Introduction à l'Optimisation Numérique

7/21/2019 Introduction l'Optimisation Numrique

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D^HZTL@LETVT^I

6@L HEEL@IF

Ietraduftiae oApti`ishtiae Eu`riqul

Hudl Zaedlpilrrl & Vjhstile Tardlux

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3/43

Thjol dls `htirls

Ietraduftiae |

y

R

J(x, r) 8, c(x)

c(y). (8.:)

Ae dit hoars qulc(x)lst ue ìeiù` oafho dlcsurR. x Ze lst ue paiet dl ìeiù` koajho dlcsurRssi

xR lt yR, c(x) c(y). (8.6)

Ae dit hoars qulc(x)lst ue ìeiù` koajho dlcsurR.

Ols eatiaes dl `hxiù` oafho lt koajho saet deils dl chae taut chit siìohirl. Le chit, aeplut chfiol`let dàetrlr qul ols prajo`ls (hvlf au shes faetrhietls) 7

ìex

c(x) lt `hxx

c(x)

saet quivholets dhes ol sles a ios aet ``l lesl`jol dl saoutiaes lt 7

ìex

c(x) 0 `hxx

c(x) au lefarl `hxx

c(x) 0 ìex

c(x).

Hiesi oh rlfblrfbl due `hxiù` pauvhet sl rh`lelr oh rlfblrfbl due ìeiù`, eaus par-tlraes uel httletiae pous phrtifuoirl oh rlfblrfbl du ìeiù`.

< > < 8>

>.>

>.

8.

:.

6.

Lxl`pol 8.:.8 8. Ols lesl`jols ltZe saet auvlrts.:. Ols jauols auvlrtls saet dls auvlrts.

6. Ols ietlrvhools dl oh car`lPh, j], h ? j+, saet dls auvlrts dl Z.Deitiae 8.3 (Lesl`jol clr`) VaitC Ze. Olesl`jolClst ue clr` si sae fa`po`le-thirl lst ue auvlrt.

Lxl`pol 8.:.:

8. Ols lesl`jols ltZe saet clr`s, olurs fa`po`lethirls rlsplftics thetZe lt.:. Ols ietlrvhools dl oh car`l]h, jP, ? h ? j ?+, saet dls clr`s dl Z.

^rapasitiae 8.8 VaitC Ze.C lst ue clr` si lt sluol`let si tautl suitl faevlrkletl do-`lets dlCh sh oiìtl dhesC.

^rluvl. ()Vuppasaes qul Clst clr` lt eataes A sae fa`po`lethirl. Vait (xe)eE uelsuitl do`lets dlCqui faevlrkl vlrsx Ze. Vuppasaes phr ohjsurdl qul x>.

Fa``l Alst auvlrt, io lxistl r 5>tlo qul 7 J(x, r)A. Ar(xe)eEfaevlrkl vlrs x da 7E E, eE, xe x? r,

Flfi i`poiqulxeJ(x, r)Alt daefxeA phrtir due flrthie rhek, fl qui lst i`passijolfhrxeCqulo qul saite E phr deitiae.

()Vuppasaes qul tautl suitl faevlrkletl dl C hd`lt uel oiìtl dhes C. @aetraes qulA,ol fa`po`lethirl dlC, lst auvlrt.^hr ohjsurdl, suppasaes qul A el sait phs auvlrt, i.l. 7 io lxistl x A tlo qul 7r 5>, J(x, r)A. Hutrl`let dit 7

r 5> J(x, r) C0.âurr 0 8

e+8, eaus faestruisaes uel suitl (xe)eEdo`lets dl C tlool qul

xeJ(x, 8e+ 8

).

Flfi plut trl trhduit lexe x 8e+8 . Io suit qulxefaevlrkl vlrsx. Fa``lClst clr`,xC, fl qui lst i`passijol fhrxA.

Zlvleaes `hietlehet oh fhrhftrishtiae dl olesl`jol dls faetrhietls dhes ol fhs dl faetrhietlsdkhoit lt/au diekhoit.

^rapasitiae 8.: Vailetk 7 Ze Zq ltb7 Ze Zp dlux caeftiaes faetieuls. R0{x Ze 7b(x) 0 >, k(x)>} lst ue lesl`jol clr` dl Ze. R0{x Ze 7k(x)? >} lst ue auvlrt dl Ze.


12/43

8: 8.6. Faevlxit lt aptiìshtiae

8.6 Faevlxit lt aptiìshtiae

Ols prajo`ls daet ols daeels saet faevlxls, faestitulet uel fohssl i`parthetl le aptiì-shtiae, fhr crqul``let rlefaetrs dhes ols hppoifhtiaes lt oh jhsl dl ea`jrlusls `tbadlsdvloappls paur dls prajo`ls pous kerhux.

8.6.8 o`lets dhehoysl faevlxl

Deitiae 8.; (Lesl`jol faevlxl) VaitF Ze. Olesl`jolFlst faevlxl ssi(x, y)F:, P>, 8], x+ (8 )yF,

flst--dirl, sixltysaet dlux o`lets dlFhoars ol slk`let qui rloilxylst iefous dhesF.

Lxl`pol. Ze lst faevlxl.

Deitiae 8.2 (Caeftiae faevlxl/striftl`let faevlxl) VaitF Ze faevlxl ltc 7F Z. clst faevlxl ssi(x, y)F:, P>, 8], c(x+ (8 )y) c(x) + (8 )c(y).

clst striftl`let faevlxl ssi(x, y)F:, x0y,P>, 8], c(x+ (8 )y)? c(x) + (8 )c(y).

Tbar`l 8.8 VaitF Ze faevlxl ltc 7F Z diccrletihjol. Oh caeftiaeclst faevlxl ssi 7(x, y)F:, c(y)c(x) + c(x), y x, (8.4)

au dl chae quivholetl, ssi 7(x, y)F:, c(y) c(x), y x >. (8., 8]7

c((8 t)x+ty)(8 t)c(x) +tc(y) 0c(x) +t(c(y) c(x)),

sait 7c(x+t(y x)) c(x)

t c(y) c(x).Le phsshet oh oiìtl paurt>+, io suit (8.4).

Zfipraqul`let, ae hppoiqul (8.4) tx+ (8 t)yltx, puis tx+ (8 t)ylty, da 7c(x)

c(tx+ (8

t)y) + (8

t)

c(tx+ (8

t)y), y

x

c(y) c(tx+ (8 t)y) t c(tx+ (8 t)y), y xLe fa`jiehet fls dlux iekhoits, ae ajtilet 7 tc(x) + (8 t)c(y)c(tx+ (8 t)y),lt daefoh faevlxit dlc.

Le fbheklhet ols rols dlx lt y dhes (8.4), puis le sa``het ols dlux iekhoits ajtleuls,ae dàetrl shes prajo`l qul (8.4) i`poiqul (8., 8Ptc(x) + (8 t)c(y) c(tx+ (8 t)y),lt ae àetrl qullst pasitivl sur]>, 8P. Dàestrhtiae ohissl le lxlrfifl (ae paurrh trhflr olthjolhu dl vhrihtiae dlsur]>, 8P).


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Fbhpitrl 8. Carùohtiae lt hehoysl due prajo`l daptiìshtiae 86

Zl`hrqul 8.8 Viclst striftl`let àeatael i.l. si ols iekhoits (8.4) lt (8., 8P). Vi(:)(x)5 >, hoars()? (8 )(>) +(8)paurP>, 8]. Vi

(:)

(x)>, hoars()(8 )(>) +(8).^rluvl. Fa``l (:)(x) 5 > (rlsp.) paur > x 8, lst striftl`let fraisshetl (rlsp.fraisshetl) lt phr faesqulet

() (>) 0 >

(t)dt ? () (rlsp.),

(8) () 0 8

(t)dt 5(8 )() (rlsp.).

Le krauphet fls dlux iekhoits, eaus tiraes

() (>)

? ()? (8) ()8 (rlsp.)=

flst--dirl 7()? (8 )(>) +(8) (rlsp.). ^rluvl du tbar`l 8.:.Vuppasaes B]cP(z)sy`triqul deil pasitivl. Vailet xlt y dluxo`lets distiefts dl Ze. Ietraduisaes oh caeftiae7 ]>, 8P Z dl fohssl F: deil phr

() 0c(x+ (8 )y).\tioisaes ol dvloappl`let oardrl dlux dl paur fhofuolr oh drivl slfaedl dl

(+b) 0c((+b)x+ (8 b)y) 0c(x+ (8 )y+b(x y)).Hvlft 0 x+ (8 )y, eaus hvaes

(+b) 0c(t) + ]c(t)Pb(x y) +8:

b(x y)B]cP(t)b(x y)+b:x y:(t, b(x y)),

lt, phr faesqulet

(+b) 0() +b

]c(t)P(x y)

+

b:

: (x y)B]cP(t)(x y)

+b:(b)


14/43

84 8.6. Faevlxit lt aptiìshtiae

Fa``llst dl fohssl F:, oh drivl prlìrl lt oh drivl slfaedl dl saet daef daeels phr() 0 ]c(x+ (8 )y)P(x y),(:)() 0 (x

y)B]cP((x+ (8

)y) Zee (x y).

Daef oh caeftiae(:)()lst striftl`let pasitivl fhrB]cP(x + (8 )y)lst sy`triqul deilpasitivl ltx0y. Ol ol``l 8.8 sfrit 7

c(x+ (8 )y)? (8 )c(y) +c(x).puisqul 7(>) 0c(y),(8) 0c(x)lt() 0c(x+ (8 )y).

