Arithmetic-geometric mean: Difference between revisions

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This page uses content from Wikipedia. The original article was at Arithmetic-geometric mean. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

Write a function to compute the arithmetic-geometric mean of two numbers. [1] The arithmetic-geometric mean of two numbers can be (usefully) denoted as $\mathrm {agm} (a,g)$ , and is equal to the limit of the sequence:

a_{0}=a;\qquad g_{0}=g

a_{n+1}={\tfrac {1}{2}}(a_{n}+g_{n});\quad g_{n+1}={\sqrt {a_{n}g_{n}}}.

Since the limit of $a_{n}-g_{n}$ tends (rapidly) to zero with iterations, this is an efficient method.

Demonstrate the function by calculating:

\mathrm {agm} (1,1/{\sqrt {2}})

Ada

<lang Ada>with Ada.Text_IO, Ada.Numerics.Generic_Elementary_Functions;

procedure Arith_Geom_Mean is

  type Num is digits 18; -- the largest value gnat/gcc allows
  package N_IO is new Ada.Text_IO.Float_IO(Num);
  package Math is new Ada.Numerics.Generic_Elementary_Functions(Num);

  function AGM(A, G: Num) return Num is
     Old_G: Num;
     New_G: Num := G;
     New_A: Num := A;
  begin
     loop
        Old_G := New_G;
        New_G := Math.Sqrt(New_A*New_G);
        New_A := (Old_G + New_A) * 0.5;
        exit when (New_A - New_G) <= Num'Epsilon;
        -- Num'Epsilon denotes the relative error when performing arithmetic over Num
     end loop;
     return New_G;
  end AGM;

begin

  N_IO.Put(AGM(1.0, 1.0/Math.Sqrt(2.0)), Fore => 1, Aft => 17, Exp => 0);

end Arith_Geom_Mean;</lang>

Output:

0.84721308479397909

C++

<lang cpp>/*Arithmetic Geometric Mean of 1 and 1/sqrt(2)

 Nigel_Galloway
 February 7th., 2012.

/

include "gmp.h"

void agm (const mpf_t in1, const mpf_t in2, mpf_t out1, mpf_t out2) { mpf_add (out1, in1, in2); mpf_div_ui (out1, out1, 2); mpf_mul (out2, in1, in2); mpf_sqrt (out2, out2); }

int main (void) { mpf_set_default_prec (65568); mpf_t x0, y0, resA, resB;

mpf_init_set_ui (y0, 1); mpf_init_set_d (x0, 0.5); mpf_sqrt (x0, x0); mpf_init (resA); mpf_init (resB);

for(int i=0; i<7; i++){ agm(x0, y0, resA, resB); agm(resA, resB, x0, y0); } gmp_printf ("%.20000Ff\n", x0); gmp_printf ("%.20000Ff\n\n", y0);

return 0; }</lang>

The first couple of iterations produces:

0.853
0.840

Then 7 iterations produces:

