Asymptotic distribution of global errors in the numerical computations of dynamical systems

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Asymptotic distribution of global errors in the

numerical computations of dynamical systems

Giorgio Turchetti, Sandro Vaienti, Francesco Zanlungo

To cite this version:

Giorgio Turchetti, Sandro Vaienti, Francesco Zanlungo. Asymptotic distribution of global errors in the

numerical computations of dynamical systems. Physica A: Statistical Mechanics and its Applications,

Elsevier, 2010, 389 (21), pp.4994-5006. �10.1016/j.physa.2010.06.060�. �hal-00476240�

numeri al omputations of dynami al systems G.Tur hetti

∗

,S. Vaienti

†

,F. Zanlungo

‡

Mar h7, 2010 Abstra t

Weproposeananalysisoftheee tsintrodu edbynitea ura yand

roundoarithmeti onnumeri al omputationsofdis retedynami al

sys-tems. Ourmethod,thatusesthe statisti altoolofthede ayof delity,

omputestheerror omparingdire tlythenumeri alorbitwiththeexa t

one (or, more pre isely, with another numeri al orbit omputed with a

mu hhighera ura y). Furthermore,asamodelofthe ee tsofround

oarithmeti onthemap,wealso onsiderarandomperturbationofthe

exa torbitwithanadditivenoise,forwhi hexa tresults anbeobtained

for someprototype maps. Weinvestigatethe de ay laws of delity and

their relationshipwiththe errorprobability distributionfor regular and

haoti maps, both for additive and numeri al noise. Inparti ular, for

regularmapswendanexponentialde ayforadditivenoise,andapower

lawde ayfornumeri alnoise. For haoti mapsnumeri alnoiseis

equiv-alenttoadditivenoise,andourmethodissuitabletoidentifyathreshold

for thereliability of numeri al results,i.e. a numberofiterationsbelow

whi h global errors an be ignored. This threshold grows linearly with

thenumberofbitsusedtorepresentrealnumbers.

1 Introdu tion

The reliability of numeri al omputations in dynami al systems is a relevant

longstandingquestion. For ontinuoussystemstheglobalerrorontheorbitis

theresultof thea umulationof twotypes oferrors: therstoneis thelo al

errordue to the dis retization algorithms, the se ond oneis due to the nite

pre isionrepresentationofreals inany omputationaldevi eand totheround

oarithmeti . Inthis paperwepropose astatisti al analysisof theglobal

er-rorsdue to nite a ura yandround o in dis retedynami al systems. Even

∗

DepartmentofPhysi s,UniversityofBologna

†

Centre de Physique Théorique, CNRS, Universités d'Aix-MarseilleI,II, Universitédu Sud,Toulon-VarandFRUMAM

‡

DepartmentofPhysi s,Universityof BolognaandATRIntelligent Roboti sand Com-muni ationLaboratories,Kyoto,Japan

the omparisonwithrandomlyperturbedmaps,forwhi hanalyti alresultsare

available,allowsto understandthekeyfeatures. Formapswith hyperboli

at-tra torstheexisten eoftrueorbits losetopseudo-orbitsisofgreattheoreti al

importan ebutdoesnotprovideanystatisti alinformationonthedis repan ies

between apseudo-orbit and a true orbitwith the same initial ondition. We

showthatfor haoti mapsthereisasharptransitionbetweentworegions,one

inwhi htheerrorisnegligibleandoneinwhi hitsvarian eis omparablewith

thesize of the attra tor,bothfor the numeri al and the randomlyperturbed

map. Forregularorquasiregularaniso hronousmapsthetransitionissmooth

and thea ura y de ay followsapowerlawfor thenumeri al map and

expo-nentiallawfortherandomlyperturbedmap.

Inapreviouswork[1℄,we omparedthevalueofanobservable omputedalong

theorbitofagivenmapandtheorbitofthesamemapwitharandom

pertur-bationforthe sameinitial ondition. Thedieren eis arandompro essand,

forlargeiterationtimes,itsprobabilitydensityfun tion(pdf) onvergestoward

awidespreaddistribution,whi h depends onthe hoi e ofthe observable. For

haoti maps,therelaxationissharp,in thesense thatthereexistsathreshold

timebelowwhi hthewidthofthepdfisverysmallapproximatinga

δ

fun tion andabovewhi hithasalreadyrea heditsasymptoti value. Theseresultswere

obtainedanalyti allyand numeri allyby omputing theinverse Fourier

trans-form of asuitable orrelation integralbetween the twoobservables, alled

-delity. For haoti systemsthedelityde aysatleastexponentiallyfastandthis

allowstore overtheaverageoftheobservableswithrespe ttotheinvariantand

thestationarymeasures. FortheBernoullimapsthede ayissuper-exponential

and there is numeri al eviden e for this type of de ay for most haoti maps

numeri allyinvestigated. Forregularmapssu hasthetranslationsonthetorus

thede ayissmootherandfollowsanexponentiallaw. Numeri aleviden ewas

foundthatthesameexponentialde aypersists forquasiintegrablemaps.

In the subsequent letter [2℄ we applied the previous theoreti al framework to

ompare the exa t orbitof amap with its numeri al omputation (on a

ma- hine). The latter ould be onsidered as a pseudo-random orbit where the

noiseisgivenbytheroundoerrors.

In the present paper wedevelop with more details and formal onsiderations

theresultsannoun edin[2℄andprovidesomenewresults.

Thedieren e between the valuesof an observable omputed along the exa t

andthenumeri alorbitsshowsagaintheexisten eofanabrupttransitiontime

n

∗

forthe orrespondingpdf,whi hhasaninterpretationof pra ti alinterest.