Flfi prauvl qulclst striftl`let faevlxl. Vi paur taut x,B]cP(x)lst pasitivl, uel prluvlsiìohirl eaus plr`lt dl àetrlr qulclst faevlxl.

8.6.: Zsuothts dueifit le apti ishtiae faevlxlTbar`l 8.6 (Faeditiae sucshetl dapti`hoit koajhol) VailetF Ze ue faevlxl ltc 7F Z uel caeftiaeelool. Vaitx ue paiet dl ìeiù` oafho dlc.

i. Viclst faevlxl, hoarsx lst ue paiet dl ìeiù` koajho dlc.

ii. Viclst striftl`let faevlxl, hoarsx lst oueiqul paiet dl ìeiù` koajho dlc.

^rluvl.i. ^hr ohjsurdl 7 sait x Fue paiet qui rhoisl ue ìeiù` oafho dl csurF, i.l. 7r 5>| yF hvlfy x? r, c (y)c(x). (8.3)

tlo qulx el rhoisl phs ue ìeiù` koajho, i.l. quio lxistl ue paietx+ dlFtlo qul 7

c(x+)? c(x). (8.;)Ae ietraduit ol paiet 7yr/:0x+ + (8 )x, hvlf 7 0 r

:x+ xP>, 8].Dhprs ols bypatbsls (8.3) lt (8.;), ae h 7 x+ /J(x, r), fl qui i`poiqul 7r ?x+ x,

lt 7 P>, 8]. ^hr faevlxit dl F, ol paiet yr/:hpphrtilet daef hu slk`let ]x+, xPoui-``lfaetleu dhesF. Dl pous 7

c(yr/:) c(x+) + (8 )c(x) phr faevlxit dlc? c(x) dhprs (8.3).

Flfi faetrldit (8.3) fhr 7yr/:x 0 r: . Ol paietyr/:hpphrtilet daef khol`let oh jauol

auvlrtlJ(x

, r)i.l. 7c(yr/:)c(x

).ii. Zhisaeeaes eauvlhu phr ohjsurdl. Vailet x8 lt x: dlux o`lets dl F rhoishet ol

ìeiù` dlc. ^hr faevlxit dlF,x8+x:

: F, lt fa``lclst striftl`let faevlxl, io suit

c

x8+x:

:

?

8

:c(x8) +

8

:c(x:)

? 8

:iecyF

c(y) + 8

:iecyF

c(y) 0 iecyF

c(y),

Fl qui lst i`passijol.


15/43

Fbhpitrl :

Aptiìshtiae eu`riqul shes faetrhietls

Eaus eaus ietrlssaes dhes fl fbhpitrl oh faeflptiae dl `tbadls eu`riquls paur oh

rlfblrfbl dls paietsx Ze qui rhoislet ol ìeiù` duel caeftiae c 7 Ze Z 7(^) ìe

xZec(x),

aclst suppasl hu àies diccrletihjol. Ae phrol daptiìshtiae shes faetrhietl.

:.8 Faeditiae sucshetl dlxistlefl due paiet ìeiù`

Hvhet dtudilr oh saoutiae (au ols saoutiaes) dl (^), io chut shssurlr dl olur lxistlefl.

Deitiae :.8 \el hppoifhtiaec 7R Ze Z lst ditl ieeil oieei (au falrfivl) ssiH Z, Z 5>| xR, ]x Z 0c(x)HP (:.8)

Ae eatl 7 oi`x+

c(x) 0 +.

Lxl`pols

Lxl`pol dl caeftiae falrfivl Lxl`pol dl caeftiae eae falrfivl

8. c(x) 0x:lst falrfivl.

8


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83 :.:. Faeditiaes dapti`hoit

:. c(x) 0x:8 x::elst phs falrfivl 7 le lcclt, oh suitl dl tlr`l kerhoxe 0 (>, e),e E,lst tlool qul 7 oi`

e+xe0 oi`

e+e0 + `his 7 oi`

e+c(xe) 0 oi`

e+e: 0.

âur àetrlr qulclst ieeil oieei ae utioisl sauvlet oh prapasitiae suivhetl 7

^rapasitiae :.8 Vaitc 7R Ze Z uel hppoifhtiae ltk 7 Z Z vrihetc(x)k(x) hvlf oi`

t+k(t) 0 +.

Hoars,clst ieeil oieei.

^rluvl.Fa``lktled vlrs+ le+H Z, Z 5>| t Z tZ 0k(t)H.

Hvlft 0

x

lt fa``lk(x)

c(

x

), eaus ajtleaes (:.8).

Tbar`l :.8 Vaitc 7 Ze Zuel caeftiae faetieul lt ieeil oieei. Hoars io lxistl uepaietx Ze qui rhoisl ol ìeiù` dlcsurZe. Hutrl`let dit, io lxistlx Ze tlo qul

c(x)c(y), y Ze.^rluvl.Vaitd 0 iec

xZec(x)? +. Vait(xe)eEuel suitl ìeiìshetl flst--dirl tlool qul 7

oiè+

c(xe) 0d ?+. (:.:)

@aetraes qul oh suitl (xe)eElst jarel. ^hr ohjsurdl, ae suppasl qulool el olst phs flst-

-dirl quio lxistl uel saus-suitl eatl (x(e))edl(xe)etlool qul 7 oiè+ x(e) 0 +. ^hr

falrfivit dlc, ae h hoars 7 oiè+

c(x(e)) 0 +, fl qui faetrldit (:.:).Oh suitl(xe)eElst daef jarel 7 io lxistl hoars uel suitl lxtrhitl eatl (x(e))edl (xe)e,

qui faevlrkl vlrs x Ze. Le utioishet `hietlehet oh faetieuit dl c, ae h hoars 7c(x) 0 oi`

e+c(x(e)) 0d.

Ae le dduit hoars dlux fbasls 7d 5 ltxsaoutiae du prajo`l(^).

:.: Faeditiaes dapti`hoitTbar`l :.: (Faeditiaes eflsshirls dapti`hoit oafhol) Vaitc 7 Ze Z uel hppoifhtiaediccrletihjol. Vix Ze rhoisl ue ìeiù` oafho (rlsp. `hxiù` oafho) dl c, hoars 7

c(x) 0 > (FE dapti`hoit du8lr ardrl)Vi, dl pous,clst dlux cais diccrletihjol dhes ue vaisiehkl auvlrt dl x, hoars 7

B]cP(x)slìdeil pasitivl (FE dapti`hoit du:ed ardrl)(rlsp.B]cP(x)slìdeil ekhtivl)


17/43

Fbhpitrl :. Aptiìshtiae eu`riqul shes faetrhietls 8;

^rluvl.Vaitb Ze,b0 >. âurs hsslz pltit, ae deit 7 s Z c(x +sb). hd`ltdaef ue ìeiù` oafho les 0 >, da 7(>) 0c(x)b0 >. Flfi thet vrhi paur tautb, aele dduit 7 c(x) 0 >.

Vuppasaes `hietlehet cdlux cais diccrletihjol. Ae frit ol dvloappl`let dl Thyoar dardrl: dl oh caeftiae. Fa``l c(x) 0 >, ae ajtilet 7

c(x +sb) c(x) 0 s:

:bB]cP(x)b+a(s:).

sait 7s:

:bB]cP(x)b+a(s:)>puisqul x lst ue paiet dl ìeiù` oafho dl c. Hprs divisiae

phrs:, ae chit tledrlsvlrs>lt ae ajtilet 7bB]cP(x)b>. Oh faeditiae du prlìlr ardrl gaul ue rol fletrho le aptiìshtiae eu`riqul 7 lool plr`lt

dl solftiaeelr ue flrthie ea`jrl dl paiets fhedidhts trl dls lxtrl`h oafhux, hpplospaietsfritiqulsau paiets sthtiaeehirls. ^hrì lux, kurlet dls ìei`h oafhux, dls `hxi`h oafhuxlt dhutrls qui el saet ei oue, ei ohutrl, hpplos paiets slool.

Deitiae :.: (âiets fritiquls) Vaitc 7 Ze Zuel hppoifhtiae diccrletihjol. Taut paietx Ze vrihet 7

c(x) 0 >,lst hpplo paiet fritiqul (au paiet sthtiaeehirl) dlc.

@his httletiae ! Ols faeditiaes du tbar`l :.: el saet qul eflsshirls 7 taut paiet a olkrhdilet lst euo elst phs eflsshirl`let ue lxtrlù`. Le lcclt, oh caeftiaeelool 7c 7xZx6 hd`lt lex0 >ue paiet dl drivl euool, qui elst phs ue lxtrlù` ``l oafho dlc.

Tbar`l :.6 (Faeditiae Vucshetl dapti`hoit oafhol) Vait A ue auvlrt dl Ze

. Vait c 7Ze Z uel hppoifhtiae suppasl dl fohsslF: surA. Vi xA vril 7

c(x) 0 > lt B]cP(x)sy`triqul, deil pasitivl (rlsp. deil ekhtivl)Hoarsxlst ue paiet dl ìeiù` oafho (rlsp. `hxiù` oafho) dlc.

Zl`hrqul :.8

8. Fa``lclst dl fohsslF:, phr ol tbar`l dl Vfbwhrz, io elst phs eflsshirl dl vrilrqul oh `htrifl blssileelB]cPlst sy`triqul.

:. Due paiet dl vul ka`triqul, oh faeditiae du slfaed ardrl 7 B]cP(x)deil pasitivl,rlvilet dirl qulclst oafhol`let faevlxl lex, i.l. faevlxl dhes ue vaisiehkl auvlrt dlx. Le prhtiqul, lool lst dicfiol vrilr syst`htiqul`let fhr lool eflssitl dl fhofuolrols drivls slfaedls lt dtudilr ols vholurs praprls dl oh `htrifl blssileel.

