Generalizing the arithmatic geometric mean — a hapless computer experiment

GENERALIZING THE ARITHMETIC GEOMETRIC MEAN

A HAPLESS

COMPUTER

EXPERIMENT

JAAK PEETRE Matematiskainstitutionen

Box

6701

S-113 85 Stockholm Sweden

(Received

May 1, 1988 andinrevised form

Sept.20,1988)

ABSTRACT. The paper discusses the asymptoticbehavior ofgeneralizations of theGauss’s

arithmetic-geometricmean, associated with the names Meissel

(1875)

and Borchardt

(1876).

The"hapless computer experiment"inthetitlerefers to the fact that the author atanearlier

stagethought that onehad genuine asymptotic formulae butit isnow shown that ingeneral

"fluctuations"arepresent.

However,

novery conclusiveresultsareobtainedsothe paper ends

in a conjecture concerning the special rSle of the algorithms ofGauss and Borchardt. The paperdiscussesthe asymptoticbehaviorofgeneralizationsof the

Gauss’s

arithmetic-geometric

mean,associated withthenamesMeissel

(1875)

and Borchardt

(1876).

The"hapless computer experiment"inthetitlerefers tothe fact that the author at anearlierstage thought that one

had genuine asymptotic formulae butit is nowshown thatingeneral"fluctuations"arepresent.

However,

novery conclusiveresultsareobtainedsothe paper endsinaconjecture concerning

thespecialrSleof thealgorithmsofGaussandBorchardt.

KEY

WORDS AND PHRASES.Arithmetic-geometric mean, iteration,algorithm,asymptotic formula

1980

AMS

SUBJECT

CLASSICATION CODES.

26E99, 26-03, 01A55

0.

INTRODUCTION.

Ihavenowworkedon algorithmsof thetype ofGauss’sarithmetic-geometricmean (agM.)

foraperiod of nearly4 years(startingaround 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 theVarna conference inMay/June 1984

[P1].

Thesameyear

I

also preparedanoverallsurveyof

"means

andtheiriterations" for the XIXth Nordic Mathematical

Congress

in Reykjav

[ACJP]

(with

J.

Arazy,

T. Claesson andS. Janson as

coauthors).

Itookupthesamesubject the yearafter formyaddresstothe A.Haar MemorialConferenceinBudapest

[P2],

whichisa collection of"unsolved problems",

someofthempertainingtothe

agM.

Inparticular, suggestedthere acertain approximation

fora2-dimensional algorithmderivedfrom the

"agM."

correspondingtothe cyclicgroup

C3

withthreeelements analogoustoanasymptoticformulain anotebyE.Meissel

[M]

andalso

numerical data. Hence the second link ofthe title.

(I

believe now that the claim in

[P1]

concerningtheasymptoticbehaviorof iterated powermeansexperiencesthesame

fate.)

Thus

the question offinding good (asymptotic) approximations for the algorithm ofthe type in question isby andlargeopen.

The main purpose of this paper is to provide the reader a general background for this

problem, and to outline themeagerprogress myself havemade onit. Maybe, canthereby inspireotherpeopleto continuewhere stopped

(failed)...

As

Grunert’s

Arehiv isnowadaysvirtuallyinaeeesibleformostreadersandasthecommonof

knowledgeofWestern languages,Englishexcluded,amongmathematicians is in suchasorrow

state, haveincluded atranslationof Meissel’s note

[M]

in eztenso

(see

Appendix).

1.

THE

agM.

G.F. Gauss

[1777-1855]

studiedthe

agM.

from anearly age on

(some

say

14).

Let a,bbe tworealnumbers, 0

<

b

_<

a

<

cz. Taking succesivelyarithmeticand geometricmeans weget

a",

andb,

b’,

b" with two sequences a,

a’,

a+b

b’

2

a’

b’

a"-

+

b"-2

Itis easy toseethattheyconvergetoa commonlimit,

M(a,

b)tlim

a,, limb,,,

called the

agM.

ofa and b.