We ould in fa t argue that below thetransition time the numeri al orbitsis

faithfultotheexa torbit. Bytheway,thisapproa hrequirestheknowledgeof

theexa tmapwhi hisnota essible. Thisdi ultyisover ameby omparing

thenumeri almap omputedforagivena ura y(simpleoating point

pre i-sion)withthesamemapevaluatednumeri allywithahighera ura y(double

orhigher oatingpoint pre ision), thatwe onsider asthereferen emap. We

showthattheresultsofthesingle-higherpre ision omparisonareequivalent,at

asinglepre isionandtheexa tmap. Weshowthatthethreshold

n

∗

growslike

− ln ε

,where

ε

isthea ura yspe iedbytheleastsigni antbitusedto rep-resentareal number. Asa onsequen ethedelitythreshold

n

∗

growslinearly

with the length of the string of bits used to represent real numbers. Beyond

thisthresholdtheerrordistributionspreadsqui klyoveritsa essiblerange.

Thebehaviorjustdes ribedistypi alof haoti systems. Forregularmapsthe

situation is slightlydierent: for iso hronousmaps the delityerror does not

de ay,whereasforaniso hronousmapsthede ayfollowsapowerlaw. Forthese

systemsitisnotpossibleto identifyasharptransitionfromthedelta fun tion

totheasymptoti one;thetransitionisagradualpro esswhoselengthdepends

on

ε

, i.e. getslonger with thenumberof bits used to representreal numbers. Asimilar de ayo urs forthe orrelationswhere itis due to thelo al mixing

properties,apropertytypi alofaniso hronousmaps.

2 Additive noise

Webeginbyre allingthemainresultsobtainedapplyingthedelitytorandom

perturbations ofdynami alsystems. We onsideramap

T

dened onaphase spa e

X

whi hisasubsetof

R

d

,endowedwithaninvariantphysi almeasure

µ

denedby

lim

n→∞

Z

X

Ψ(T

n

(x))dm(x) =

Z

X

Ψ(x)dµ(x)

(1)

where

m

denotes the Lebesgue measure and

Ψ

a ontinuous observable. Let usthen onsiderasequen eofindependentandidenti allydistributedrandom

variables

ξ

i

withvaluesintheprobabilityspa e

Ξ

andwithprobabilitydensity distribution

η(ξ)

su hthat

T x + ε ξ

stillmaps

X

intoitself. Theiterationofthe map

T

isthereforerepla edbya ompositionofmaps hosenrandomly loseto it(notethat

T

itselfisre overedwhen

ε = 0

):

T

n

ε

(x) = (T + εξ

n

) ◦ (T + εξ

n−1

) ◦

. . . ◦ (T + εξ

1

) (x)

andthestationarymeasure

µ

ε

ofthepro essisdenedby[3℄

lim

n→∞

Z

X,Ξ

dm(x)

Y

i

η(ξ

i

)dξ

i

Ψ(T

ε

n

(x)) =

Z

X

Ψ(x)dµ

ε

(x)

(2)

Wewant to study thestatisti al properties of the errorat the

n

-th iteration, dened as

∆

n

ε

(x) = f (T

n

x) − f(T

ε

n

x)

, where

f

isasmoothobservable; in the following we take

f (x) = x

for one dimensional maps and

f (x) = x

i

, i =

1, · · · , d

,

x

≡ (x

1

, · · · , x

d

)

, for multidimensional maps. We are in parti ular interestedinstudying

ρ

n

ε

,theprobabilitydistribution fun tionof

∆

n

ε

(x)

. This probabilitydistribution anbestudieddire tly,throughaMonteCarlosampling

over initial onditions

x

and random perturbations

ξ

, or indire tly, using the delity.

Fidelityisdened throughthefollowingintegral:

F

ε

n

=

Z

X,Ξ

dm(x)

Y

i

η(ξ

i

)dξ

i

Ψ(T

n

(x))Φ(T

ε

n

(x))

(3)

the delity onvergesto

R

X

Ψ(x)dµ(x)

R

X

Φ(x)dµ

ε

(x)

, and the absolute value ofthedieren ebetweentheintegral(3) anditslimitingvalueintermsof the

invariantand stationary measure will be alled thedelity error and denoted

with

δF

n

ε

. Note that by theasymptoti hara terization of theinvariantand stationarymeasures,thedelityerror ouldbeequivalentlydened as:

δF

ε

n

= F

ε

n

−

Z

X

Ψ(T

n

(x))dm(x)

Z

X,Ξ

Y

i

η(ξ

i

)dξ

i

dm(x)Φ(T

ε

n

(x))

(4)

whi h emphasizesthe role of the Lebesgue measure in the numeri al

ompu-tations. Thisquantityis relevantbe ause theinverse Fouriertransformof the

delity

F

n

ε

(u)

, omputed by hoosing

Ψ(x) = e

iuf (x)

,

Φ(x) = e

−iuf (x)

is the

probabilityerrordistribution

ρ

n

ε

(x)

. Ifthedelityerror onvergestozero,then thedelity onvergestotheprodu tof theFouriertransformsofthemeasures

R

X

e

iux

_dµ(x)

R

X

e

−iux

_dµ

ε

(x)

anditsinverseFouriertransformisjust

ρ

∞

ε

. The initialerror distributionis theDira distribution

ρ

0

_{(s) = δ(s)}

, whereasit an

beshownthatiftheinvariantandstationarymeasuresareLebesguethe

asymp-toti distribution isthe triangular fun tion

ρ

∞

(s) = (1 − |s|)ϑ(1 − |s|)

. Sin e thede aytimes alesof thedelityerrordepend veryweaklyonthe hoi eof

theobservables

Ψ

and

Φ

, the study of

δF

n

ε

using

Ψ(x) = Φ(x) = x

is usually the most e ient tool to investigate the onvergen e to the asymptoti error

distribution

ρ

∞

ε

.