Vi dl pous oh caeftiaeelool aptiìslr lst faevlxl au striftl`let faevlxl, le hppoiquhet oltbar`l 8.6 hu faevlxlF0 Ze, ae ajtilet 7

Tbar`l :.4 (Faeditiae Vucshetl dapti`hoit koajhol) Vaitc 7 Ze Zuel hppoifhtiaediccrletihjol ltxue paiet fritiqul dlc.

i. Viclst faevlxl, hoarsxlst ue paiet dl ìeiù` koajho dlc.

ii. Viclst striftl`let faevlxl, hoars xlst oueiqul paiet dl ìeiù` koajho dlc.


18/43

82 :.6. Kerhoits sur ols hokaritb`ls dl dlsfletl

:.6 Kerhoits sur ols hokaritb`ls dl dlsfletl

^hrthet due paietx>hrjitrhirl`let fbaisi, ue hokaritb`l dl dlsfletl vh fblrfblr kerlr

uel suitl ditrs(xm)mEtlool qul 7m E, c(xm+8)c(xm).

Fa``leaes phr deir pous prfis`let oh eatiae dl dlsfletl.

:.6.8 Eatiae dl dirlftiae dl dlsfletl

Ol krhdilet gaul ue rol lssletilo le aptiìshtiae. Dhes ol fhdrl dls `tbadls daptiìshtiae,io slrh khol`let i`parthet dhehoyslr ol fa`partl`let dl oh caeftiae ajglftic dhes flrthiels

dirlftiaes. Fa``leaes paur floh phr rhpplolr ol faeflpt dl drivl dirlftiaeelool 7

Deitiae :.6 Vaitc 7 Ze Z uel hppoifhtiae faetieul. Vaitx Ze ltd Ze.Oh drivl dirlftiaeelool dlclexdhes oh dirlftiaedlst deil phr 7

dc(x= d) 70 oi`t>+

c(x+td) c(x)t

,

si flttl oiìtl lxistl.

^rapasitiae :.: Viclst diccrletihjol le ue paietx

Ze, hoars paur tautd

0 >,chd`lt uel

drivl dhes oh dirlftiaedlexlt 7

dc(x= d) 0Dc(x)(d) 0c(x)d.

Ae rhpplool qul oh rfipraqul lst chussl ! Oh drivhjioit sloae taut vlftlur lex ei`poiqulphs eflsshirl`let oh diccrletihjioit dlclex.

Oh drivl dirlftiaeelool daeel dls iecar`htiaes sur oh pletl dl oh caeftiae dhes oh dirlftiaed, taut fa``l oh drivl daeel dls iecar`htiaes sur oh pletl dls caeftiaes uel vhrihjol. Lephrtifuoilr,

sidc(x= d)5 >hoarsclst fraisshetl dhes oh dirlftiae d. sidc(x= d)? >hoarsclst dfraisshetl dhes oh dirlftiae d.

Dhes fl dlreilr fhs, ae dirh quldlst uel dirlftiae dl dlsfletl dl c.

Deitiae :.4 (Dirlftiae dl dlsfletl) Vailetc 7 Ze Zltx Ze. Ol vlftlurd Ze lstuel dirlftiae dl dlsfletl paurc phrtir du paietxsitc(x+td)lst dfraisshetl let 0 >,flst--dirl sio lxistl 5>tlo qul 7

tP>, P, c(x+td)? c(x). (:.6)


19/43

Fbhpitrl :. Apti`ishtiae eu`riqul shes faetrhietls 89

>.> >.8 >.: >.6 >.4 >.< >.3 >.; >.2 >.9 8.>

>

:

4

3

2

8>

8:

84

83

82

:>

d80c(8, 8)

d60 88

d:0

86

(t)0

c(x

+

tdi)

CIK\ZL :.8 Hoourl dl oh caeftiae c 7 x 8:

x:8 + :x:: hu paiet x 0 (8, 8)

dhes pousilursdirlftiaes.

^rapasitiae :.6 Vailetc 7 Ze Zdiccrletihjol ltx Ze tlo qul 7c(x)0 >. Ol vlftlurd Ze lst uel dirlftiae dl dlsfletl paurc phrtir du paietxssi oh drivl dirlftiaeelool dlclexdhes oh dirlftiaedvril 7

dc(x= d) 0c(x)d ?>. (:.4)

Dl pous paur taut ?8, io lxistl 5>tlo qul 7

tP>,P, c(x+td)? c(x) +tc(x)d. (:.

>.

Vhfbhet qul0c(x)d 5>, io lxistl 5>tlo qul si |t| , hoars |(t)|? . Da 7

tP>, P, c(x+td) c(x) 0tc(x)d+(t)

? t

c(x)d+

0 >.


20/43

:> :.6. Kerhoits sur ols hokaritb`ls dl dlsfletl

Ae dàetrl dl oh ``l chae oiekhoit (:.hrjitrhirl`let fbaisi, ue hokaritb`l dl dlsfletl vh fblrfblr kerlruel suitl ditrs(xm)mEtlool qul 7

m E, c(xm+8)c(xm)Dhprs oh fhrhftrishtiae dl oh dlsfletl (fc prapasitiae :.6), io shkit daef fbhqul itrhtiaem,dl trauvlr ue paietxm+8dhes uel dirlftiaedvrihet 7 c(xm)d ?>.

Ol sfb`h kerho due hokaritb`l dl dlsfletl lst ol suivhet 7

HOKAZITB@L DL DLVFLETL @ADOL.

Daeels7 c 7 Ze Z suppasl hu àies diccrletihjol, x> paiet ieitiho hrjitrhirl`letfbaisi.

Vartil7 uel hppraxi`htiae dl oh saoutiae du prajo`l 7 ìexZe

c(x).


21/43

Fbhpitrl :. Aptiìshtiae eu`riqul shes faetrhietls :8

8. m70 >

:. Thet qul tlst dl faevlrklefl eae shtischit,

(h) Trauvlr uel dirlftiae dl dlsfletldmtlool qul 7

c(xm)

dm?>.

(j) Zlfblrfbl oiehirl 7Fbaisir ue phssm 5> chirl dhes flttl dirlftiae lt tlo qul 7

c(xm+smdm)? c(xm).

(f) @isl gaur 7xm+8 0 xm+smdm=m 70 m + 8 =

6. Zltaurelrxm.

Arhfol/Jaitl eairl. âur ajtleir ol prafbhie itr, ohokaritb`l hurh jlsaie diecar`htiaes suroh caeftiae ajglfticc7 oh vholur eu`riqul dl cle ue paiet daee x, lt sauvlet khol`let ol

krhdiletc(x). Fls iecar`htiaes saet caureils le jaitl eairl, i.l. phr ue saus-prakrh``liedpledhet dl ohokaritb`l daptiìshtiae fbaisi 7 rautiel dl fhofuo du krhdilet phr diccrleflseils oarsqul floui-fi elst phs fhofuohjol lxpoifitl`let, au siùohtlur rlevayhet ols vholurseu`riqulsc(x)lt c(x)shes carùol `htb`htiqul lxpoifitl phr lxl`pol.

Tlst dl faevlrklefl/Tlst dhrrt. Vaitx ue paiet dl ìeiù` oafho du fritrlc aptiìslr.Vuppasaes qul oae fbaisissl fa``l tlst dhrrt dhes ohokaritb`l dl dlsfletl àdol, ol fritrlidho 7 xm 0 x. Dhes ue àedl idho (i.l. le suppashet taus ols fhofuos lxhfts lt oh fhphfitdl fhofuo iooiìtl), sait ohokaritb`l shrrtl hprs ue ea`jrl ei ditrhtiaes, sait io faestruit(tbariqul`let) uel suitl ieeilx8, x:, . . . , xm, . . . dl paiets dl Ze qui faevlrkl vlrsx.

Le prhtiqul, ue tlst dhrrt dlvrh trl fbaisi paur khrhetir qul ohokaritb`l shrrtl taugaurshprs ue ea`jrl ei ditrhtiaes lt qul ol dlreilr paiet fhofuo sait sucsh``let prafbl dl x.Vait 5>oh prfisiae dl`hedl. ôusilurs fritrls saet eatrl dispasitiae 7 taut dhjard (lt

flst ol pous ehturlo), ue fritrl dapti`hoit jhs sur ols faeditiaes eflsshirls dapti`hoit duprlìlr ardrl prsletls dhes oh slftiae :.: 7 ae tlstl si

c(xm)? , (:.;)huqulo fhs ohokaritb`l shrrtl lt caureit oitr faurhetxmfa``l saoutiae.

Le prhtiqul, ol tlst dapti`hoit elst phs taugaurs shtischit lt ae dlvrh chirl hpplo dhutrlsfritrls (caeds sur olxprilefl du eu`riqul) 7

Vthkehtiae dl oh saoutiae 7 xm+8 xm? xm. Vthkehtiae dl oh vholur faurhetl 7 c(xm+8) c(xm)? |c(xm)|. Ea`jrl ditrhtiaes dphsshet ue sluio x ohvhefl 7m ?Itlr@hx.

lt kerhol`let uel fa`jiehisae dl fls fritrls 7

Fritrl dhrrt 0 Tlst dapti`hoit shtischitA\ (Vthkehtiae dl oh vholur faurhetl & Vthkehtiae dl oh saoutiae)A\ Ea`jrl ditrhtiaes `hxiù` hutaris dphss.

