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longues" } {MPLTEXT 1 204 0 "" }}{PARA 202 "" 1 "" {TEXT 207 49 "Warning, the nam e changecoords has been redefined" }{TEXT 207 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 11 "Question a)" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 25 " Cauchy:=proc(f,a,epsilon)" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 29 "\n local aa,bb,compteur,milieu;" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 32 "\naa:=a;bb:=f(aa)+aa;compteur:=1;" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 50 "\nwhile is(abs(bb-aa)>epsilon) and compteur " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 11 "Question b)" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 209 15 "Premier exemple" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 42 "f:=x->exp(-0 .5*x)-1;a:=4;epsilon:=10^(-8);" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,&- I$expGF%6#,$*&$\"\"&!\"\"\"\"\"9$F5F4F5F4F5F%F%F%" }{TEXT 210 0 "" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"\"\"%" }{TEXT 210 0 "" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#>I(epsilonG6\"#\"\"\"\"*++++\"" } {TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 26 "sol:=C auchy(f,a,epsilon); " }{TEXT 205 8 "solution" }{TEXT 206 32 " propos\3 51 par la proc\351dure Cauchy" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 " " {XPPMATH 20 "6#>I$solG6\"7$\"#K$\"$S\"!#5" }{TEXT 210 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 30 "'f'(sol[2])=evalf(f(sol[ 2])); " }{TEXT 205 12 "valeur de f " }{TEXT 206 11 "en ce point" } {MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#/-I\"fG6\"6#$\" $S\"!#5$!#qF*" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 209 16 "Deuxi\350me exemple" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 41 "f:=x->x^3-4*x+1;a:=1.86;epsilon :=10^(-4);" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I \"fG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,(*$9$\"\"$\"\"\"F.!\"% F0F0F%F%F%" }{TEXT 210 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"aG6 \"$\"$'=!\"#" }{TEXT 210 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I(ep silonG6\"#\"\"\"\"&++\"" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 26 "sol:=Cauchy(f,a,epsilon); " }{TEXT 205 8 "solution " }{TEXT 206 32 " propos\351 par la proc\351dure Cauchy" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I$solG6\"7$\"#H$\"\"\"I)i nfinityGI*protectedGF+" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 30 "'f'(sol[2])=evalf(f(sol[2])); " }{TEXT 205 12 "vale ur de f " }{TEXT 206 11 "en ce point" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#/-I\"fG6\"6#$\"\"\"I)infinityGI*protectedGF +$F)I*undefinedGF+" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 209 17 "Troisi\3 50me exemple" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 42 "f:=x->x^3-4*x+1;a:=0.254;epsilon:=10^(-4);" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I\"xGF%F%6$I)oper atorGF%I&arrowGF%F%,(*$9$\"\"$\"\"\"F.!\"%F0F0F%F%F%" }{TEXT 210 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"$\"$a#!\"$" }{TEXT 210 0 " " }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I(epsilonG6\"#\"\"\"\"&++\"" } {TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 26 "sol:=C auchy(f,a,epsilon); " }{TEXT 205 8 "solution" }{TEXT 206 32 " propos\3 51 par la proc\351dure Cauchy" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 " " {XPPMATH 20 "6#>I$solG6\"7$\"#L$\"\"\"I)infinityGI*protectedGF+" } {TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 30 "'f'(so l[2])=evalf(f(sol[2])); " }{TEXT 205 12 "valeur de f " }{TEXT 206 11 " en ce point" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#/ 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I%**fV7!