GENERALIZING THE ARITHMETIC GEOMETRIC MEAN A HAPLESS COMPUTER EXPERIMENT

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 rSle 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 rSle of the algorithms of Gauss and Borchardt.


GENERALIZING THE ARITHMETIC GEOMETRIC MEAN
A HAPLESS COMPUTER EXPERIMENT JAAK PEETRE
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 rSle 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 rSle of the algorithms of Gauss and Borchardt.
I have now worked on algorithms of the type of Gauss's arithmetic-geometric mean (agM.) for a period of nearly 4 years (starting around the turn of the year 83]84).Strangely enough some of the impetus for getting interested in this field came from the theory of (abtract) interpolation.This connection is described in my talk to the Varna conference in May/June 1984 [P1].The same year I also prepared an over all survey of "means and their iterations" for the XIXth Nordic Mathematical Congress in Reykjav [ACJP] (with J. Arazy, T. Claesson and S. Janson as coauthors).I took up the same subject the year after for my address to the A.Haar Memorial Conference in Budapest [P2], which is a collection of "unsolved problems", some of them pertaining to the agM.In particular, suggested there a certain approximation for a 2-dimensional algorithm derived from the "agM."corresponding to the cyclic group C3 with three elements analogous to an asymptotic formula in a note by E. Meissel [M] and also the classical asymptotic formula due to Gauss [G].However, numerical evidence produced later by P. Borwein [Borw] indicates that it here can't be question of a true asymptotic formula.
think now that fell in the trap of relying too much on information derived from unsufficient numerical data.Hence the second link of the title.(I believe now that the claim in [P1] concerning the asymptotic behavior of iterated power means experiences the same fate.)Thus the question of finding good (asymptotic) approximations for the algorithm of the type in question is by and large open.
The main purpose of this paper is to provide the reader a general background for this problem, and to outline the meager progress myself have made on it.Maybe, can thereby inspire other people to continue where stopped (failed)...As Grunert's Arehiv is nowadays virtually inaeeesible for most readers and as the common of knowledge of Western languages, English excluded, among mathematicians is in such a sorrow state, have included a translation of Meissel's note [M] in eztenso (see Appendix).
1. THE agM.G.F. Gauss [1777-1855] studied the agM.from an early age on (some say 14).Let a,b be two real numbers, 0 < b _< a < cz.Taking succesively arithmetic and geometric means we get a", and b, b', b" with two sequences a, a', It is easy to see that they converge to a common limit, M(a, b)tlim a,, limb,,, called the agM. of a and b. (We have b _< b' _< b" _< _< a" _< a' _< a and .,_e (/z-) The agM. and other related "means" are discussed in the survey [ACJP].(For a more ex- tensive treatment see e.g. the book [BB].)From there we recall only the following.
Basic properties of the Gauss agM.: 1) Extremely rapid convergence ("quadratic" in the techrfical sense).2) Integral representation.As Gauss discovered, one has - [,,/2   M(a,b) ao de v/a2 cos + b 2 sin 2 ' where the integral to the left is a complete elliptic integral of the first kind.In standard notation, writing k b/a ("modulus"), k' y/1-b2/a 2 ("complementary modulus"), the latter is just g(k')/a (cf.[BB], theorem 1.1).Thus the formula can also be written