Le prhtiqul, ae prcrlrh trhvhioolr hvlf ols lrrlurs rlohtivls poutt quhvlf ols lrrlurs hjsa-ouls, trap dpledhetls dl ofblool.


22/43

:: :.6. Kerhoits sur ols hokaritb`ls dl dlsfletl

:.6.6 Faevlrklefl lt vitlssl dl faevlrklefl

tudilr oh faevlrklefl due hokaritb`l, flst tudilr oh faevlrklefl dl oh suitl dls itrskers phr ohokaritb`l. \e hokaritb`l dl dlsfletl sloae ol àdol prfdlet, lst dit faevlrklet

si oh suitl dl sls itrs(xm)mEfaevlrkl vlrs ue paiet oiìtlx, saoutiae du prajo`l 7

ìexZe

c(x).

Dl pous, oh faevlrklefl lst ditl oafholsi lool eh oilu qul paur dls paiets ieitihux x> dhes uevaisiehkl dlx. Vieae lool lst ditlkoajhol.

Le prhtiqul, ol jut due hokaritb`l daptiìshtiae el slrh qul dl trauvlr ue paiet fritiqul(i.l. ue paiet vrihet oh faeditiae dapti`hoit du prlìlr ardrl 7c(x) 0 >). Ae ietraduithoars oh eatiae dl faevlrklefl koajhol due hokaritb`l daptiìshtiae 7

Deitiae :.< Vait ue hokaritb`l itrhtic qui kerl uel suitl(xm)mEdhes Ze he dl rsaudrlol prajo`l 7

ìexZe

c(x),

ac 7 Ze Z lst uel hppoifhtiae dl fohsslF8. Ohokaritb`l lst dit koajhol`let faevlrklet siqulo qul sait ol paiet ieitiho x> Ze,

oi`m+

c(xm)0 >.

Flttl praprit khrhetit qul ol fritrl dhrrt c(xm) 1 slrh shtischit phrtir due flrthierhek qulool qul sait oh prfisiae 5>dl`hedl.

Io lst jile letledu trs i`parthet dl khrhetir oh faevlrklefl due hokaritb`l saus flrthiels

bypatbsls, `his oh vitlssl dl faevlrklefl lt oh fa`polxit saet khol`let dls chftlurs prledrlle fa`ptl oars dl oh faeflptiae au dl outioishtiae due hokaritb`l = le lcclt, ae h taut ietrt flqul oh `tbadl fbaisil sait oh cais rhpidl, prfisl lt sthjol. âur floh, ae ietraduit ols eatiaesdl vitlssl (au thux) dl faevlrklefl qui `lsurlet ovaoutiae dl olrrlur fa`ìsl xm x.Deitiae :.3 Vait(xm)mEuel suitl ditrs kers phr ue hokaritb`l faevlrklet daee. Aeeatl x oh oiìtl dl oh suitl (xm)mE lt ae suppasl 7m E, xm 0 x (sieae ohokaritb`lfaevlrklrhit le ue ea`jrl ei ditrhtiaes). Oh faevlrklefl dl ohokaritb`l lst ditl 7

oiehirl si olrrlurlm 0xm x dfrat oiehirl`let i.l. sio lxistlP>, 8]tlo qul 7

oi`m+

xm+8 xxm x

0 .

suplroiehirl si

oi`m+

xm+8 xxm x 0 >.

dardrlpsio lxistl >tlo qul 7

oi`m+

xm+8 xxm xp 0 .

Le phrtifuoilr, sip 0 :, oh faevlrklefl lst ditl quhdrhtiqul (krassa àda phrtir dueflrthie rhek, ol ea`jrl dl fbiccrls sikeifhtics lxhfts daujol fbhqul itrhtiae).


23/43


Jile letledu, ae h ietrt fl qul oh faevlrklefl due hokaritb`l sait oh pous olvl passijolhe dl faevlrklr vlrs oh saoutiae le ue ìeiù` ditrhtiaes paur uel prfisiae daeel.

Lxl`pol :.6.8 Oh caeftiaec 7 x

x6

3x+ 8hd`lt ue ìeiù` oafho surZ lex 0

:.

^hrthet duel hppraxi`htiae krassirlx> 0 :dlx, fa`phraes pousilurs hokaritb`ls dl fhofuohpprafb dlx hvlf ? ? 8:, flt hokaritb`lfaevlrkl oiehirl`let hvlf ue thux0|:: 8|.

:6 >.< 8

68

::

0|:: 8| >.22< >.484 >.>Ej ditrhtiaes 8>< 8< 3 4

Ej fbiccrls sike. lxhfts < < ; 8>

Vi 0 8

:

:, oh faevlrklefl lst ditl suplroiehirl lt flst oh `lioolurl faevlrklefl pas-

sijol dl ohokaritb`l le qulstiae.

Vait ohokaritb`l 7 xm+8 0 8:

(xm + :

xm) daet oh faevlrklefl lst quhdrhtiqul. Hoars 4

itrhtiaes sucslet paur fhofuolr uel vholur hpprafbl dlx hvlf < fbiccrls sikeifhticslxhfts = le rhoit, ae h ``l 88 fbiccrls sikeifhtics lxhfts ds oh quhtri`l itrhtiae.

:.4 ^rlìlrs hokaritb`ls dl dlsfletl\e hokaritb`l dl dlsfletl lst dtlrìe phr ols strhtkils dl fbaix dls dirlftiaes dl dlsfletl

sufflssivls, puis phr ol phs qui slrh lcclftu dhes oh dirlftiae fbaisil. Faefletraes eaus dhesflttl phrtil sur ol fbaix dl oh dirlftiae dl dlsfletl 7 oidl lst dl rl`pohflr cphr ue àdol oafhopous si`pol, daet oh ìeiìshtiae eaus daeelrh uel dirlftiae dl dlsfletl dl c.

:.4.8 Hokaritb`ls dl krhdilet phs xl/phs apti`ho

Vait xm Ze oitr faurhet. thet daees oh vholur c(xm) lt ol krhdiletc(xm) (eatrlarhfol), ae rl`pohflchu vaisiehkl dlxmphr sae dvloappl`let dl Thyoar dl prlìlr ardrl 7

c(xm+d)c(xm) + c(xm)d.Ae vaudrhit qul oh drivl dirlftiaeelool c(xm)d sait oh pous pltitl passijol dhes ue vaisiehkldld 0 >. Ae fblrfbl daef rsaudrl 7

ìedZe

c(xm)d s.f. d0 8,daet oh saoutiae eaus hit daeel phr ol tbar`l :.


24/43

:4 :.4. rlìlrs hokaritb`ls dl dlsfletl

Ol fbaix dl oh dirlftiae dl pous cartl dlsfletl deit uel chìool dhokaritb`ls hpplos hoka-ritb`ls dl dlsfletl dl krhdilet daet ol sfb`h lst ol suivhet 7

HOKAZITB@L DL DLVFLETL DL KZHDILET.

Daeels7 c,x>prlìrl hppraxi`htiae dl oh saoutiae fblrfbl, 5>prfisiae dl`hedl.Vartil7 uel hppraxi`htiaex dl oh saoutiae dl 7 c(x) 0 >.

8. m70 > =

:. Thet qul fritrl dhrrt eae shtischit,

(h) Dirlftiae dl dlsfletl 7dm0c(xm)/c(xm).(j) Zlfblrfbl oiehirl7 trauvlr ue phs smtlo qul 7c(xm+smdm)? c(xm).

(f) xm+8 0 xm sm c(xm)

c(xm)

=m 70 m + 8 =

6. Zltaurelrxm.

Io rlstl `hietlehet deir uel strhtkil dl rlfblrfbl oiehirl paur ol fhofuo du phs. Eaustudiaes ifi le prlìrl hpprafbl uel `tbadl phs apti`ho, puis uel phs xl.

@tbadl dl pous pracaedl dlsfletl (Vtllplst dlsflet)

\el idl ehturlool faesistl suivrl oh dirlftiae dl pous cartl dlsfletl lt chirl ue phs quirledl oh caeftiae ìeiìslr oh pous pltitl passijol dhes flttl dirlftiae. Flttl `tbadl lsthpplol `tbadl dl krhdilet phs apti`ho au lefarl `tbadl dl pous pracaedl dlsfletl.Othpl :(h) dl ohokaritb`l dl dlsfletl dl krhdilet lst hoars rl`pohfl phr 7

ZLFBLZFBL OIEHIZL LRHFTL.

:. (h) Fhofuolr ue phs apti hosmsaoutiae dl 7ìes5>

c(xm+sdm).

Oh `tbadl dl pous pracaedl dlsfletl lst uel sartl didhoishtiae 7 duel phrt, eaus el shvaesphs le prhtiqul fhofuolr dl chae lxhftl ue paiet ìeiù`sm dl oajglftic dhes uel dirlftiaedaeel lt ol prajo`l elst le kerho phs triviho. Dhutrl phrt, oh rsaoutiae du prajo`l dlìeiìshtiae ueidi`lesiaeelo dl othpl : (h), ``l dl chae hpprafbl, fatl fblr le tl`psdl fhofuo. âur fls rhisaes, ae plut oui prcrlr phrcais ohokaritb`l dl krhdilet phs faesthet.

Hokaritb`l dl krhdilet phs xl

Oidl lst trs si`pol 7 ae i`pasl uel cais paur tautls oh thiool du phs lcclftu sloae ohdirlftiae dl dlsfletl fhofuol fbhqul itrhtiae. Ols itrhtiaes : (h) lt (j) dl ohokaritb`l dldlsfletl dl krhdilet saet hoars rl`pohfls phr 7

xm+80xm s c(xm)c(xm) .