0.847213084793979086606499123482191636481445910326942185060579372659734004834134759723200293994611229942122285625233410963097962665830871059699713635983384251176326814289060389706768601616650048281188721897713309411767462019944392962902167289194499507231677897346863947606671057980557852173140349398304200422119216039839553595098193641293716340646029599967970599434351602031842648756950242174863855405981954581601742417887854192758804162719012085587685648326834140431218400804035809204559494313877815120926522254574397124286820766340954733674599621792665535348625686118543308626287287287563010835563193570668714785639088982115108836352147696979612621832943228417868113768445170018146021913694027020945996683513596327880804274345481744587363220025153952936265806614198365616491626259607434723706616902353080017375312847852558430631907454274934152685790655269406003147591020332746719686124796325510554648902820855297439651249940096625528660675804487353892185701401167716976535014084952476848993257321337028984668939194665861873752966387562266045914777044204681089256584408380320409106190031537067341195941010074743310599055058205243260099516927924174782169767810616836977141107392733439215501430220070873673659622721492587761928510523803670268904639096219076636442355380859029452340651900133423451058383417121805142550039237011113254111446126289062541335505266436535958245521562933975182514706501346410470569793556813066063293733450387109770972948759171790158173202815782884871499313408154933423677970447127859376185950851466773645546792016159342239971429840707888822790326567515965284358177957272848083564899635044041407342261101833835469759626633304220849998523007427039302772434749797179732645525465430198316949684610986907439050680137661192529197709384412997070158894931666611619945922650113111839663525025305616464315872084545229887754751772727476567216489829182392388952072076428397108847059603569219929218319015481412807665926982944644571492396663299730758139049576224389624231752095073190184244624423709864272811495111808228260538624846176751801409831274972576519837564923569028002161749055314272081534395405955635763711272816570597373374429700390560401563886630722257003892301591123769601215800817790778633512408624310735715837659265045466527873378744448344063102447570396812554539822664303534164130356138016341655752655897529445211668734512201912274667331915712407637538211069681410769263900748331757433967523196603308649735713838741960989838322028826948821913028193669499544222406972761686213695116578388850121990961606554546115432531481642493326947970041594914763231129205935165189979433500459762882172926259180894055084314663937825483351395501906533708720620640240770560758487964998436515927282645344286366154191425857771067561850172780332871751951893050318055052454260223355229007714181287986543511879180063562795936247682677864122494603381260826282540988953125276775346562432792145112295555160318184331336929617230417838551571255674049834166659269695800089537245730576945422753721602096871914703988784663672432627061911270717165908246400416799411204056571036408300024192943985530739946565396778104927010554103595133394321999250666762020783946955537605517964010097492188563113010178138885787938131720959480625392013009836502879176958279859052799477219417979970249430621584194688853281154977215799601944096234776861440850757392842988237593968232236705803341347746231128976258593243766317789749110772619097044895222045096307255155900938249040213648077920347672150485684460225544099928261631743126422857876289833806507220230103717531492635046310601885737725670066183812905806389545081270313113710437161358334880658339554312179013483988332164130576352447125115394720666703301013487165163241138288176398396295261211412632197959650986567867552507607604240959075175230219461045325643332496149012535333292237238689481278850201359663053760558493589283916304694038878549600274714871978014576595790495858022600660995249673643249668334617601066081567069751423818665036108388522097616550025160731149921612947757901997292486896382206038087602762816723701668191066335857751546503813342367223476420265585655884641601021054048985561871147358849763784064864267981865044863190774703822867114351511230036070865742988647714667473375011434581885279700605621172469217484718069486625119947289344427037830462070735493805287272062156063071882868580564521110696708028569906982576917722099867195996850779068144349493280497681154368046325993869307623507099951829512958112123570724538335482619075239515827309824818054966589790916886798407170779370595904577584091047341310960419411135775662072733779783320379730113767265853574771027978140972130961214239385473746276961504130795283737288205065871915225976508402779699176117539300672549249122984508236297556872271106584943553385049453263873648980460665597995436016950309279009245005785647723587619884898603441219534079536900299641197454906074160097885953766072290516077242859007090115663913836429904122082676962979786764903235649998199076599743987054864876909102491192709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00

The limit (19,740) is imposed by the accuracy (65568). Using 6 iterations would produce a less accurate result. At 7 iterations increasing the 65568 would mean we already have 38,000 or so digits accurate.

Go

<lang go>package main

import (

   "fmt"
   "math"
   "math/cmplx"

)

const ε = 1e-14

func agm(a, g complex128) complex128 {

   for cmplx.Abs(a-g) > cmplx.Abs(a)*ε {
       a, g = (a+g)*.5, cmplx.Rect(math.Sqrt(cmplx.Abs(a)*cmplx.Abs(g)),
           (cmplx.Phase(a)+cmplx.Phase(g))*.5)
   }
   return a

}

func main() {

   fmt.Println(agm(1, 1/math.Sqrt2))

}</lang> Output:

(0.8472130847939792+0i)

J

This one is worth not naming, in J, because there are so many interesting variations.