Fora oupleof systems, translations onthe torus (a regular system) and the

Bernoullimap (a haoti system), analyti al results are available also for the

transient. Choosing the probability distribution

η(ξ) =

1

2

χ

[−1,1]

(ξ)

we found thatthedelityfortranslationsisgivenby[1℄

F

ε

n

=

X

k∈Z

Ψ

k

Φ

−k

S

n

(kε)

S(x) =

sin(2πx)

2πx

(5)

where

Ψ

k

and

Φ

k

are theFourier omponentsoffun tions

Ψ

and

Φ

, whilefor theBernoullimap

T x = qx

mod 1,withinteger

q ≥ 2

wehave

F

ε

n

=

X

k∈Z

Ψ

k

Φ

−k

S

n,q

(kε)

S

n,q

(x) =

n−1

Y

j=0

S(q

j

x)

(6)

Intherst asethede ayofdelityisexponential,withtimes ale

ε

−2

,whilein

these ond asewehaveaplateauoflength

n

∗

∝ − ln ε

,belowwhi hdelityis almost onstant,followedbyan

ε

independentsuper-exponentialde ay. Below thethreshold

n

∗

theerrorprobabilitydistribution anbeapproximatedbya

δ

fun tion,andtheperturbedsystem onsideredasequivalenttotheunperturbed

one. Theasymptoti errordistribution is thesamefor thetwosystems(sin e

thephysi alandstationarymeasure oin idewithLebesgue),andresultstobe

thetriangularfun tion(gures1and2). Thetransitionbetweenthe

δ

andthe triangularfun tions issharpand

ε

independentfortheBernuollisystem,while

0

25

50

75

100

n

0.001

0.01

0.1

1

δ

F

0

10

20

n

0.0001

0.001

0.01

0.1

1

δ

F

Figure1: Comparisonbetweenanalyti alresultsandMonte Carlointegralsfor

thede ayof delity. Left: translations,

ε = 0.1

(analyti al result: bluedashed line;MonteCarlo: ir les). Right:

3x

mod 1.

ε = 10

−4

in bla kand

ε = 10

−8

in green(analyti alresult: dashed lineand rosses;Monte Carlo: ir les). We

used

N = 10

7

integrationpointsintheMonteCarlomethod. Thetestfun tions

Ψ = Φ

∗

weredened inFourierspa eas

Ψ

k

= k

−1

, trun atedat

k = 30

.

-1

-0,5

0

0,5

1

0

25

50

75

100

∆

ρ

-1

-0,5

0

0,5

1

0

5

10

∆

ρ

-1

-0,5

0

0,5

1

0

0,5

1

∆

ρ

Figure 2:

ρ

n

ε

for

3x

mod 1,

ε = 10

−8

. Left:

n = 13

; enter:

n = 15

, right:

itisgradualand

ε

dependentforthetranslations(see[1℄).

Our numeri al study of maps for whi h analyti al results are not available

(Hénon,Baker's,Intermittent,Logisti andStandardmaps)[1℄showsthat the

behavior of translations and of Bernoulli maps anbe onsidered as a

proto-typeof,respe tively,regularand haoti maps. Inparti ularforallthestudied

haoti mapsispossibletoidentifyathreshold

n

∗

∝ − ln ε

belowwhi hthe per-turbedsystem anbe onsideredasfaithfultotheunperturbedone;athreshold

followedbyan

ε

independent,super-exponentialde ay. Tothisthreshold orre-spondsasharptransitionoftheerrorprobabilitydistributionfroma

δ

fun tion totheasymptoti distribution(that ingeneralisnotatriangularfun tion but

dependsontheinvariantandstationarymeasures).

3 Numeri al noise

Sin e real numbers have to be represented as strings of bits on a omputer,

to ea h dis rete map

T

orresponds a numeri al map

T

∗

. The a tion of the numeri almapdependsonthelengthofbitstringsusedtorepresentreal

num-bersandonthedetails ofroundoalgebra,whi harehardwaredependent[4℄,

nevertheless somegeneral results anbe stated. Inour omputational devi e

anyrealnumber

x ∈ R

isrepresentedbyaoatingpointnumber

x

∗p

,that isa

binarystringwith

p

signi antbits

x

∗p

_{= f 2}

e

_{f = ±0.f}

1

f

2

. . . f

p

=

p

X

k=1

f

k

2

−k

e = ±

q

X

k=1

e

k

2

k

where

f

k

, e

k

are0or1. Thesux

p

willbenegle tedwheneverthereisno am-biguity. Theresultofanyarithmeti operationsu has

x

∗p

_{⊕ y}

∗p

orresponding

to

x + y

is aoating pointnumber

z

∗p

_{∈ F}

p

and sum

⊕

implies a round o. Therelativeerror

r

p

isdenedby

x

∗p

= x(1 + r

p

)

|r

p

| ≤ ǫ = 2

−p

Forinstan ein oatingpointsimplepre isionwehave

ǫ = 2

−25

. We anwrite,

usingthenotationintrodu edforadditivenoise,

T

∗

x = T

ε

x = T x + εξ(x)

where

ξ

is now determined in a deterministi way by the initial ondition and

ε

is a onstant whose magnitude is of the order of the last signi antbit used to

represent

x

. The iterated map

T

n

∗

anbewritten as

T

n

ε

using the previously introdu ednotation,butinthis asethesinglesteperrors

ξ

i

willbe

n

dierent fun tions oftheinitial ondition

x

. Noti ethat

T

∗

is dened bytheround o rules of the omputer, and the notation

T

n

∗

orrespond to apply

n

times the mapasdened bythoseround orules;while

εξ

n

= x

∗

n

− T (x

∗

n−1

) = T

∗

n

(x) − T T

∗

n−1

(x)