(We

have b

_<

b’

_<

b"

_<

_< a"

_<

a’ _<

a and

.,_

e

(/z-

)

2

5

(

v-v

-

)=

<5

l(a-)

1

([G],

p.

28).

The

agM.

and other related

"means"

arediscussed inthe survey

[ACJP]. (For

a more ex-tensivetreatment seee.g. the book

[BB].)

Fromtherewerecallonlythe following.

Basic properties of theGauss

agM.:

1)

Extremelyrapid convergence("quadratic" inthetechrfical

sense).

2)

Integralrepresentation. As Gauss discovered,onehas

-

[,,/2

M(a,b)

ao

de

v/a2

cos

+

b2_sin2

’

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

latterisjust

g(k’)/a

(cf. [BB],

theorem

1.1).

Thusthe formulacanalsobe written

M(a,b)=a

1

3) Differential

equation. The function y satisfies a differential equation vith

M(1, k)

algebraiccoefficients

(known

as Legendre’sdifferentialequation)

k(k

1)y"

+

(3k

1)y’

+

ky O,

whichafterachangeofvariablebecomesaspecialcaseof thehypergeometricequation,likewise consideredbyGauss.

4) Uniformization.

Writinga and binthe form

a

Mp2(x),b= Mq2(x)

(M

M(a,b)),

wherep and q are, what Gauss

calls,

summatoricfunctions

(=

thetavalues; in conventional notation

p(x)

Ooo(O,t),

q(x)

Ool(O,t)

wherex

eit),

the algorithm reducestoz x

2.

5)

Complexvalues. Using theuniformization it ispossibletoextend the algoritmtothecase of complexvaluesofa and b. Then thelimit

M(a, b)

is not any longerunique and the values

corresponding to different"determinations" ofit are related by a modular transformation. This isalsooneofthehistoric rootsofthetheory of modularfunctions.

6)

Asymptotic

formula.

CentralinGauss’streatmentof the

agM.

isthe asymptotic formula

M(1,

k)

(k

0).

log

I

canbereadilyderivedfrom thefollowingformulaforthe uniformizing parameter, likewise

duetoGauss

(see

again thediscussion in

[nn]):

2. BORCHARDT’S GENERALIZATION OF

THE agM.

WITH FOUR "ELEMENTS". C.W.Borchardt

[1817-1880]

was astudent and closefriendof Jacobi’s. Healso editedvolume

oneof Jacobi’s collected works and became theeditorof"Crelle’sJournal" after Crellc’sdeath;

therefore thisjournal wasfor sometimeknown as "Borchardt’s Journal". He proposed as a generalization of the

agM.

thefollowingschemebasedontheiterationof the transformation

bf

C

a+b+c+e

4 2 2

Itturnsout thatthisalgorithmhasatheory entirelyparallelto

Gauss’s.

Inparticular,onehas propercounterpartsof all the properties

1)-6)

mentionedinSec. 1,with thepossible exception of

6).

The "uniformization" isnowobtained by thetafunctions in two variables andinplace

of the elliptic curve now enters

Kummer’s

quartic surface. Borchardt also briefly indicates a2’*-dimensionalgeneralization, but forvarious reasons thisalgorithmis far more defective,

largelybecause thetafunctions innvariablesdo notsuffice forpurposes ofuniformization. We quote

([Bore],

p.

621):

"... but of which kind thesetranscendentalfunctions are, in terms of which thelimitcanberepresented,isaquestion whoseanswermustbeleft for thefuture".

3. THE

"MONSTER"

ALGORITHM.

In

[ACJP]

it ispointed out that both

Gauss’s

andBorchardt’s algorithm are specialcases

ofthefollowingvery generalconstruction. Let Gbeany compact group. Let

f

be apositive

type

(in

fact, in

L’(G)).

Similarly,

f"

v/-f

v/

is a continuous function

(in

C(G)).

Continuingweobtainasequence of functionswhich,as is provedin

[ACJP],

tends pointwise,

in fact uniformly, toa constant functions, denoted

M(f)

or

Ma(f) (and

identified with the corresponding

number).