M(a,b)=a
The importance of the agM.therefore stems in part from the fact that it connects with "elliptic" theory (elliptic integrals, differentials, functions, curves).k(k 1)y" + (3k 1)y' + ky O, which after a change of variable becomes a special case of the hypergeometric equation, likewise considered by Gauss. 4)Uniformization.Writing a and b in the form a Mp2(x),b= Mq2(x) where p and q are, what Gauss calls, summatoric functions (= theta values; in conventional notation p(x) Ooo(O,t), q(x) Ool(O,t) where x eit), the algorithm reduces to z x2. 5) Complex values.Using the uniformization it is possible to extend the algoritm to the case of complex values of a and b.Then the limit M(a, b) is not any longer unique and the values corresponding to different "determinations" of it are related by a modular transformation.This is also one of the historic roots of the theory of modular functions.6) Asymptotic formula.Central in Gauss's treatment of the agM. is the asymptotic formula M(1, k) (k 0).log I can be readily derived from the following formula for the uniformizing parameter, likewise due to Gauss (see again the discussion in [nn]): 2. BORCHARDT'S GENERALIZATION OF THE agM.WITH FOUR "ELEMENTS".
C.W. Borchardt [1817-1880] was a student and close friend of Jacobi's.He also edited volume one of Jacobi's collected works and became the editor of "Crelle's Journal" after Crellc's death; therefore this journal was for some time known as "Borchardt's Journal".He proposed as a generalization of the agM. the following scheme based on the iteration of the transformation It turns out that this algorithm has a theory entirely parallel to Gauss's.In particular, one has proper counterparts of all the properties 1)-6) mentioned in Sec. 1, with the possible exception of 6).The "uniformization" is now obtained by theta functions in two variables and in place of the elliptic curve now enters Kummer's quartic surface.Borchardt also briefly indicates a 2'*-dimensional generalization, but for various reasons this algorithm is far more defective, largely because theta functions in n variables do not suffice for purposes of uniformization.We quote ([Bore], p. 621): "... but of which kind these transcendental functions are, in terms of which the limit can be represented, is a question whose answer must be left for the future".
In [ACJP] it is pointed out that both Gauss's and Borchardt's algorithm are special cases of the following very general construction.Let G be any compact group.Let f be a positive meast/rable function in the Lebesgue space L1/2(G).Then f' vr] v/]", where stands for convolution (with Haar measure so normalized that 1 1 1), is a function of the same type (in fact, in L'(G)).Similarly, f" v/-f v/ is a continuous function (in C(G)).
Continuing we obtain a sequence of functions which, as is proved in [ACJP], tends pointwise, in fact uniformly, to a constant functions, denoted M(f) or Ma(f) (and identified with the corresponding number).
It is likewise mentioned in [ACJP] that if we pass to complex valued functions f, assuming that Ref > 0, the same convergence holds true, but this is much harder to prove.
EXAMPLE.For instance, if G ZI" and if f is analytic inside the unit disk D (we make the identification OD Z/') then M(f)= f(0).
We are however mostly interested in the case of finite groups.
EXAMPLE.If G C2 (the cyclic group with two elements) we get back Gauss's algorithm and if G C2 x 6'2 we get Borchardt's algorithm.Borchardt's generalization with 2" elements similarly corresponds to G C'.The first non-classical case is thus G C3 (the cyclic group with three elements).Spelled out explicitly it is thus question of iterating the map a+b+c Let us return to the case of a general group G.For simplicity we take G finite.Let H be any subgroup of G.We restrict attention to functions f which are constant on H and on the complement HC G\H, i.e.
{a /-/ f= b H Then the iterates are of the same type: where N la" HI IGI/IHI is the index of H in G. (By Lagrange's theorem we know that N is an integer.)Thus we are lead to consider, quite generally, the 2-dimensional algorithm (1) where 0 is any number in the interval (0, 1/2].
Assume first that G1.Then we have the following possibilities: j Ga, k G1 coesponds to a te al, j G2G1, k G2G corresponds to a te 2a j GG-I, k GG_i corresponds to a term a.
This accounts for the foula for a.
Next sme that G2G.Then we have the following possibilities: j GG2, kG2 G orj j =, a=ka opona to re= (= -)== j GsG2, k GzG corresponds to a te sas j GG_x, k GG_a corresponds to a term aa.
This accounts for the foula for a.
In the se way one treats the ce GsG2.The proof is concluded by induction argument.# 5. MEISSEL.
In a short note [M] (of.Appendix to this paper) published in 1874 in Grunert's Archiv (D.E.)F.Meissel [1826Meissel [ -1895]], who was headmaster of a secondary school in Kiel, Germany, toward the end of the last century, considers the iteration a+b+c +ac + bc v/ab 3 In particular, he states without proof the following asymptotic formula for the corresponding limit M(a, b, c): ( 2 (The exact meaning of is not clear from the context; actually, Meissel himself writes simply in formula (2).) In Sec. 8 will give an attempt to justify Meissel's formula (1).
REMARK.In the excellent book [BB], p. 268-269, this algorithm is called Schl6milch's algorithm and in this context reference is made to a paper by Schoenberg's [Scho], which however has been inaccessible to me.have checked with the index of Jahrbuch up to and including the year 1903 (the year of my mother's birth) and found only one paper of SchlSmilch's dealing with iteration of means, namely [SchlS], but it does not seem to be very relevant in the present context.
Guided by the asymptotic formulae by Gauss and Meissel suggested in [P2] the following approximtion of the limit M(a, b) of the iterations of the transformation (1) in Sec. 4" made also numerical experiments on a minicomputer, which to some extent seemed to support my guess.Here are some values for the constant A obtained: N= 2 A 1.570796327... N= 3 A 2.3410... N= 4 A 3.289868133... N= 5 A 4.420... N= 6 A 5.738... N= 7 A 7.251... N= 8 A 8.96...I thought at the time that had a rigorous proof of (3) interpreted as a genuine asymptotic formula so did not pay much attention to the fluctuations in the numerical data, which seemed to increase with N. Of course, as (3) reduces to Gauss's formula if N 2 there are no such fluctuations in this case and also not in Borchardt's case (N 4).Therefore Borwein's letter [Borw] came as a surprise, if not a shock.However, now quickly realized that the fluctuations really were part of the picture (in all other cases but N 2, 4) and not something connecting with the insufficiency of the numerical device available to me.This will be explained in the next Section.
The numerical experiment thus suggests only the following: There is an exact asymptotic formula only if N 2,4 and there can be an integral formula .for the limit of the Gauss- Borchardt type only in these two cases.But, emphasize, this is something that have not proved so it is therefore just question of another conjecture.In the former case A r/2 exactly and the latter case probably A 2/3.
7. EXPLANATION OF THE FLUCTUATION.
In inhomogeneous notation (M(b) M(1, b)) the functional equation for the limit/ll of the algorithm (1) can be written where we for convenience have put ()