Oh qulstiae lst hoars 7 fa``let fbaisir ue phs qui khrhetissl oh faevlrklefl dl ohokaritb`l1


25/43

Fbhpitrl :. Aptiìshtiae eu`riqul shes faetrhietls :<

Xuloquls ajslrvhtiaes eu`riquls. Ae saubhitl ìeiìslrc 7 (x, y) Z: 8:

x: + ;:

y:,le utioishet ols hokaritb`ls dl dlsfletl dl krhdilet phs xl lt phs apti`ho.

Fa``leaes phr hehoyslr ol prajo`l dl ìeiìshtiae 7 duel phrt, oh caeftiae clst dluxcais diccrletihjol sur Z: lt striftl`let faevlxl. Dhutrl phrt, ol paiet(>, >)vril ols faeditiaessucshetls dapti`hoit du tbar`l :.4. Daef(>, >)lst oueiqul paiet dl ìeiù` koajho dlc.

VaitRm 0 (xm, ym) Z: oitr faurhet tlo qul 7c(xm, ym)0 >. Fhofuoaes phr oh `tbadldl pous pracaedl dlsfletl, oitr suivhet 7

Dirlftiae dl pous cartl dlsfletl 7dm 0c(Rm) 0 xm

;ym

.

Fhofuo du phs apti`hosmsaoutiae, si lool lxistl, du prajo`l uel di`lesiae 7

ìes5>

c(Rm+sdm) 0 ìes5>

8

:

x:m(8

s): +

;

:

y:m(8

;s):.

Oh saoutiae sl fhofuol dl chae i``dihtl 7sm 0 (x:m+ ;:y:m)/(x

:m+ ;

6y:m).

H fbhqul itrhtiae, oh `tbadl kerl daef ol paiet 7 xm+80xm+x:m+ ;

:y:mx:m+ ;

6y:m

xm;ym

.

Hppoiquaes `hietlehet fls dlux `tbadls phrtir du paiet x> 0 (;, 8., sait 7

c(xm)c(xm smc(xm)) 0 >.Ol paietxm+80 xm smc(xm)vril daef 7 c(xm)c(xm+8) 0 >.

Dlux dirlftiaes dl dlsfletl sufflssivls fhofuols phr ohokaritb`l dl pous pracaedl dls-fletl saetartbakaeholsfl qul trhduislet ols zikzhks dls itrs, ajslrvs sur oh kurl :.:.

Lee, ols daeels du thjolhu :.: iooustrlet oi`parthefl du fbaix du phs dhes ohokaritb`l

dl phs xl 7 ue phs jile fbaisi daeel dls rsuothts fa`phrhjols flux ajtleus phr oh pouspracaedl dlsfletl, ue phs pous pltit htteul ols zikzhk dls itrs `his huk`letl sikeifhtivl`letol ea`jrl ditrhtiaes lt lee, ue phs trap krhed chit divlrklr oh `tbadl.

:.4.: @tbadl dl Elwtae oafhol

âur faestruirl ols `tbadls dl krhdilet, eaus hvaes rl`pohf cphr sae hppraxi`htiaeoiehirl hu vaisiehkl dl oitr faurhet. Eaus hvaes vu qul fls `tbadls el saet phs trs plrcar-`hetls, le phrtil phrfl qulools el tileelet phs fa`ptl dl oh faurjurl qui lst uel iecar`htiaedl slfaed ardrl.


26/43


CIK\ZL :.: Itrhtiaes dls hokas dl krhdilet phs xl lt apti`ho, kerls phrtir du paiet(;, 8., sait 7

c(xm) +B]cP(xm)(xm+8 xm) 0 >,au lefarl, le suppashet qulB]cP(xm)lst deil pasitivl 7

xm+8 0 xm B]cP(xm)8c(xm). (:.2)Ae rlfaeeht dhes oh car`uol (:.2) ols itrhtiaes dl oh `tbadl dl Elwtae vul le faurs

dhehoysl eu`riqul, hppoiqul ifi oh rsaoutiae dl oquhtiae 7c(x) 0 >. Oh `tbadl eldait flpledhet gh`his trl hppoiqul le utioishet uel ievlrsiae dl oh `htrifl Blssileel (qui pluttrl dl trs krhedl thiool lt `ho faeditiaeel) `his poutt le utioishet 7


27/43

Fbhpitrl :. Aptiìshtiae eu`riqul shes faetrhietls :;

m c(xm, ym) c(xm, ym): sm xm ym> 6:.6;< 8>.9>:94 :.4826;:;3 >.699336< 8.:346>.894>:99 8..8432 8.; :.96463< >.6.6< 8>.8: 8>.82 8>.2:9:;32 8>< >.894>:99 >.39 8>< >.>; 8> >.:9 >.69 >.>6 8>9;; 3.36>l :8 >.86>624> 8>9 >.894>:99 >.88 8>9 >.>8 8>9;2 6.433l :8 >.86>624> 8>9 >.68> >.83 8>8>;9 8.28:l :8 >.3929:;2 8>8> >.894>:99 >.8> >.>< 8>8>

THJOL :.8 Itrhtiaes dl oh `tbadl dl pous pracaedl dlsfletl. Ol fritrl dapti`hoit lstshtischit le46itrhtiaes paur uel prfisiae 0 8>< lt le;9itrhtiaes si 0 8>8>.

phs >.6:< >.:< >.8:< >.>< >.>8Ej ditrhtiaes DS 49 8>8 :36 864>

THJOL:.: Ea`jrls ditrhtiaes dl ohokaritb`l dl krhdilet phs xl paur hpprafblr oueiqulhrku`let ìeiù` dlc8>< prs, le caeftiae du phs fbaisi - âiet ieitiho 7 x>0 (;, 8.


28/43


Hokaritb`l

@TBADL DLEL[TAE OAFHOL.

Daeels7 c 7 Ze Zdl fohssl F:, x> prlìrl hppraxi`htiae dl oh saoutiae fblrfbl, 5>prfisiae dl`hedl.

Vartil7 uel hppraxi`htiaex dl oh saoutiae.

8. m70 > =

:. Thet qul c(xm)5 ,(h) Fhofuolrdmsaoutiae du syst`l 7 B]cP(xm)dm 0c(xm) =(j) xm+8 0 xm+dm=

(f) m70 m + 8 =6. Zltaurelrxm=

Zl`hrqul :.: Dlux ajslrvhtiaes i`parthetls 7

8. Oh `tbadl dl Elwtae lst ue hokaritb`l dl dlsfletl phs xl kho 8.

:. Vi oh caeftiaeelool clst quhdrhtiqul, striftl`let faevlxl, hoars ohokaritb`l faevlrkl leuel itrhtiae.

Lxlrfifl :.4.8 Dàetrlr ols dlux hsslrtiaes dl oh rl`hrqul :.:.

Faevlrklefl dl oh `tbadl dl Elwtae oafhol

Ohokaritb`l britl dls praprits dl ohokaritb`l dl Elwtae vu le faurs dhehoysl eu`-riqul paur oh rsaoutiae dls quhtiaes eae oiehirls 7

^rapasitiae :.4 Vaitcdl fohsslF6 ltx ue paiet dl ìeiù` oafho dlc. Ae suppasl le autrlqul oh `htrifl BlssileelB]cP(x)lst deil pasitivl.

8. Io lxistl ue vaisiehkl S dlx tlo qul six> S, hoars oh suitl dls itrs(xm)mEkers phrtir dlx>phr oh `tbadl dl Elwtae oafhol, faevlrkl vlrsx.

:. Oh faevlrklefl lst hu àies quhdrhtiqul.

Oh `tbadl plut divlrklr si ol paiet ieitiho elst phs sucsh``let prafbl due paiet dl ì-eiù` oafho, lt lool elst phs deil si ols `htrifls B]cP(xm) el saet phs deils pasitivls.\tioisl dhes ol fhdrl dl oaptiìshtiae, oh `tbadl dl Elwtae oafhol prsletl ue hutrl iefae-veilet 7 oh saoutiae idletil oh e dl ohokaritb`l elst phs carf`let ue paiet dl ìeiùòafho, `his ueiqul`let ue paiet fritiqul dl c.


29/43


:.4.6 @tbadl dl Khuss-Elwtae

Vi `hietlehetCdsikel uel hppoifhtiae dl Ze dhes Z`, hvlf phr lxl`pol` 5 e, ol sys-t`l dquhtiaesC(x) 0 >eh kerhol`let phs dl saoutiaes. Ol prajo`l dl àiedrls fhrrs

hssafi Cfaesistl rlfblrfblrxtlo qul

r(x) 0 ìe

r(x) 0

8

:

i08

Ci(x): 0

8

:C(x)::, x Ze

. (:.9)

Dl tlos prajo`ls sl rlefaetrlet crqul``let dhes ol fhdrl dl oidletifhtiae dl phrh`trls.Ols vhrihjolsxisaet ols e phrh`trls due àdol pbysiqul eae oiehirl. Ae lcclftul` 5 e`lsurls, lt ae fblrfbl olsxiqui plr`lttlet dhgustlr hu ìlux fl àdol hux `lsurls.

Oh saoutiae dl (:.9) lst fhrhftrisl phr r(x) 0 >. âur hppoiqulr oh `tbadl dl Elwtae,ae dait rsaudrl dls syst`ls dl oh car`l

Br(x) d0r(x) GC(x)T GC(x) + ì08

Ci(x) BCi(x) d0GC(x)T C(x), (:.8>)aGC(x) 0C(x) 0 ]C8(x) . . . C (x)Pdsikel oh `htrifl Ghfajileel dl C lex. Olsfhofuos plr`ltthet dajtleir oh rlohtiae (:.8>) saet dthioos le heelxl.