(7)

is introdu ed to ompare the round o results with the previous results for

additivenoise. Weintrodu ealsotheglobalerror

∆

n

∗

whi h anbewrittenas

Theerror

∆

n

∗

is the umulativeresultofthe lo al errorfor

n

iterationsof the map. It is interesting to ompare these denitions with the lo al and global

errorswhen the perturbation to the map

T (x)

is given by theadditive noise. Inthis asethemap

T

∗

(x)

isrepla ed by

T

ǫ

= T (x) + ǫ ξ

sothat letting

x

n

∗

=

T

n

ǫ

(x) = (T (x) + ǫ ξ

n

) ◦ . . . (T (x) + ǫ ξ

1

)

wehave

ǫξ

n

= T

ǫ

(x

n−1

∗

) − T (x

n−1

∗

) = T

ǫ

n

(x) − T T

ǫ

n−1

(x)

Inthis ase we an write an expli it expression for theglobal error, whi h at

rstorder in

ǫ

reads

∆

n

ε

= ǫ

n

X

k=1

DT

n−k

_(x

1

) ξ

k

+ O(ǫ

2

)

(8)

where

DT (x)

isthe ja obianmatrixof themap

T

, the

ξ

k

arerandom ve tors andthe hainrule appliesto ompute

DT

n

. Forthe translationsonthetorus

DT = 1

andwe anwrite(8)as

∆

n

ε

= ǫ(nξ + w

n

)

(9)

where

ξ ≡ hξi

istheaverageofthesto hasti pro essand

w

n

aretheu tuations whose absolute value is of order 1. As a onsequen e in this ase we have

lim

n→∞

∆

n

ε

/n = ǫξ

. Therandomperturbationswe onsider haveusually zero meansothat

ξ = 0

;thismaynotbethe aseofroundoerrorswhoseaverage anbenonzero. Inthe aseoftheBernoullimaps

DT

isa onstantandif the pro ess

ξ

haszero meanagain

ξ = 0

unlikethe ase ofround errorsdis ussed below. Wealsooutlinethatwhendealingwithroundoerrors,thesummation

in eq. 8 starts with

k = 0

be ause also the initial error

x

0

_{− x}

0

∗

is present. Here

x

0

∗

standsforthenitea ura yrepresentationoftheinitial ondition

x

0

,

whi h is in generaldierent from

x

0

. As we will see, this may have relevant

onsequen es.

4 Generalizations of Fidelity

Thisobservableisusedto hara terizestatisti allythespreadbetweentheorbit

anditssto hasti perturbation. Aspreviouslyintrodu ed(3)delitywasdened

byanintegraloverallthepossiblesinglesteperrorrealizations

ξ

i

. Nevertheless, from anumeri alpointof view, integralswere performed with aMonte Carlo

method,i.e. hoosing

N

representativerandomve tors

(x, ξ

i

)

,apro edurethat ledtoarelativeerroroforder

N

−1/2

(seegure1).

Wesupposethatifthedeterministi

ξ

i

(x)

fun tionaldependen eofsinglestep errorson the initial onditionis omplex enough, theve tor

(x, ξ

i

(x))

an be onsidered as equivalent to a random sequen e, i.e. the Monte Carlo integral

overinitial onditionsandnoise anbesubstitutedforroundobyanintegral

omputethedelityerrorforasystemperturbedwithnumeri alnoiseas

F

_∗

n

=

Z

X

Ψ(T

n

x)Φ(T

_∗

n

x)dm(x)

(10)

δF

∗

n

= F

∗

n

−

Z

X

Ψ(T

n

x)dm(x)

 Z

X

Φ(T

∗

n

x)dm(x)



(11)

Thisdenition requirestheknowledgeoftheexa tmap

T

, whi h isin general notavailable. This problem anbesolvedby omparingtheroundomap

T

∗

, realizedwithagivenpre ision,i.e. asastringofbitsof agivenlength,with a

map realizedat anhigher pre ision,

T

†

, that we allthe referen e map. For example

T

∗

ouldbeasinglepre ision(8digits)map,and

T

†

adoublepre ision (16digits)map. Thenumeri ally omputeddelitywillthusbe

F

_∗

n

=

Z

X

Ψ(T

_†

n

x)Φ(T

_∗

n

x)dm(x)

(12)

To he ktherelevan eoftheseresultswe an omparethemwiththoseobtained

substituting

T

†

with

T

‡

(forexampleamap realized using24 or32signi ant digits). If the results do not depend on the pre ision of the referen e map,

we an assumethat they areequivalent to those that ould be obtainedifwe

had a ess to the exa t map. We have applied this pro edure to the results

shown in this paper. Typi ally, the resultsobtainedusing asreferen e map a

doublepre ision

T

†

ora24(32)digitmap

T

‡

areequivalentbelowagiventime s ale. This time s ale, that orrespondsto the times ale under whi h

T

†

an be onsidered equivalentto

T

‡

, as anbe he kedthroughadire t omparison between

T

†

and

T

‡

,is onsiderablylongerthanthetimes aleatwhi htheerror probability distribution of

T

∗

has rea hed its asymptoti form and thus the resultsof thesingle-double pre ision omparison anbe onsidered equivalent

tothosethatwouldbeobtainedbya omparisonbetweenasinglepre isionand

anexa tmap.

Anotherway to avoid the problems related to the ina essibility of the exa t

mapwouldbetorely,forinvertiblemaps,onadierentdenitionofdelityas

˜

F

∗

n

=

Z

X

Ψ(x)Φ(T

∗

n

T

∗

−n

x)dm(x)

(13) where

T

−1

∗

isthenumeri al realizationof theinversemap, whi h is in general dierentfromtheinverseof

T

∗

. Notethatinthedeterministi settingandwith invertiblemaps

T

∗

preservingthemeasure

m

, eq. (13)is equivalentto (3)[5℄. We generalize it by integrating on given dierent realizations respe tively of

T

−n

∗

and

T

n

∗

, and by supposing that this integral onvergesto

R Ψdm R Φdm

withthesamerateasthedelityerror(11). Forthenumeri alrealizationofan

invertiblemapasthestandardmap,theequivalen ebetweenthetwodenitions

hasbeen he kedwithgoodresults,atleastinthe haoti regime(gure 3).