First, the basic approach (with display precision set to 16 digits, which slightly exceeds the accuracy of 64 bit IEEE floating point arithmetic):

<lang j>mean=: +/ % #

  (mean , */ %:~ #)^:_] 1,%%:2

0.8472130847939792 0.8472130847939791</lang>

This is the limit -- it stops when values are within a small epsilon of previous calculations. We can ask J for unique values (which also means -- unless we specify otherwise -- values within a small epsilon of each other, for floating point values):

Another variation would be to show intermediate values, in the limit process:

<lang j> (mean, */ %:~ #)^:a: 1,%%:2

                1 0.7071067811865475

0.8535533905932737 0.8408964152537145 0.8472249029234942 0.8472012667468915 0.8472130848351929 0.8472130847527654 0.8472130847939792 0.8472130847939791</lang>

Another variation would be to use arbitrary precision arithmetic in place of floating point arithmetic. (out of time, maybe later)

Mathematica

To any arbitrary precision, just increase PrecisionDigits <lang Mathematica>PrecisionDigits = 85; AGMean[a_, b_] := FixedPoint[{ (Plus@@#)/2, Sqrt[Times@@#] }&, N[{a,b}, PrecisionDigits]]1</lang>

AGMean[1, 1/Sqrt[2]]
0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597232

Perl 6

<lang perl6>sub agm ($a, $g) {

   sub iter ($old) {
       my $new := [ 0.5 * [+](@$old), sqrt [*](@$old) ];
       last if $new ~~ $old;
       $new;
   }

   ([$a,$g], &iter ... 0)[*-1][0];

}

say agm 1, 1/sqrt 2;</lang>

Output:

0.84721308479397917

Obviously the "fixed point" detector here is relying on the floating-point representation running out of bits, or this algorithm would not terminate before using up all memory.

Ruby

The thing to note about this implementation is that it uses the [2] library for high-precision math. This lets you adapt context (including precision and epsilon) to a ridiculous-in-real-life degree.

There's probably a more Ruby Way-like implementation for this; this was written in ten minutes well after midnight. It was, however, fully tested.

The flt package (http://flt.rubyforge.org/) is useful for high-precision floating-point math.
It lets us control 'context' of numbers, individually or collectively -- including precision
(which adjusts the context's value of epsilon accordingly).

require 'flt' include Flt

BinNum.Context.precision = 512 # default 53 (bits)

def AGM(a,g)

 new_a = BinNum a
 new_g = BinNum g
 while new_a - new_g > new_a.class.Context.epsilon do
   old_g = new_g
   new_g = (new_a * new_g).sqrt
   new_a = (old_g + new_a) * 0.5
 end
 new_g

end

puts AGM 1, 1 / BinNum(2).sqrt

</lang>

Output:

0.84721308479397908660649912348219163648144591032694218506057937265973400483413475972320029399461122994212228562523341096309796266583087105969971363598338426

Adjusting the precision setting (at about line 9) will of course affect this. :-)

Tcl

The tricky thing about this implementation is that despite the finite precision available to IEEE doubles (which Tcl uses in its implementation of floating point arithmetic, in common with many other languages) the sequence of values does not quite converge to a single value; it gets to within a ULP and then errors prevent it from getting closer. This means that an additional termination condition is required: once a value does not change (hence the old_b variable) we have got as close as we can. Note also that we are using exact equality with floating point; this is reasonable because this is a rapidly converging sequence (it only takes 4 iterations in this case). <lang tcl>proc agm {a b} {

   set old_b [expr {$b<0?inf:-inf}]
   while {$a != $b && $b != $old_b} {

set old_b $b lassign [list [expr {0.5*($a+$b)}] [expr {sqrt($a*$b)}]] a b

   }
   return $a

}

puts [agm 1 [expr 1/sqrt(2)]]</lang> Output:

0.8472130847939792