Properties:

O) M(1)=

1.

1) M(Af)

AM(f) (A

>

0)

(homogeneity).

2)

inf

f

< M(f) <

sup

f.

3) M(f)

<

M(9)

if

f

<

9 a.e. (monotonicity).

4)

M(f

+

9) >

M(f)

+

M(9)

(concavity).

Itislikewise mentioned in

[ACJP]

that ifwepass tocomplex valuedfunctionsf, assuming that

Ref >

0,thesameconvergence holdstrue, butthis ismuch harderto prove.

EXAMPLE. Forinstance, ifG

ZI"

andif

f

isanalyticinsidetheunit disk

D

(we

make the

identification

OD

Z/’)

then

M(f)= f(0).

Wearehowevermostlyinterested inthe caseoffinitegroups.

EXAMPLE. IfG

C2 (the

cyclic groupwithtwo

elements)

weget backGauss’s algorithm

andifG

C2

x

6’2

weget Borchardt’s algorithm. Borchardt’s generalizationwith 2"elements similarlycorrespondstoG

C’.

Thefirst non-classicalcaseisthusG

C3 (the

cyclic group

withthree

elements).

Spelledoutexplicitlyit isthus questionofiterating themap

a+b+c

a

3

2v+c

3

c’=

2/"+

b

3

4.

ALGORITHMS

DERIVED

FROM

THE

"MONSTER"

ALGORITHM.

Let usreturn to the case ofageneralgroup G. For simplicitywe take Gfinite. Let

H

be

anysubgroup of

G.

Werestrict attentiontofunctions

f

which areconstant on

H

andonthe

complement HC

G\H,

i.e.

{a

/-/

f=

_b

H

Then theiteratesareof thesametype:

f,=

{

a

+

(1

-)b

fe=d

a’

’

+

(1

-)b

fe_d

b’

f’=

7+

(1

)b’

whereN

la"

HI

IGI/IHI

isthe indexof

H

in G.

(By

Lagrange’s

theoremwe knowthat Nisaninteger.) Thuswe arelead toconsider, quitegenerally,the2-dimensional algorithm

(1)

a’=

0(1

a)

+

0b

b’

20V/’

+

(1

20)b

0(1

a)

+

Ob

0(

v/)

2’ where0is anynumberin the interval

(0,

1/2].

Restricting attention to functions

f

which are constant on the sets

G,\G,-1

(i

1,...,r),

that is,

f[al

al,

f[a,\al

a2,

f[a.ka.-I

ar

with

[GI[--

o1,

[G2\GI[

a:,

IG\G-ll

,

1

+-.-+r

Ial,

we refacedwiththe r-dimensional gorithm

(0,

,/Ial)

a

01

a

+

+

Ora

201+

(1’)

a

201 + 202 +

(03

01

02)

3

+

04a

4

+

+

Or

a=

20a+202+...+20_a_aa+(O-Oa-...-O_a)a

Noticethat one cMsowrite

=’

011

+

+

0=

0

(

)=

0=(

)=

-0_,(-

)

PROOF OF

(1’).

Indeedconsidertheequation jk.

Assumefirstthat

G1.

Thenwehave thefollowingpossibilities:

j

Ga,

k

G1

coesponds toateal, j

G2G1,

k

G2G

correspondstoate2a j

GG-I,

k

GG_i

corresponds toaterm

a.

This accountsfor the foula for

a.

Next smethat

G2G.

Thenwehave thefollowingpossibilities: j GG2,

kG2

G

orj

j

=,

a=ka

opona

to

re= (= -)==

j

GsG2,

k

GzG

correspondstoate

sas

j

GG_x,

k

GG_a

corresponds toaterm

aa.

Thisaccountsforthe foula for

a.

In the se way one treats the ce

GsG2.