0

10

20

n

0,01

0,1

1

F

δ

Figure3: De ayofdelityforthestandardmapwith

K = 10

,

ε = 10

−8

,Monte

Carlointegralsusing

N = 10

6

;redand rosses: omparisonbetweensingleand

doublepre ision,bla kand ir les: singlepre isionusinginversion.

2GHz Intel Core Duo on a Ma OSX v10.5 (Darwin 9) operating system, a

pro essorIntelPentium 43.00GHzonaLinux Fedora4OS, and apro essor

1500MHZ IP35 ona IRIX 6.5 OS (whennot spe ied, the results shown in

guresareobtainedonthe Darwinar hite ture). Thedetails of theround o

pro ess depend on the ar hite ture, nevertheless our studies show that some

generalrulesabouttheerrordistribution anbestated.

5 Regular maps

Forregularmapsthebehaviorissigni antlydierentfromadditivenoise,

show-ingthat theintegralovertheonlyinitial onditionsis notequivalentforthese

systemstoanintegralalsoonthenoise.Fortranslationsonthetorus,

T x = x+ω

mod1,whi haretheprototypesforintegrablemaps,wehavefoundthatis

pos-sibletowritetheglobalerror

∆

n

∗

(x, ω) = T

n

x − T

∗

n

x

asin (9)

∆

n

∗

(x, ω) = ε(ξ(ω)φ(n) + w

n

(x)),

φ(n) = n + φ

0

+ φ

1

n

−1

+ ...

(14) Here

ε

is a onstant that represents the last signi ant bit a ura y,

ξ

is a onstant that depends in a non trivial (and ma hine dependent) way on

ω

, whi haswewillseeisequivalenttoanaverageofthesinglesteperrors

ξ

i

,while

w

n

isabounded,periodi fun tionwhi hdependsontheinitial ondition

x

and haszeroaveragewithrespe tto theinitial onditions. Noti ethat inthis ase

ξ

isnotzeroin ontrastwiththe aseofrandomnoise. Wethushave

lim

n→∞

∆

n

∗

(x, ω)

0

25

50

75

100

n

-1,5e-06

-1e-06

-5e-07

0

∆

0

250

500

750

1000

n

-1e-05

-5e-06

0

∆

Figure 4:

∆

n

∗

(x, ω)

in translations on the Torus for two dierent, randomly hosen,initial onditions

x

(redandbla k), omparedwithanaverageover

10

4

values of

x

(green).

ω =

√

2/10

. The two gures show the same results on dierenttimes ales.

Figure 4shows the global error

∆

n

∗

(x, ω)

for twodierent initial onditions

x

and the same

ω

, on two dierent time s ales, showing that the ontributions of

w

n

goes to zero in the limit. Theperiodi nature of thesingle step errors

ξ

i

(and thus of

w

n

) is shown by gure5 (periodi ity emerges after somenon periodi iterations). Theasymptoti averageerror

ξ(ω)

is learly orrelatedto theroundoerror

δω = ω − ω

∗

ofthetranslationparameterasshownbygure 6. Thisgureshowsatleft thedistribution of

ξ(δω)

andat rightthe

ξ

and

δω

distributionsobtainedusingaMonteCarlosamplingoftheparameter

ω

. From thegure at left we an see that

ξ(δω)

is givenby adistribution law entred around

ξ = δω

. Thedistributionlawfortheroundo erroronthetranslation parameter

η(δω) = 2

p−1

∞

X

k=p

χ

[−2

−

k

,2

−

k

]

(δω)

where

χ

I

(x)

denotesthe hara teristi fun tion oftheinterval

I

,isstepwiseas a onsequen e of the exponent in the nite pre isionrepresentations of reals,

with a ura y

ǫ = 2

−p

The stepwise shape of

η(δω)

is learly ree ted also in the distributionlaw for

ξ

, asshown in gure 6. These results suggestthat anuniform integration over

ω

implies an integration over

ξ

. This integration over

ξ

should orrespondmoreorless toarandomorpseudo-randompro ess, sin eis relatedto atrun ation of theless signi ativedigits. The probability

distributionfor

ξ

followinganuniformintegrationfor

ω

isnotuniformasshown ingure6,butin thefollowingdis ussionswewilloftenapproximateitwitha

uniformone.

Thepresen eofanonzeroaverage ontributionofnoise(15)leadstoaquite

dierent situation with respe t to that obtained with random white noise, in

whi hthis ontributionisnotpresent.

Even if these results do not show a qualitative dependen e on the hardware

0

25

50

75

100

i

-2e-08

0

2e-08

4e-08

ξ

ε

Figure5: Singlestep errors

ξ

i

, fortwodierent, randomly hosen,initial on-ditions

x

. TranslationsontheTorus,

ω =

√

2/10

.

-1e-07

0

-5e-08

0

5e-08

1e-07

4e+07

8e+07

η(δω)

η(εξ)

_

Figure6: Left:

ξ(δω)

distribution(translationontheTorus). Right: omparison betweenthe

ξ

(red)and

δω

(bla k)distributions. Bothguresareobtainedwith aMonteCarlosamplingover

ω

(

10

5

valuesfortherstgure,

10

7

forthese ond

0

2500

5000

7500

10000

n

-4e-06

-2e-06

0

∆

0

2e+05

4e+05

6e+05

n

0

5e-13

1e-12

1.5e-12

2e-12

<w >

2

Figure7: Left:

∆

n

∗

(x, ω)

fortwodierent,randomly hosen,

x

values(bla kand red), omparedwith anaverageover

10

4

valuesof

x

(green).