Oh `htrifl BlssileelBr(x)dlr(x)h uel lxprlssiae hsslz fa`poiqul. Flpledhet ol tlr`li08

Ci(x) BCi(x)lst tlo qul oarsqul ol rsiduC(x)dlvilet pltit, flst--dirl oarsqul oae slrhpprafbl dl oh saoutiae, io dlvilet oui ``l ekoiklhjol.

Oh `tbadl dl Khuss-Elwtae faesistl rl`pohflr (:.8>) phr 7

GC(x)GC(x)d0GC(x)C(x). (:.88)

HOKAZITB@L DLKH\VV-E L[TAE.

Daeels7 Ccaeftiae diccrletihjol,x>paiet ieitiho, 5>prfisiae dl`hedl.Vartil7 uel hppraxi`htiae dl oh saoutiae du prajo`l dl àiedrls fhrrs 7

ìexZe

r(x) 08

:C(x)C(x).

.8. m70 > =

:. Thet qul ,(h) Fhofuo duel dirlftiae dl rlfblrfbl 7fhofuolrdm+8saoutiae dl 7

GC(x)T GC(x) d0GC(x) C(x).

(j) xm+8 0 xm+dm+8=

(f) m70 m + 8 =6. Zltaurelrsm.


30/43

6> :.4. rlìlrs hokaritb`ls dl dlsfletl

Hppoifhtiae hux àiedrls fhrrs oiehirls. Dhes ol fhs a oh caeftiaeClst oiehirl, i.l. 7

C(x) 0H x j, hvlf H@e,p(Z),

ae ajtilet 7 GC(x) 0H, lt oquhtiae dl Khuss-Elwtae (:.88) dlvilet 7 HHdm+8 0H(Hxmj).Fa``ldm+80xm+8 xm, ae ajtilet 7

HHxm+8 0 Hj

lt flfi qulo qul sait xm. Ae rlfaeeht ifi ol syst`l dquhtiaes ear`hols du prajo`l dlàiedrls fhrrs oiehirl 7

ìexZe

8

:Hx j::, (:.8:)

Ae rhpplool qul dhprs ol tbar`l :.4 du faurs dhehoysl eu`riqul,x lst saoutiae du pra-jo`l (:.8:) si lt sluol`let si

x

vril ols quhtiaes ear`hols 7HHx 0 Hj

. Dl pous siHlst dl rhek polie,x lst oueiqul saoutiae dl (:.8:).

Flfi sikeil daef qul oh `tbadl dl Khuss-Elwtae idletil oh saoutiae le uel sluol itrhtiaeoarsqul oh caeftiaeClst oiehirl.

Lxlrfifl :.4.: VaitC 7 Z6 Ze uel hppoifhtiae deil phr 7

Ci(x>, x8, x:) 0fi x> x8 lx:ti , i0 8, . . . , e .

8. FhofuolrCiltB]CiP.:. Le dduirl 7GrltBrar 0 8:C(x)

C(x).


31/43

Fbhpitrl 6

Ietraduftiae oaptiìshtiae sausfaetrhietls

Fl fbhpitrl lst uel faurtl ietraduftiae oaptiìshtiae saus faetrhietl. Ae sietrlssl ohrsaoutiae dl prajo`ls daptiìshtiae dl oh car`l 7

ìe c(x) s.f. xR, (6.8)

aRlst ue saus-lesl`jol eae vidl dl Ze.

TLZ@IEAOAKIL7 Olesl`jolRlst hpplolesl`jol au da`hiel dls faetrhietls.

Taut paietxZe

vrihet 7xR, lst hpplopaiet hdìssijoldu prajo`l (6.8).Fblrfblr uel saoutiae du prajo`l hvlf faetrhietls (6.8) rlvilet fblrfblr ue paiet dlìeiù` oafho dlcdhes olesl`jol dls paiets hdìssijols.

6.8 Faeditiae sucshetl dlxistlefl due paiet ìeiù`

Oh prlìrl qulstiae lst flool dl olxistlefl du paiet dl ìeiù` koajho dl oh caeftiae csurF. Io lxistl priefiphol`let dlux tbar`ls qui plr`lttlet dl rpaedrl flttl qulstiae. Olprlìlr hcr`l olxistlefl due paiet dl ìeiù` oarsqul olesl`jol dls faetrhietls lst clr`jare. Ol slfaed lst sae quivholet paur ols lesl`jols dl faetrhietls clr`s `his eae jares.

Tbar`l 6.8 (Tbar`l dl [lilrstrhss) VaitRue fa`phft (i.l. ue clr` jare) eae vidldl Ze ltc 7R Ze Z uel hppoifhtiae faetieul surR.

Hoarsclst jarel lt httliet sls jarels. Hutrl`let dit, io lxistl x Rpaiet dl ìeiù`koajho dlcsurRi.l. 7

yR, c(x)c(y).

^rluvl.Ietraduisaes oi`hkl dirlftl dlRphr c

c(R) 0

c(y)

yR

.

68


32/43

6: 6.8. Faeditiae sucshetl dlxistlefl due paiet ieiù`

Faesidraes uel suitl ìeiìshetl dhes c(R), flst--dirl uel suitl dl (xe)eEdo`lets dlRtlool qul

c(xe) iecyR

c(y).

Fa``l Rlst clr` jare, io lxistl uel saus-suitl lxtrhitl (x(e))eE qui faevlrkl vlrs uexR. Flttl suitl lxtrhitl vril

x(e)x lt c(x(e)) iecyR

c(y).

Arclst faetieul, da phr ueifit dl oh oiìtl, io suit

c(x) 0 iec yR

c(y)hvlfxR,ltcrhoisl sae ìeiù` surR.

Tbar`l 6.: VailetCue clr` eae vidl dlZe ltc 7C Ze Z uel hppoifhtiae faetieulieeil oieei. Hoarschd`lt ue paiet dl ìeiù` surC. Hutrl`let dit, io lxistlxCtlo qul

yC, c(x)c(y).^rluvl. (i)DeissaesMfa`phft. Fa``l C0, eaus hvaes 7 iec(C) Z {}. VaitH Z, tlo qul H 5 iec

yCc(y). Fa``l clst ieeil oieei, io lxistl Z85>tlo qul paur y Ze

y5 Z8 0 c(y)5 H.Dl pousClst eae vidl 7 io lxistl daef Z: 5 >tlo qul

J(>, Z:) C0.FbaisissaesZ 0 `hx(Z8, Z:)

y5 Z 0 c(y)5 H lt J(>, Z) C0.Ae ietraduit hoars 7M 0 J(>, Z) C. Olesl`jolMlst ue fa`phft eae vidl fhr io lst jare(y Z) lt clr` (ietlrslftiae dl dlux clr`s).(ii)@ieiìsaesc surM. Fa``lclst faetieul lt Mlst fa`phft,chttliet sae ìeiù` surM, flst--dirl

xM | c(x) 0 iecyM

c(y), (6.:)

(iii)@aetraes qul ìeiìslr surMrlvilet ìeiìslr surC. Duel phrt, eaus hvaes

iecyC

c(y) 0 iec iecyMc(y)= iecyC_Mc(y).Dhutrl phrt, paurzCltz /M, ae h 7z Z, sait 7

c(z)5 H 5 iecyC

c(y)

^hr faesqulet 7iecyCc(y) ? iecyC_M

c(y). Io suit

iecyC

c(y) 0 iec yM

c(y)

lt dhprs (6.:) io vilet 7 xMC | c(x) 0 iecyC

c(y).


33/43

Fbhpitrl 6. Ietraduftiae oaptiìshtiae saus faetrhietls 66

6.: Faeditiaes dapti`hoit

Ofriturl dls faeditiaes dapti`hoit le prslefl dl faetrhietls lst jhsl sur oh ``l ie-tuitiae qul dhes ol fhs shes faetrhietls, shvair quio lst i`passijol dl dlsfledrl phrtir dueìeiù`. Tautlcais ols faeditiaes dapti`hoit vuls hu fbhpitrl : el saet pous vhohjols fa``lol àetrl olxl`pol suivhet.

Lxl`pol 6.:.8 Vait ol prajo`l 7

@ieiìslrc(x) 0x:,x Ze, saus oh faetrhietl 7x8.Oh saoutiae dl fl prajo`l lst 7x0 8, lt paurthet 7 c(8) 0 :0 >.

Dhes ol fhs a ol da`hiel dls faetrhietls lst faevlxl, ae h oh faeditiae eflsshirl dapti`h-

oit oafhol suivhetl 7Tbar`l 6.6 (Faeditiae eflsshirl dapti`hoit oafhol) Vaitc 7 Ze Zuel caeftiae dic-

crletihjol ltRue faevlxl clr`, eae vidl dl Ze. Vaitx Ze ue paiet dl ìeiù` oafho duprajo`l 7

ìexZe

c(x) saus oh faetrhietl 7 xR.

Hoars, paur tautxR, c(x)(x x)>.^rluvl.Vait xR. hr faevlxit dl R, x +(xx)lst ue o`let dl Rpaur taut ]>, 8P,da 7

]>, 8P, c(x

+(x x

))c(x

)puisqul x lst ue paiet dl ìeiù` oafho dlcsurR. Ae divisl lesuitl phr (5 >) lt ae chittledrlvlrs>+.

Zhpploaes qul dhes ol fhs faevlxl (i.l. si clst le pous faevlxl) hoars oh faeditiae prfdletldlvilet eflsshirl lt sucshetl.