#<$\"0t^o%o509!#:" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "C urve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 1 4" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" " Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" }}{TEXT 212 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 203 0 "" }}}{EXCHG {PARA 203 "" 0 " " {TEXT 203 62 "sur l'axe des abbscissdes apparaissent les points de l a suite " }{TEXT 208 15 "x(n+1)=xn+f(xn)" }{TEXT 203 0 "" }{TEXT 203 4 "\nla " }{TEXT 208 6 "courbe" }{TEXT 203 13 " est cell de " }{TEXT 208 1 "f" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 29 "points:=seq([i,s[i]],i=1..N):" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 51 " plot([points],color=black,thickness=2,style=point);" }{MPLTEXT 1 204 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6(-%'CURVE SG6#7.7$$\"\"\"\"\"!$\"3/++++++SD!#=7$$\"\"#F*$\"3#******fsOtM\"F-7$$ \"\"$F*$\"3O+++Iteep!#>7$$\"\"%F*$\"3B+++!=D\"RNF87$$\"\"&F*$\"3/++++u 7&y\"F87$$\"\"'F*$\"3G++++ANl*)!#?7$$\"\"(F*$\"3K++++$3F\\%FH7$$\"\")F *$\"37++++`()[AFH7$$\"\"*F*$\"3-++++'p]7\"FH7$$\"#5F*$\"3h*********Hpi &!#@7$$\"#6F*$\"3')********4'Q\"GFgn7$$\"#7F*$\"3\"**********GqS\"Fgn- %+AXESLABELSG6$Q!6\"Feo-%*THICKNESSG6#F0-%&STYLEG6#%&POINTG-%&COLORG6& %$RGBG$F*!\"\"FbpFbp-%%VIEWG6$;$\"1-++++++y!#;$\"2-+++++?A\"!#:;$!+UI[ O\\!#7$\"-Uf=x!f#Faq" 1 5 2 0 10 2 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve \+ 1" }}{TEXT 212 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 203 31 "Les poin ts trac\351s sont ceux de " }{TEXT 208 18 "coordonn\351es (n,xn)" } {TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 208 "> \+ " 0 "" {TEXT 213 21 "Quelques Explications" }{MPLTEXT 1 204 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 203 2 "Po" }{TEXT 203 21 "ur que la m\351thode du " }{TEXT 208 28 "point fixe de Cauchy-Picard " }{TEXT 203 51 "marche correcteme nt encore faut-il que la fonction " }{TEXT 208 11 "g(x)=f(x)+x" } {TEXT 203 6 " soit " }{TEXT 208 12 "contractante" }{TEXT 203 8 " sur u n " }{TEXT 208 29 "intervalle stable contenant a" }{TEXT 203 0 "" } {TEXT 203 106 "\nA d\351faut de pouvoir justifier rigoureusement le d \351faut de la m\351thode pour les deux derniers exemples, la " } {TEXT 208 25 "grande valeur de |g'(a)| " }{TEXT 203 37 "perment de voi r comment la propri\351t\351 " }{TEXT 208 40 "\"g est contractante\" e st mise en d\351faut " }{TEXT 203 4 "(cf " }{TEXT 208 33 "th\351or\350 me des accroissements finis" }{TEXT 203 1 ")" }{TEXT 203 0 "" }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 204 "" 1 "" {TEXT 210 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 211 16 "Deuxi\35 0me Exemple" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 24 "f:=x->x^3-4* x+1;a:=1.86;" }{MPLTEXT 1 204 1 "\n" }}{PARA 204 "" 1 "" {XPPMATH 20 " 6#>I\"fG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,(*$9$\"\"$\"\"\"F. !\"%F0F0F%F%F%" }{TEXT 210 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I \"aG6\"$\"$'=!\"#" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 21 "g:=unapply(f(x)+x,x);" }{MPLTEXT 1 204 0 "" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"gG6\"f*6#I\"xGF%F%6$I)operatorGF% I&arrowGF%F%,(*$9$\"\"$\"\"\"F.!\"$F0F0F%F%F%" }{TEXT 210 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 38 "Diff('g'(a),x)=subs(x=a, diff(g(x),x));" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#/-I%DiffG6$I*protec tedGF'I(_syslibG6\"6$-I\"gGF)6#$\"$'=!\"#I\"xGF)$\"&)yt!