ω =

√

2/5

. Map

realizedasa2D rotation. Right: varian eof

w

n

for

ω =

√

2/10

, averageover

10

4

dierentinitial onditions,maprealizedasa2Drotation. Bla kanddashed

orrespondsto theresultsobtainedontheDarwin ar hite ture,while redand

ontinuousto thoseobtainedontheIRIXar hite ture.

anbefoundsee therightframeof gure7),they andependstronglyon the

algorithm. Theresults just shown, for whi h the period of

w

n

is of order 100 andits magnitudeof order 10,where obtainedby implementing themap asa

translationonthe torus

x

′

_{= x + ω}

mod 1. Fromananalyti al pointof view,

thisisequivalenttoa2Drotationona ir le,butfromanumeri alpointofview

thelatterleadsto asequen e

w

n

whi h hasinitially,foranumberofiterations of order

10

5

, properties similar to a randomsequen e (i.e. its varian egrows

approximatelywithalinearlaw),seegure7. Forthetranslationonthetorus

ξ(ω)

isgivenby

h∆

n

∗

(x, ω)i

x

/n

withani ea ura yevenforlowvaluesof

n

. Con erning the de ay of delity, for regular maps we distinguish between

iso hronous maps, for whi h the frequen y is xed, and aniso hronous maps,

su h as the skew map on the ylinder

x

′

_{= x + ω(y)}

,

y

′

_{= y}

, for whi h the

frequen ydependsontheinitial onditions. Intheformer asedelitydoesnot

de ay, whilein the latterit hasapowerlawde ay. Forthe iso hronousmaps

thesystemis basi allyequivalentto adeterministi rotationwitha frequen y

ω + εξ

, and it an be easily shown that in this ase delity does not de ay, but os illates with onstant amplitude. Indeed, ignoring the ontribution of

theterm

w

n

, we anwrite thea tion of

n

iterationsof thenumeri al map as

x

n

∗

= x

0

+ n(ω + εξ)

. Substituting intheFourieranalysis ofappendix Bin[1℄ andkeeping

ξ

xed(i.e.,notperformingtheintegralinthenoise),thisleadsto anos illatingphasetermbutnottoade ayterm(a tuallythepresen eofthe

os illatingterm

w

n

an auseade ayofdelityforthehighestFouriermodes, those orrespondingtodistan esoftheorderofthemagnitudeof

w

n

).

Fortheaniso hronousmap with

ω(y) = y

(to whi h anymap withmonotoni

ω(y)

anberedu ed),apowerlawde ayfor

F

n

−F

0

isa tuallyobserved. Indeed themapinthis aseis

T (x, y)

denedonthe ylinder

T

× [a, b]

orthetorus

T

T

x

= x + y

mod

1

T

y

= y

andthedelitywithrespe ttoperturbedmapis omputedbyintegratingover

x, y

. If

T

∗

denotes the map with numeri al noise, the map be omes

T

∗

(x, y)

whoseiterate anbewritten

x

n

∗

= x

0

+ ǫξ

x

(x

0

) + n(y

0

+ ǫξ

y

(y

0

))

y

n

∗

= y

0

+ ǫξ

y

(y

0

)

InwritingthisequationweusethepreviousresultsfortranslationsontheTorus,

were the role of

ξ

is nowplayedby

ξ

y

and that of

ω

by

y

0

. Wehave already

ingnoredtheboundedterm

w

n

butwehaveespli itelywrotetheroundoerror of

x

0

forsimmetrywithrespe tto

y

0

. Letus onsider

Φ

and

P hi

tobefun tions oftheonlyvariable

x

,andusetheusualFourieranalysisapproa h. Theintegrals anbe omputed ifwerepla e, followingagainthe previousdis ussion,

ξ

x

(x

0

₎

and

ξ

y

(y

0

₎

with

ξ

x

and

ξ

y

assuming that they are random variables ranging in

[−1, 1]

sothat theintegration over

x

0

, y

0

isrepla ed byan integrationover

x

0

, y

0

and

ξ

x

, ξ

y

. Thedelity

F

n

isgivenby

F

_∗

n

=

X

k∈Z

Ψ

k

Φ

−k

sin(2πnkε)

2πnkε

sin(2πkε)

2πkε

(16)

Thisratherstrongassumptionprovidesaresultwhi hisingoodagreementwith

thenumeri al omputationsseegure8. A mapwhereatanystepboth

x

and

y

where ae tedbyarandomerror

T

ǫ x

= x + y + ǫξ

x

T

ǫ y

= y + ǫξ

y

would givea ompletely dierent result. Indeedtaking into a ount that the

iterateoforder

n

is

x

n

= x

0

+ ny

0

+ ε



n

X

j=1

ξ

j

x

+

n−1

X

j=1

(n − j)ξ

y

j



y

n

= y

0

+ ǫ

n

X

j=1

ξ

y

j

thefollowingresultforthedelityis found

F

n

∗

=

X

k∈Z

Ψ

k

Φ

−k



sin(2πkε)

2πkε



n n

_Y

j=1

sin(2πjkε)

2πjkε

(17)

Figure 8showsthat this resultagrees with the delity omputed numeri ally

for a randomly perturbed map and diersdrasti ally from the delity of the

mapwithroundonoise.

6 Chaoti maps

Fidelity de ay an be easily studied for haoti numeri almaps (for example

0

1e+05

2e+05

3e+05

4e+05

n

0

0.025

0.05

0.075

F

δ

0

5e+06

1e+07

n

0,03

0,06

0,09

δ

F

Figure8: Left: De ayofdelityfortheskewmap. Bla k: additivenoise,

om-paredwithitsanalyti alpredi tioneq17(red);greenroundonoise ompared

withitsanalyti alpredi tioneq16(blue). Right: omparisonofroundonoise

andanalyti alpredi tiononalongertimes ale.

N = 10

4

with

K ≫ 1

)and showsalwaysagood qualitativeagreementwiththe results obtainedforadditivenoise,i.e. thepresen eofathresholdbelowwhi hdelity

is onstant and the error fun tion is qualitatively a

δ

fun tion (its support is manyorder of magnitude smaller than thesize of the phase spa e, asit

hap-pened in gure2). Beyondthis threshold,that we an all

n

∗

andthat grows as

− ln ε

, i.e. linearly in the number of bits used to represent real numbers (gure9),theerrordistributionspreadsqui klyoveritsa essiblerange,as an

be he ked using also a Monte Carlo sampling of the error distribution. The

resultsfor haoti maps donotshowasigni antdependen e onthe hoi eof

thear hite ture.