The proofis 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

[1826-1895],

whowas headmaster of a secondary schoolin Kiel, Germany,

toward the end of the last century, considerstheiteration a+b+c

+ac

+

bc

v/ab

3

Inparticular,he states withoutproof the followingasymptotic formulafor thecorresponding

limit

M(a,

b,

c):

(2)

M(1

1

c)(

A

B

)x

c-O

log

3

3

I+

V3

and

A

and

B

arecertainsconstants

A

0.3951642...,

B

10

’299z49"’’.

(The

exactmeaningof isnot clearfrom thecontext; actually,Meissel himself writessimply

informula

(2).)

In

Sec. 8 willgiveanattempttojustifyMeissel’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 toand

includingtheyear1903

(the

yearofmymother’s

birth)

andfoundonlyonepaperof SchlSmilch’s

dealingwith iterationof means,namely

[SchlS],

butitdoesnot seemtobeveryrelevantinthe present context.

6. DISCUSSION

(THE

COMPUTER

EXPERIMENT).

Guidedbythe asymptoticformulae by Gaussand Meissel suggestedin

[P2]

the following

approximtion ofthelimit

M(a,

b)

of theiterationsof thetransformation

(1)

inSec. 4" 1

(3)

M(1,b)A

₄

(b-,O)

(og

)

made also numerical experiments on a minicomputer, which to some extent seemed to

support myguess. Herearesomevalues 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 thetime that had arigorous proofof

(3)

interpreted as agenuine asymptotic formulaso didnotpaymuchattention tothefluctuations inthenumericaldata,whichseemed to increase withN. Of course, as

(3)

reduces to Gauss’sformulaifN 2thereare nosuch

fluctuations in thiscase and also not in Borchardt’scase

(N

4).

ThereforeBorwein’s letter

[Borw]

cameas asurprise, if notashock.

However,

nowquicklyrealized that the fluctuations

reallywerepart of the picture

(in

all othercasesbutN 2,

4)

and not something connecting

withthe insufficiency of the numerical device available tome. This will be explained in the next Section.

Thenumerical experiment thussuggests 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 somethingthat have not

provedsoitistherefore just question of another conjecture.

In

the formercase

A

r/2

exactly and the lattercaseprobably

A

2/3.

7. EXPLANATION OF

THE

FLUCTUATION.

Ininhomogeneousnotation

(M(b)

M(1, b))

thefunctionalequationfor thelimit/llof the

(4)

M(b)-g(b)M(b’),

wherewefor conveniencehave put

()

+

(

).

It is easy to seethat the only solution ofthe functional equation

(4)

whichis analyticnear b 1 normalizedby the requirement

M(1)

1comes as aninfiniteproduct

(5)

M(b)

g(b)g(b’)g(b")...

This is connected with the fact that b 1 is anattractive

(even

hyperattractive) fixed point

ofthe map

20vf

+

(1

20)

b-b’

+

(

)

andthat

g(1)-

1,whichguarantees the convergence of the productin

(5). (In (5)

b’,b",...,

of

course,aretheiteratesofbunderthis

map.)

Next,

consider insteadthefunctionalequation

(6)

Z(b)

h(b)Z(b’),

wherenow

h(b)

g(b)

0

+

(1

0)b

b 0 is arepulsive fixedpoint for themapb

-

b’,

hence an attractive (againeven

hyperat-tractive)

onefor the inverse map bH’b.

(If

b issmall then b’

2v/

so that fortheinverse

holdsbH ’b

.

b2/4.)

Also

h(0)

1. It follows, by thesamereason asbefore, thatthere is a

uniquesolutionof

(5),

analyticat b 0 and normalizedby

Z(0)

1. Onecanthuswrite

Z(b)

1

+

Zlb

+

z2b

2

+

andit ispossible to write down arecursion forthe coefficients zn,whichgeneralizes the one

usedby Gauss

[G].

Finally,consider

N()

(N(’))

.

This isessentially Bhttcher’ equation forthe inverse map

(see

e.g.

[ACJP]).

It follows that there existsaunique solution which admits the expansion

4

N(b)

-

+

no

+

nb

+

n2b

:

+

thusinpartcular satisfying

4

N(b)

-

(b

0).