\el hutrl hpprafbl 7 eatiae dl dirlftiae hdìssijol \el dicfuot i parthetl le apti i-shtiae saus faetrhietl faesistl shvair sl dpohflr dhes olesl`jol dls faetrhietls, i.l. thetdaeel uel dirlftiae dl rlfblrfbl fa``let khrhetir qul oae rlstl dhes olesl`jol F. âur

floh, ae ietraduit oh eatiae dl dirlftiae hdìssijol.Deitiae 6.8 (Dirlftiae hdìssijol) Vaitx Ze ue paiet hdìssijol du prajo`l (6.8).

\el dirlftiae d Ze slrh ditl hdìssijol le x sio lxistl 5>tlo qul x +sd sait hdìssijolqulo qul saitsP>, P.Dhes ol fhs phrtifuoilr a ol da`hiel dls faetrhietls lst faevlxl, dtlrìelr uel dirlftiae dhdìssijol lex rlvilet dtlr ielr ue paiet hdìssijoly, diccrlet dlx7 d 0 y xlst hoarsuel dirlftiae hdìssijol. Le lcclt, qulo qul sait ]>, 8P,x+ d 0 (8)x+ y lst uelfa`jiehisae faevlxl do`lets du faevlxlR, lt daef ue o`let dlR.


34/43

64 6.6. Hokaritb`l du krhdilet praglt

Hutrl prluvl du tbar`l 6.6.Zhisaeeaes phr ohjsurdl. Vuppasaes quio lxistl ue paiet xRtlo qul 7

c(x)(x x)? >.Dhprs oh deitiae :.4 vul hu fbhpitrl :,d 0 x x lst daef uel dirlftiae dl dlsfletl dlchu paietx. Le utioishet oh fhrhftrishtiae daeel phr oh prapasitiae :.6 7 io lxistl 5>tlo qul 7

sP>, P, c(x +sd)? c(x).

Ar 7x R. Dl pous,d 0 x x deit uel dirlftiae hdìssijol du prajo`l (6.8). Daef paurtautsP>, ìe(, 8)P,x +sdlst ue paiet hdìssijol dl (6.8).

6.6 Hokaritb`l du krhdilet praglt

Faestruisaes `hietlehet ue hokaritb`l dl rsaoutiae du prajo`l 7

@ieiìslrc(x),x Ze, saus oh faetrhietl 7xR, (6.6)

dhes ol fhs aRlst ue saus-lesl`jol faevlxl clr` eae vidl dl Ze.

Oh `tbadl du krhdilet praglt siespirl dls `tbadls dl krhdilet dfritls dhes ol fbhpitrl

prfdlet. Oidl dl jhsl faesistl suivrl oh dirlftiae dl pous pracaedl dlsfletl, fa``l dhesol fhs shes faetrhietl 7

xm+80xm smc(xm)asm 5>lst fbaisi dl sartl qul 7 c(xm+smdm)? c(xm). Tautlcais, sixm R, rile el khrhetitqulxm+8hpphrtileel khol`let R. Ds qul oae ajtilet ue paiet eae hdìssijol, ae praglttlfloui-fi sur olesl`jol dl faetrhietlsR.

^raglftiae sur ue faevlxl. Vait Rue faevlxl clr`, eae vidl dlZe. Oh praglftiae due paietx

Ze surR, eatlpR(x), lst ajtleul fa``l saoutiae du prajo`l daptiìshtiae suivhet 7

@ieiìslr 8

:x y:: saus oh faetrhietl 7 yR (6.4)

Oh caeftiaeelool 7y Ze 8:x y::thet faevlxl lt c(y) 0y x, ol tbar`l 6.6 eaus

daeel uel faeditiae eflsshirl lt sucshetl paur qul x 0pR(x)sait saoutiae dl (6.4) 7

yR, (x x)(y x)>. (6.


35/43

Fbhpitrl 6. Ietraduftiae oaptiìshtiae saus faetrhietls 6<

^riefipl dl ohokaritb`l. Vait xm oitr faurhet. Ae kerl oitrhtiae suivhetl ol paiethdìssijol 7

ym0pR(xm smc(xm))

apRdsikel oaprhtlur dl praglftiae sur olesl`jol R. Hvhet dhoolr pous oaie, vriaes quloh dirlftiaedm0ym xm, si lool lst eae euool, lst jile uel dirlftiae dl dlsfletl dlclexm.

Ol``l 6.8 Vaitc 7 Ze Zsuppasl diccrletihjol ltR Ze ue faevlxl clr`, eae vidl.Eataesxmoitr faurhet lt 7

d(s) 0pR(xm sc(xm)) xm, s 5>Vid(s)lst eae euool, hoarsd(s)lst uel dirlftiae dl dlsfletl paur tauts 5>.

^rluvl.Vait s 5>x. Vuppasaes 7 d(s) 0pR(xm sc(xm))xm0 >. Io shkit dl dàetrlrquld(s)lst uel dirlftiae dl dlsfletl dlclexm, hutrl`let dit qul c(xm)d(s)? >.Dhprs oh fhrhftrishtiae (6.,

Da, paur tauty R7d(s) + sc(xm)y xm d(s)>. ûisqulxm R, ae fbaisity0xm, sait 7

d(s) +sc(xm)

d(s)>, au lefarl 7 c(xm)d(s) 8s

d(s)d(s)>.

^hr bypatbsl,d(s)0 >fl qui i`poiqul 7 c(xm)d(s)? >.

Zl`hrqul 6.8 Oh dirlftiaed(s)passdl ols praprits suivhetls 7

8. Vid(s) 0 >, hoars 7pR(xm sc(xm)) 0 xm. Floh sikeil si`pol`let qul oh dirlftiaefbaisil phr ohokaritb`l dl krhdilet lst artbakaehol olesl`jol Rdls faetrhietls le xm.

Ol paietxm lst hoars ue paiet sthtiaeehirl fhr oh faeditiae eflsshirl dapti`hoit (6.6)lst shtischitl.

:. Vuppasaesd(s)0 >. Hoarsxm ltpR(xm sc(xm))saet dls paiets hdìssijols du pra-jo`l (6.6). Oh faevlxit dlReaus khrhetit hoars 7

]>, 8P, xm+d(s)R.

HOKAZITB@L D\ KZHDILET ^ZAGLT.

Daeels7 c,pRue aprhtlur dl praglftiae surR,x>prlìrl hppraxi`htiae dl oh saoutiaefblrfbl, 5>prfisiae dl`hedl.

Vartil7 uel hppraxi`htiaex dl oh saoutiae.

8. m70 > =

:. Thet qul fritrl dhrrt eae shtischit,


36/43

63 6.6. Hokaritb`l du krhdilet praglt

(h) ^raglftiae surR7 ym0pR(xm sc(xm))aslst ol phs fhofuo phr oh `tbadl dl krhdilet fbaisil (s0 8phr lxl`pol) =

(j) Dirlftiae dl dlsfletl 7dm0ym

xm=

(f) Zlfblrfbl oiehirl7 trauvlr ue phs mtlo qul 7c(xm+mdm)? c(xm) =(d) xm+8 0 xm+mdm=m 70 m + 8 =

6. Zltaurelrxm.

Io lst i`parthet dl rl`hrqulr qul ol fhofuo othpl :(h) du praglt surR, plut phrcais trlhussi dicfiol qul ol prajo`l ieitiho. Le lccltymlst ajtleu le rsaovhet ol prajo`l 7

ìeyZe

8

:xm sc(xm) y::

s.f. xR.Io shkit daef dl rsaudrl ue prajo`l daptiìshtiae sur ue faevlxl, hvlf uel caeftiae ajglfticfaevlxl. Oarsqul ol da`hielRdls faetrhietls lst si`pol (faetrhietls dl jarels le phrtifuoilr),flst chishjol. Ds qul ols faetrhietls el saet phs dls faetrhietls dl jarels, ol fhofuo dl ohpraglftiae dlvilet jlhufaup pous doifht.

Lxl`pol 6.6.8 Ae vlut rsaudrl phr uel `tbadl dl krhdilet praglt ol prajo`l suivhet 7

ìe(x,y)Z:

c(x, y) 08

:x: +

;

:y:

s.f. x+y 0 8.Ol da`hiel dls faetrhietlsR 0{(x, y) Z:/x + y 0 8} lst ue faevlxl clr`. âurohvair rsaou hu fbhpitrl :, ae shit qul ol prajo`l bars faetrhietl hd`lt ue ìeiù` koajhole(>, >). Flpledhet(>, >)el shtischit phs oh faetrhietl, fl elst daef phs ue paiet hdìssijoldu prajo`l hvlf faetrhietl.

He dl `lttrl le aluvrl ohokaritb`l dl krhdilet praglt, io eaus chut fbaisir ols `tbadlsdl fhofuo dls phssmltmhux thpls :(h) lt :(f) dl eatrl hokaritb`l 7

thpl :(h) 7 ae fbaisit uel `tbadl dl krhdilet paur ol fhofuo du phssm. thpl :(f) 7 le prlìrl hpprafbl, ae fbaisit ue phs xlm0 8, fl qui i`poiqul 7xm+8 0

ym 0pR(xm sc(xm)).Ol fa`partl`let eu`riqul dl oh `tbadl dl krhdilet praglt hiesi deil, lst iooustr phr

oh kurl 6.8 lt ol thjolhu 6.8 dls itrhtiaes. Ae ajslrvl uel faevlrklefl dl oh suitl ditrs

kers phrtir du paietx>0 (4,


37/43


38/43


39/43

Heelxl H

Fa`po`lets

H.8 Zhpplos dl fhofuo diccrletiloH.8.8 Diccrletihjioit lt krhdilet

Deitiae H.8 VaitG 7 Ze Z uel caeftiaeelool.Glst diccrletihjol hu paietxssi io lxistlG(x) Ze ltb(b, x)vrihet

G(x+ b) 0G(x) + ]G(x)Pb +b(b, x), b Ze, (H.8)hvlf

(b, x) b>

>. (H.:)

Oarsquio lxistl, ol krhdilet plut trl lxpri` ohidl dls drivls phrtilools

G(x) 0

x8G(x)x:G(x)

xeG(x)

. (H.6)Deitiae H.: VaitG 7 Ze Z uel caeftiaeelool.Glst dl fohsslF8.