Stating that the de ay law of delity is qualitatively similar to the one

ob-servedforadditivenoiseisnotenoughtoprovethatthesequen eofsinglestep

errors

ξ

i

(seeequation 7) anbe onsidered equivalenttoarandomone,aswe laimedinouransatz. Inordertobetterunderstandthenatureofthissequen e

wearegoingto usetwoapproa hes. Therstapproa hwouldbetostudy the

distribution of

ξ

i

fordierent

i

values. In the aseof additivenoisewe would havea ontinuous uniformdistribution identi al forea h

i

, andthus themore dierent is the a tual

ξ

i

distribution from a ontinuous uniform one, the less valid is ourassumption. The se ond approa h onsists in studying in greater

detailthede aylaw, omparingitwiththeoneforadditivenoise,toseeifsome

dieren e emerges. In order to do that, an be useful to study the de ay of

delity for quen hed noise. By quen hed noise we mean a perturbation that

usesthe same sequen e

ξ

i

for ea h initial ondition(the sequen e is obtained randomly hoosing thevalues

ξ

i

but without integrating onthem). Ifour as-sumptionisvalid,thedelityforroundonoiseshouldbemoresimilartothat

obtainedusing additive noisethan to the oneobtained using quen hed noise.

Letus startwiththemap

3x

mod 1. Fromadetailed observationofgure9is possibleto see, in parti ular forsinglepre ision,that somedieren ebetween

0

10

20

30

n

0.0001

0.001

0.01

0.1

F

δ

Figure9: De ayofdelityfor

3x

mod 1representedin singlepre ision(bla k, ir les)and doublepre ision(red, diamonds) ompared to areferen emap

T

†

using 32 digits; the results are ompared to the de ay of delity for random

(additive)noisewith

ε = 2

−25

(blue, rosses)and

ε = 2

−53

(green,squares),the

valueof singleand doublepre isionlast signi antbits forreals onthe torus.

MonteCarlointegralswith

N = 10

7

roundo and additivenoiseis present. Forthis map, wehavefound that the

globalerror anbedes ribedas

∆

n

∗

(x) =

n

X

i=0

3

n−i

_εξ

i

(x)

(18)

(this equation, aseq. (14), doesnot takein a ount boundary ee ts) where

theinitial onditionroundo

ξ

0

hasastep-wise ontinuumspe trum

η(εξ) =

2

p−1

P

∞

k=p

χ

[−2

−

k

,2

−

k

]

(εξ)

distribution on e sampled over the spa e of initial ongurations(duetotheexponentin theoatingpointrepresentation,

2

−p

is

thevalueof theleastsigni antbit,that we analso onsiderasequivalentto

the onstant

ε

),whilethe

ξ

i

with

i ≥ 1

haveadis retespe trum,whi hresults to be almost ompletely redu ed to zerofor

i ≥ 2

(i.e. norelevant errorsare madeafter therst iteration). (gure 10). Due to this ee t,whi his aused

bytheextremely simplealgorithmi nature ofthemap, the sequen eof single

steperrors annotbeusedasarepresentativesequen eforanintegraloverthe

noise.

Itisthanevidentthatonlytheinitial roundohasa ontinuous(evenifnot

uniform)distribution,and anbein someway onsidered asequivalentto the

additivenoise. Inthedis ussionoftheskewmap,wehaveseenthattheinitial

roundodeterminedthede aylawofdelity,andthisistruealsoforthemap

-4e-08

0

-2e-08

0

2e-08

4e-08

25000

50000

75000

1e+05

ξ

η

ε

-2e-07

-1e-07

0

1e-07

2e-07

0

1e+06

2e+06

3e+06

4e+06

ξ

η

ε

-2e-07

-1e-07

0

1e-07

2e-07

0

2e+06

4e+06

6e+06

8e+06

ξ

η

ε

Figure10: Singlesteperrordistributionforthemap

3x

mod1insinglepre ision. Left:

ξ

0

; enter

ξ

1

;right

ξ

2

. Monte Carlosamplingwith

N = 10

7

theskew map, that the integralontheinitial ondition

x

0

is equivalent toan

uniformintegralontheinitialroundoerror,andletusignorethe ontribution

of therest of thesingle steperror sequen e,

ξ

i

with

i ≤ 1

. . It is possible to show using Fourier analysis that the de aylawfor the delity of a

qx

mod 1 mapperturbedattheonlyinitialstep,withanuniformintegralovernoiseis

δF

ε

n

=

X

k6=0

Ψ

k

Φ

−k

sin(2πεkq

n

₎

2πεkq

n

(19)

(thisresultistriviallyobtainedfollowingtheFourieranalysispro edurein

ap-pendix Bof [1℄ and performing only therst integral over noise). The de ay

law(19)des ribesbetterthede aylaw fordelity dueto roundo noisethan

to(6),showingthatthemain ontributiontode aydelityisduetotheinitial,

ontinuousspe trum perturbation (gure 11). Toanalyse the ontribution of

ξ

i

with

i ≤ 1

letusremovetheinitial round o ( hoosing rationalinitial on-ditionsthat anberepresentedexa tlywiththe pres ribedpre ision). Inthis

asedelitypresentsathreshold,butdoesnotde aytozero(afterthethreshold

theerrordistributionexpandsto thewhole phasespa ebut doesnot onverge

to an asymptoti distribution, gure 12). In this ase, i.e. when only initial

onditionpointsthat anberepresentedexa tlyusingthenumeri alpre ision

under examinationand thus theinitial roundo isremoved,the delity

os il-latesafterthethresholdwithoutgoingtozero(gure13). Isnotsurprisingthat

this behaviour is quite similar to that obtained studying the quen hed noise

perturbationofthismap. Indeedit anbeshown,(generalizingaresultin[6℄),

that thedelity for quen hed noise os illates but does notde ayto zero. We

havefound that thisresultisparti ularly di ulttoverifyusing MonteCarlo

numeri alintegration. Asitisshowningure13,athemap3

x

mod1studied with numeri alpre ision

ε

1

and perturbed withnumeri al noise

ε

2

, starts os- illatingin orresponden ewiththedelitythresholdfor

ε

2

(inagreementwith theanalyti alresult)butthendropstozeroin orresponden ewiththedelity

thresholdfor

ε

1

(seegure13, omparingwithgure9toknowthesingleand doublepre isionthresholds).A ordingtousthisresultsupportsour on lusion