REMARK.

Theletters

Z

andNarepicked 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

(=

theindexof thesubgroup H thefinitegroup

G);

see

Sec. 4.

Nowit isclear that thefunction

satisfiesthe samefunctionalequation

(4)

as our meanM. The quotient isafunction which is invariantunderb b’. It follows thatwehave

whereOisan"oscillatory" function, thatis, invariant under the transformation b b’

(O(b)

O(b’)).

Nowthe presence

of

the

fluctuations

is completely explained. Ouraboveconjecture

(end

ofSee.

6)

amountstherefore toO constiffN 2,4.

REMARK. A

naturalparameternearb 0seemstobe

1/N(b)

or evenbetter

t=2

1 log log

N(b)

log2

(If

b b then

+

2r.)

Therefore one can go a step further an ask for the Fourier

development of thefunctionO:

O

A0

+

A1

cost

+ Ba

sin

+

A2

cos2t

+

B2

sin2t

+

Weseethusthat, inaway,what wehave observedinthenumericalexperimentisjust the 0-th

coefficient

A

A0

inthis expansion.

PROBLEM: To account forthe"higher"

terms!

8. ATTEMPTS TO JUSTIFY MEISSEL’S FORMULA.

Wereturn to Meissel’s algorithm

(See.

5),

denoting itslimit by

M(a,b,c).

The following propertiesof

M(a,

b,

c)

areobvious:

1) M(Aa,

Ab,

c)

,kM(a,

b,

c)

(homogeneity),

2) M(1,1,1)--

1

(normalization),

3) M(a,

b,

c)

M(a’,

b’,

c’)

(invariance).

Nowwemake the

"Ansatz":

M(a,b,c)

A(log

B---)

-’x,

where

A

A(a, b)

issupposedtobehomogeneousofdegree 1and

B

B(a, b)

homogeneous

ofdegree0. Pluggingthis intothefunctionalequationwefind

B

_

A(a.+b

b

3

B(a+b

b

3"-’

A(log

-)

3

)-(log

))-Write

A’

A(a’, b’),

B’

B(a’, b’).

Thelog-factorstotheleft andtothe right canbe writtenas

(log

1

+

log

B)

-respectively

This gives

1 1 1 1

B_a

3a(

1 1

log

-c

+

log

-

+

log log

-c

+

log

-

+

3log

B’

A

A3

].

and

logB=log-+31og

or B= ab

a’=

a+b 3

b’=

--b

3 xvhich inthe base

(t

a/b)

inducesthemap

t+l

havingonerepulsivefixedpoint k-1 _determined

(as

_in

[M])

by the equation

k(1

+

k)

3. Theb-equationcanbe written

Hencethe A-equationissatisfiedby

vith

whence

(as

in

[M])

TheB-equationcanbewritten

or, ininhomogeneousnotation,

Inparticular,

Thatis,

oriterating

3

3

lk

V3’

log(1

+

t)

log3

(b’)

a’

bB(

o

’

1)

--g-

(B( b-7,

1))3

tl/2

(t)

--.

((t’))

.

B(t)

B(k_l)

V,B(k_I))

1 1 1

B()

B(

-)

()

.(.

,)

.(.

,,)r

ThusAand

B

do"exist". Itishowevernot atall cleartomein whatsense

M

isapproximated

intermsofthe functions

A

andB.

NOTE

(added

Sept.

1988).Recently

J. and P. Borwein sent me a truly marvelous paper a+3b

)"

whereinparticular theconjecture entitled"Onthemeaniteration

(a,

b)

--4 2

aboutthe asymptoticbehaviorofthe BorchardtmeanattheendofSec. 6 isfullyestablished. Even more, the authors show that the mean in question shares with the

agM.

practically

REFERENCES

ACJP. Arazy,

J., CLAESSON, T., JANSON, S.

and

PEETRE,

J.,

Meansandtheir iterations.

In:

STEF/NSON,

JSn R.