Glst diccrletihjol hu paietx, fbhfuel dls fa`pashetls dl Glst faetieul.

H.8.: Drivls dardrl suprilur

Deitiae H.6 Ae dit qulG 7 Ze Z lst dlux cais diccrletihjol hu paietx Ze ssi Glst diccrletihjol = fbhfuel dls fa`pashetls dl Glst diccrletihjol.

Deitiae H.4 VaitG 7 Ze Z dlux cais diccrletihjol. Oh Blssileel dlGdeil ohidl dlsls drivls phrtilools dardrl :

B]GP(x) 0 ]igG(x)P8i,ge. (H.4)

69


40/43

4> H.8. Zhpplos dl fhofuo diccrletilo

Deitiae H.< \el caeftiaeeloolG 7 Ze Z lst dl fohssl F: ssi Glst dlux cais diccrletihjol = fbhfuel dls fa`pashetls dlB]GPlst faetieul.

Tbar`l H.8 (dl Vfbwhrz) VaitG 7 Ze Z uel caeftiaeelool dl fohssl F:.Oh `htriflB]GP(x), x Ze, lst sy`triqul.

H.8.6 Carùols dl Thyoar

âur uel caeftiaec 7 Z Z, ols carùols dl Thyoar sfrivlet

c(x+b) 0c(x) +b c(x) +b:

: c(x) +. . .+

be

e!c(eP(x) + A(be+8). (H.,@ 5>, |b|? h

k(b)bp@ 7

k(b)tled vlrs> hvlf b hussi vitl qul bp.

k(b) 0a(bp) oi`b>

k(b)

bp 0 >,

k(b)tled vlrs> hvlf b pous vitl qul bp

âur uel caeftiaec 7 Zp Z, ols carùols dl Thyoar sfrivlet

c(x+b) 0 c(x) +

pg08

bgxgc(x) +8

:

pg08

pm08

bgbm:xgxm

c(x) + A(b6),

0 c(x) +

pg08

bgxgc(x) +8

:

pg08

bg

pm08

bm:xmxg

c(x) + A(b6),

Gc(x) 0@htrifl Ghfajileel0

x8c(x), . . . , xpc(x)

0 (c(x))T.

Ol krhdilet dlclst ol vlftlur trhespas dl sh `htrifl ghfajileel.

Bc(x) 0@htrifl Blssileel0

:xixgc(x)8i,gp

@p(Z). Lool lst sy`triqul.

c(x+b) 0c(x) +Gc(x) b+8

:bT Bc(x) b+ A(b6). (H.3)

Oh eathtiaeH Jdsikel ol praduit `htrifilo. Dhes ol fhs du praduit duel `htrifl oikelphr uel `htrifl faoaeel, ae plut utioislr uel eathtiae dl praduit sfhohirl dl dlux `htrifls(vlftlurs) faoaeels

(H.3)c(x+b) 0c(x) + c(x), b +8:Bc(x) b, b + A(b6). (H.;)


41/43

Fbhpitrl H. Fa`po`lets 48

âur uel caeftiae c 7 Zp Ze, ols carùols dl Thyoar sfrivlet

c(x+b) 0c(x) +Oc(b) +8

:Xc(b) +

A(

b6

). (H.2)

x, b Zp ltc(x) Ze 7 (H.2) lst uel khoit dhes Ze.Oc(b)dsikel ue tlr`l (vlftlurdl Ze ) oiehirl le b, Xc(b)dsikel ue tlr`l quhdrhtiqul le b. âur ols lxpoifitlr, ae plutfrirl uel carùol dl typl (H.3) paur fbhqul fa`pashetl cidl c= paur8ie,

ci(x+b) 0 ci(x) +Gci(x) b+8

:bT Bci(x) b+ A(b6).

Le rhssl`johet tautls ols oikels lt suivhet ols rkols du praduit `htrifl-vlftlur, ae ajtiletOc(b) 0Gc(x) bhvlf

Gc(x) 0@htrifl Ghfajileel0

Gc8(x)

. . .

. . .Gce(x)

@e,p(Z). âur fbhquli, ol tlr`l quhdrhtiqul du dvloappl`let dl cilst daee phr

8

:bT Bci(x) b.

Le rhssl`johet tautls ols oikels lt le chftarishet b draitl, ae ajtilet 7

bT Bc8(x) b

. . .

. . .bT Bce(x) b

0

pm08

bm:xmx8

c8(x) . . .

pm08

bm:xmxp

c8(x)

. . . . . .

. . . . . .p

m08

bm:xmx8

ce(x) . . .

pm08

bm:xmxp

ce(x)

b8

. . .

. . .bp

. (H.9)

Ae h ajtleu uel lxprlssiae dlXc(b)qui plr`lt dfrirl (H.2) saus oh car`l

c(x+b) 0c(x) +

Gc8(x). . .

. . .Gce(x)

b+8:

bT Bc8(x). . .

. . .bT Bce(x)

b+ A(b6).


42/43

4: H.:. Xuloquls dàestrhtiaes

H.: Xuloquls dàestrhtiaes

H.:.8 @tbadl dl Khuss-Elwtae paur oh rsaoutiae dls prajo`ls dl àiedrls

fhrrs^rajo`l dl àiedrls fhrrs

C 7 Zp Ze,e 5 p7C(x) 0 C8(x8, . . . , xp). . . . . .

Ce(x8, . . . , xp)

. r(x) 0

8

:C(x): 08

:C(x), C(x).

Ae fblrfblx0 hrkìe {r(x)} r(x) 0 ìe8:e

i08

C:

i(x), x Zp.

\tioishtiae dl oh `tbadl dl Elwtae

Faeditiae eflsshirl dapti`hoit du prlìlr ardrl 7 r(x) 0 >. Zsaoutiae phr oh `tbadl dl Elwtae 7 paurm>,

x(m+8) 0x(m)

Br(x(m))

8

r(x(m)). (H.8>)

Fhofuo dl r(x)7 ae plut ol chirl leidletihet dl oh phrtil oiehirldue dvloappl`let huvaisiehkl dlx7

:r(x+b) 0 C(x+b), C(x+b) ,0 C(x) +GC(x) b+ A(b:), C(x) +GC(x) b+ A(b:), ,0 C(x), C(x) + C(x), GC(x) b + GC(x) b, C(x) + A(b:).

Fa``l le di`lesiae 8, ae hssiìol ols tlr`ls qui slraet hu àies A(b:)7C(x),A(b:)+GC(x) b, GC(x) b +. . .0A(b:).

âurH@e,p(Z), x Ze lty Zp, ae h x, H y0

HT x, y

7

r(x+b) 0 r(x) +

GC(x)T C(x), b

+ A(b:).

^hr idletifhtiae,

r(x) 0GC(x)T C(x). (H.88)


43/43

Fbhpitrl H. Fa`po`lets 46

Fhofuo dl r(x)7 ae plut ol chirlle fhofuohet ols drivls phrtilools7 ^aur tauti,8ip,

xir(x) 0xi 8:e

g08C:g(x)0 e

g08(xiCg(x)) Cg(x).

Le prlehet le fa`ptl tautl ols oikels, x8r(x). . .xpr(x)

0 x8C8(x) . . . x8Ce(x). . . . . .

xpC8(x) . . . xpCe(x)

C8(x). . .Ce(x)

0 eg08

(Cg) Cg 0GTCC.

Fhofuo dlBr(x)7 ae plut ol chirl phridletifhtiae dl oh phrtil quhdrhtiqul due dvloap-pl`let hu vaisiehkl dlx7

r(x+b) 0 8

:

C(x) +GC(x) b+

8

:Xr(b) + A(b6), C(x) +GC(x) b+8

:XC(b) + A(b6),

,

0 r(x) + r(x), b +8:GC(x) b, GC(x) b . . .

. . . +8

4C(x), XC(b) +8

4XC(b), C(x) + A(b6),

(H.8:) Oh phrtil quhdrhtiqul dl (H.8:) sidletil sloae

Br(x) b, b

0 GC(x)T GC(x) b, b+

8

:C(x), XC(b)

Vuivhet (H.9), C(x), XC(b)0

ei08

Ci(x) bT BCi(x)

b, da

Br 0GTCGC+

ei08

Ci BCi. (H.86)

Fhofuo dlBr(x)7 ae plut ol chirlle drivhet r(x)7

r0e

g08 (Cg) Cg. Ae utioisl oh rkol dl drivhtiae due praduit 7 (r) 0

eg08

BCgCg+ (Cg ) GCg

.

GCg 0 (Cg )T, lte

g08

(Cg) (Cg)T 0C8 . . . Ce

CT8 CTe

0 GTCGC.Fhofuo dl Br(x)7 ae plutohjarilusl`letol chirl le fhofuohet ols drivls phrtilools sl-

faedls = ae el ol chit phs.

Introduction à l'Optimisation Numérique

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