0 ? Cuzkatzimhut (talk) 19:03, 5 October 2016 (UTC) Do you mean K(sin(α))? Cuzkatzimhut (talk) The erroneous misuse of the arcsin of the elliptic modulus, K(α), appears to have been introduced Dec 19, 2015.Cuzkatzimhut (talk) 21:52, 5 October 2016 (UTC)
I believe I fixed the Dec 2015 damage, and am using consistent notation. You may see how the α=π/2 limit in that hapless section now works out. Cuzkatzimhut (talk) 22:19, 5 October 2016 (UTC)