+ ( ).
It is easy to see that the only solution of the functional equation ( 4) which is analytic near b 1 normalized by the requirement M(1) 1 comes as an infinite product REMARK.The letters Z and N are picked in honor of Gauss (for German Zhhler, nominator, and Nenner, denominator, respectively).Do not confuse the function N N(b) and the previous integer valued parameter N (= the index of the subgroup H the finite group G); see Sec. O A0 + A1 cost + Ba sin + A2 cos 2t + B2 sin 2t + We see thus that, in a way, what we have observed in the numerical experiment is just the 0-th coefficient A A0 in this expansion.
PROBLEM: To account for the "higher" terms! 8. ATTEMPTS TO JUSTIFY MEISSEL'S FORMULA.

B(t)
B(k_l) V,B(k_I)) 1 1 1 B() B(-) () .(. ,) .(. ,,)r Thus A and B do "exist".It is however not at all clear to me in what sense M is approximated in terms of the functions A and B. NOTE (added Sept. 1988).Recently J. and P. Borwein sent me a truly marvelous paper a+3b )" where in particular the conjecture entitled "On the mean iteration (a, b) --4 2 about the asymptotic behavior of the Borchardt mean at the end of Sec. 6 is fully established.
Even more, the authors show that the mean in question shares with the agM.practically all of its remarkable properties (see See. 1), in particular thus that the algorithm can be "uniformized" in terms of theta constants.

1 3 )
Differential equation.The function y satisfies a differential equation vith M(1, k) algebraic coefficients (known as Legendre's differential equation) DERIVED FROM THE "MONSTER" ALGORITHM.
) M(1 1 c)( A B )x c-O log where A 0.43331485... is obtained, upon eliminating of k, from the equations:

N
g(b)g(b')g(b")...This is connected with the fact that b 1 is an attractive (even hyperattractive) that g(1)-1, which guarantees the convergence of the product in (5). (In (5) b',b",..., of course, are the iterates of b under this map.)Next,consider instead the functional equation a repulsive fixed point for the map bb', hence an attractive (again even hyperat- tractive) one for the inverse map b H 'b. (If b is small then b' 2v/ so that for the inverse holds b H 'b .b2/4.)Also h(0) 1.It follows, by the same reason as before, that there is a unique solution of (5), analytic at b 0 and normalized by Z(0) 1.One can thus write Z(b) 1 + Zlb + z2b 2 + and it is possible to write down a recursion for the coefficients zn, which generalizes the one used by Gauss [G].Finally, consider N() (N(')) .This is essentially Bhttcher' equation for the inverse map (see e.g.[ACJP]).It follows that there exists a unique solution which admits the expansion 4 4. Now it is clear that the function satisfies the same functional equation (4) as our mean M. The quotient is a function which is invariant under b b'.It follows that we have M M'.O, where O is an "oscillatory" function, that is, invariant under the transformation b b' (O(b)O(b')).Now the presence of the fluctuations is completely explained.Our above conjecture (end of See. 6) amounts therefore to O const iff N 2, 4.REMARK.A natural parameter near b 0 seems to be 1/N(b) ) Therefore one can go a step further an ask for the Fourier development of the function O: Now we make the "Ansatz":M(a,b,c) A(log B---)-'x,where A A(a, b) is supposed to be homogeneous of degree 1 and B B(a, b) homogeneous of degree 0. Plugging this into the functional equation we find (a', b'), B' B(a', b').