0

10

20

n

0,001

0,01

F

δ

Figure11: De ayof delityfor

3x

mod1. Bla k, ir les: analyti alresult(6), using

ε = 2

−25

. Red, squares: analyti alresult(19). Green, diamonds: round

onoise(single pre ision omparedtodoublepre ision),

N = 10

7

-1

-0.5

0

0.5

1

0

5

10

15

20

∆

ρ

Figure12: Errordistribution

ρ

n

obtained omparingasinglepre ision

3x

mod1 mapwithadoublepre isionmap,usinginitial onditionsinsinglepre ision(no

initialround o). Bla k,

n = 30

; red,

n = 31

; green

n = 32

; ompared to the triangular fun tion (blue) obtainedusing initial ondition in double pre ision

(initialroundo). MonteCarlosamplingsusing

N = 10

7

0

20

40

n

0

0,025

0,05

0,075

δ

F

Figure 13: De ay of delity for

3x

mod 1. Bla k, diamonds: quen hed noise omputedwith 32 digits(

ε = 2

−25

). Red, squares: quen hed noise omputed

with double pre ision (

ε = 2

−25

). Green, ir les: de ay of delity obtained

omparingasinglepre isionmapwithadoublepre isionone(roundonoise)

usinginitial onditionsthat anbeexa tlyrepresentedin singlepre ision,i.e.

removingtheinitialroundo. MonteCarlointegrals,

N = 10

6

thatthedelityde aythresholdisagoodmeasureforthemaximumnumberof

iterationsunder whi hthenumeri al omputationsare ompletely reliable.

Forthe other, (slightly)morealgorithmi ally omplexmaps, the sequen eof

error

ξ

i

has a ontinuous spe trum for ea h value of

i

, and onverges qui kly toanasymptoti distribution(gure 14). Forallthesesystemstheasymptoti

errordistributionresultsto bealmostindistinguishablefromthat obtained

us-ingadditivenoise(gure 15),suggestingthat forthesemapstheassumptionof

equivalen ebetweenroundoandadditivenoise ouldbevalid(forthesemaps

theinitialround oerrorplaysnospe ialrole).

Inordertobetterverifythisassumption,wehavestudiedthede ayofdelity

formaps perturbed with quen hed noise, i.e. withoutperformingthe integral

overnoise. Nevertheless,forallthetested haoti maps(withtheex eptionof

theBaker'smap)thede aylawsforquen hedandadditivenoisewerenot

dis-tinguishable,at least using the omputationalpre isionallowedbyour Monte

Carlomethod. For theBaker'smap, is possible to see that the delity de ay

ismoreirregularforquen hednoise(duetotheabsen eofthesmoothing

inte-gral). Inthis asethede ayusingroundonoiseismoresimilartotheadditive

-2e-07

-1e-07

0

1e-07

2e-07

100

1000

10000

1e+05

ξ

η

ε

Figure14: Distributionofsinglesteperrors

ξ

i

fortheHénonmap,

x

observable,

N = 10

7

. Bla k,

i = 0

. Red,

i = 1

. Green,

i = 20

. Blue

i = 50

(greenandblue areverydi ulttodistinguish)

-3

-2

-1

0

1

2

3

0

0.1

0.2

0.3

0.4

0.5

∆

ε

8

ρ

ε

8

Figure 15: Asymptoti error distribution for the Hénon map,

x

observable. Bla k, additive noise (

ε = 10

−8

), red omparison between single and double

pre ision(thetwofun tionsarealmostidenti al). MonteCarlosamplingsusing

N = 10

7

0

10

20

30

40

n

0,001

0,01

F

δ

Figure 16: De ay of delity for the Baker'smap. Bla k, ir les:the sequen e

errors of order

2

−25

is the same for ea h initial ondition (quen hed noise),

showinganirregularde aylaw;green,squares: singlepre isionroundonoise,

showingthesamesmoothde ayofadditivenoisewith

ε = 2

−25

(red, rosses).

MonteCarlointegralswith

N = 10

7

7 Con lusions

Wehaveusedtheresultsofapreviousworkonadditivenoisetostudytheee ts

ofroundo noiseondis rete systems. Forregularmapsthebehaviordepends

onthe algorithmi realization and on its hara ter: for iso hronousmaps the

delityerrordoesnotde ay, foraniso hronousmapsithasapowerlawde ay,

whereasin presen e of anadditive noisethede ay isalwaysexponential. For

haoti systemsroundoandnoisearealmostequivalent. Wealsoshowedthat

thedelityisane ienttoolforthestudyofthetimes alesforthe onvergen e

totheasymptoti errordistribution,sin e,atleastfor haoti maps,itallowsus

tond athresholdvaluebelowwhi h thenumeri alsystem anbe onsidered

asequivalenttotheexa tone. Thisthresholdlinearlygrowsasthenumberof

Acknowledgments

G. Tur hetti a knowledges nan ial support from the PRIN 2007 grant

Sim-ulazione della produzione di elettroni dalla interazione laser-(pre)plasma nel

ontestodellaignizionevelo e oordinatedbyD. Batani

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