Arithmetic-geometric mean: Difference between revisions
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Output: |
Output: |
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<pre>-> "0.8472130847939790866064991234821916364814459103269421850605793726597340"</pre> |
<pre>-> "0.8472130847939790866064991234821916364814459103269421850605793726597340"</pre> |
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=={{header|Python}}== |
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{{libheader| Math}} |
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<lang python> |
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from math import sqrt |
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######################################################### |
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# Calculating the arithmetic-geometric |
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# mean of two numbers a0, g0. |
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# |
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# tolerance the tolerance for the converged |
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# value of the arithmetic-geometric mean |
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# (default value = 1e-10) |
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#-------------------------------------------------------- |
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def agm(a0,g0,tolerance=1e-10): |
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an=(a0+g0)/2.0 |
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gn=sqrt(a0*g0) |
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diff=abs(an-gn) |
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while abs(an-gn)>tolerance: |
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an1=(an+gn)/2.0 |
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gn1=sqrt(an*gn) |
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an=an1 |
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gn=gn1 |
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return an |
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######################################################### |
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print agm(1,1/sqrt(2)) |
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</lang> |
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Output: |
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<pre> 0.847213084835</pre> |
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=={{header|Ruby}}== |
=={{header|Ruby}}== |
Revision as of 07:57, 1 April 2012
You are encouraged to solve this task according to the task description, using any language you may know.
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 , and is equal to the limit of the sequence:
Since the limit of tends (rapidly) to zero with iterations, this is an efficient method.
Demonstrate the function by calculating:
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"
)
const ε = 1e-14
func agm(a, g float64) float64 {
for math.Abs(a-g) > math.Abs(a)*ε { a, g = (a+g)*.5, math.Sqrt(a*g) } return a
}
func main() {
fmt.Println(agm(1, 1/math.Sqrt2))
}</lang>
- Output:
0.8472130847939792
The referenced Mathworld page mentions that AGM is meaningful on the complex plane as well. <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):
<lang j> ~.(mean , */ %:~ #)^:_] 1,%%:2 0.8472130847939792</lang>
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.
PicoLisp
<lang PicoLisp>(scl 80)
(de agm (A G)
(do 7 (prog1 (/ (+ A G) 2) (setq G (sqrt (* A G)) A @) ) ) )
(round
(agm 1.0 (*/ 1.0 1.0 (sqrt (* 2.0 1.0)))) 70 )</lang>
Output:
-> "0.8472130847939790866064991234821916364814459103269421850605793726597340"
Python
<lang python> from math import sqrt
- Calculating the arithmetic-geometric
- mean of two numbers a0, g0.
- tolerance the tolerance for the converged
- value of the arithmetic-geometric mean
- (default value = 1e-10)
- --------------------------------------------------------
def agm(a0,g0,tolerance=1e-10):
an=(a0+g0)/2.0 gn=sqrt(a0*g0) diff=abs(an-gn) while abs(an-gn)>tolerance: an1=(an+gn)/2.0 gn1=sqrt(an*gn) an=an1 gn=gn1 return an
print agm(1,1/sqrt(2))
</lang> Output:
0.847213084835
Ruby
The thing to note about this implementation is that it uses the Flt library for high-precision math. This lets you adapt context (including precision and epsilon) to a ridiculous-in-real-life degree. <lang ruby>
- 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