(ed.),

Proc.

of

the Nineteenth Nordic Congress

of

MathemMicians,

Reykjavlk,1984,pp. 191-212. IcelandicMath.

Soc.,

Science

Inst.,

Univ. ofIceland, Reykjavlk,

1985.

Bore.

BORCHARDT, C.-W.,

Ueberdasarithmetisch-geometrischeMittelausvierElemcnten.

Monatshefte

Akad. Wiss. Berlin,Nov. 1876, pp. 611-621.

Borw.

BORWEIN,

P.,

personalcommunication,Jan. 1986.

BB. BORWEIN, J. and

BORWEIN,

P.,

Piand the

A

GM.

A

study in analytic numbertheory and computational complexity. Wiley,NewYork Chichester-Brisbane-Toronto-Singapore, 1987.

G.

GAUSS, C.F.,

Bestimmung der Anziehungeines elliptischen

Rin.fles.

Nachlass zur Theorie

des arithmetisch-geometrischen Mittcls und der

Modulfunktion.

Ubersetz yon _Dr. Harald

Geppert.

Ostwald’s Klassikerder exakten Wissenschaften 225. A-kademische Verlagsgesell-schaft,Leipzig, 1927.

M.

MEISSEL,

E.,

Bemerkungen zur hypergeometrischen Reihe. Arch. Math. Phys.

(R.

Hoppe)

57

(1875),

446-448.

(Translated

inthe Appendix

below.)

P1.

PEETRE,

J.,

On Asplund’s averaging method- the interpolation

(function)

way. In:

Constructive Theory

of

Functions, Proc.

of

the Internat.

Conf.

on Constructive Theory

of

Functions,

Varna,

May

27 June 2, 1984,pp. 664-671. Publishing Houseof theBulg. Acad. Sci., Sofia, 1984.

P2.

PEETRE,

J.,

Someunsolvedproblems. In: Colloquie Math. Soc. Jdnos Bolyai

49.

Alfred

HaarMemorial Conference,Budapest, 1985,pp. 711-735. NorthHolland,

Amsterdam,

19S6. Schl.

SCHLMILCH,

O.,

Hyperarithmetischeundhypergeometrische Mittel nebst geometri-scherAnwendungen. Schlmilch Z. 34

(1889),

59-63.

Scho.

SCHOENBERG,

I.J.,

Onthearithmetic-geometricmean. Delta 7

(1977),

4-67.

APPENDIX.

TRANSLATION OF

MEISSELS’S

NOTE

[M].

REMARKS ON

THE

HYPERGEOMETRIC SERIES.

Recently communicatedtoProfessorRfimkerinHamburgthe followingrelation:

de

f0

de

which I incidentally found as a byproduct in investigations concerning the Gauss hyper-geometricseries. Anotherrelation is

"

de

cos(

7rc

f0

"

(1

k:sin

)+’r

-’)

V/i-

k sin do notremember havingseenitanywhere.

What

I

reallyaimedatwasreallyanextensionof the investigation, originatingfrom

Lagrange

and further amplifiedby

Gauss,

of thelimit to whichthesequence of

(successive)

arithmetic

and geometricmeansof twonumbers aandbconverges, and consideredtheequations

3an+

=an

q-

bn

h-Cn

3b+a

=a,b,

+

anCn

+

bncn

writing

aoo

boo

coo

M(a0,

b0,

c0).

Inparticular, wasinterestedintheextremalcase

a0=l; b0=l;

co

very

small;

and found then as alimitof

aoo

boo

coo

(1)

M(1,1,

w)=(

A

B

),

log

where

A, B, A

areconstantsandinfact

*

1

log(1

+

k)

log3 where kisthe root of the equation

(

+ )’

.

The valueofAis

A

0, 43331485.

Forsimplicity wetake thelogtobeBriggian and then the values of

A

and

B

become

approx-imtely

A

0,3951642

log briggB 0,2997049

havecarriedoutthe computationsforvery smallwand, forexample, found directlythat 1

forw

1--0-,

M 0, 16783257

1

forw

106561

M

0,01483609