\"%" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 211 9 "Troisi\350me" }{TEXT 211 8 " Exemple" } }{PARA 203 "" 0 "" {TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 25 "f :=x->x^3-4*x+1;a:=0.254;" }{MPLTEXT 1 204 0 "" }{MPLTEXT 1 204 1 "\n" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I\"xGF%F%6$I)operatorG F%I&arrowGF%F%,(*$9$\"\"$\"\"\"F.!\"%F0F0F%F%F%" }{TEXT 210 0 "" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"$\"$a#!\"$" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 21 "g:=unapply(f(x)+x,x); " }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"gG6\"f*6 #I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,(*$9$\"\"$\"\"\"F.!\"$F0F0F%F%F% " }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 38 "Dif f('g'(a),x)=subs(x=a,diff(g(x),x));" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#/-I%DiffG6$I*protectedGF'I(_syslibG6\"6$-I\"gGF )6#$\"$a#!\"$I\"xGF)$!(_k!G!\"'" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 " > " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {TEXT 213 18 "Comment y rem\351dier" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 203 119 "Au lieu de r\351soudre f(x)+x=x pour trouver les zeros de f, il s'agi t de chercher les zeors de f comme \351tant solution de " }{TEXT 208 15 "\nk f(x) + x = x" }{TEXT 203 0 "" }{TEXT 203 35 "\no\371 k est cho isi de facon \340 rendre " }{TEXT 208 25 "g(x)=kf(x)+x contractante" } {TEXT 203 48 " (au moins sur un intervalle stable contenant a)" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 1 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 47 "a:=0.254;Diff('f'(a),x)=subs(x=a,diff(f(x),x));" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"$\"$a#!\"$" } {TEXT 210 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#/-I%DiffG6$I*protect edGF'I(_syslibG6\"6$-I\"fGF)6#$\"$a#!\"$I\"xGF)$!(_k!Q!\"'" }{TEXT 210 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 203 72 "Aucune valeur de k \+ ne rendra g contractante sur R, cependant en prenant " }{TEXT 208 17 " k=-1/10 et k=1/10" }{TEXT 203 12 ", on a aura " }{TEXT 208 9 "|g'(a)|< 1" }{TEXT 203 25 " (pour a=1.86 et a=0.254)" }{TEXT 203 1 " " }{TEXT 203 55 "\non peut raisonablement penser que cela restera encore " } {TEXT 208 42 "vraie en tout point d'un voisinage I de a " }{TEXT 203 22 "(par continuit\351 de g')" }{TEXT 208 2 ". " }{TEXT 203 24 "\nEnco re faut-il choisir " }{TEXT 208 42 "a est suffisament proche d'un poi nt fixe " }{TEXT 203 8 "pour que" }{TEXT 203 7 " toute " }{TEXT 208 29 "valeur successive de la suite" }{TEXT 203 28 " d\351finie par x(n +1)=g(xn) " }{TEXT 208 12 "reste dans I" }{TEXT 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 1 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 23 "fk:=unapply(f(x)/10,x);" }{TEXT 206 4 " ici" } {TEXT 205 7 " fk=k*f" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I#fkG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,(*$9$ \"\"$#\"\"\"\"#5F.#!\"#\"\"&F0F1F%F%F%" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 24 "Cauchy(fk,1.86,10^(-4));" }{MPLTEXT 1 204 0 "" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#7$\"#N$\"+:rkUD!#5" }{TEXT 210 0 "" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 26 "Cauchy(-fk,0.254,10^(- 5));" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#7$\"#P$! +g_!\\6#!\"*" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 203 93 "On voit apparaitr e un nouveau candidat pour \352tre zero de f, on calcule la d\351riv\3 51e en ce point" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 47 "a:=- 2.11;Diff('f'(a),x)=subs(x=a,diff(f(x),x));" }{MPLTEXT 1 204 0 "" }} {PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"aG6\"$!$6#!\"#" }{TEXT 210 0 "" } }{PARA 204 "" 1 "" {XPPMATH 20 "6#/-I%DiffG6$I*protectedGF'I(_syslibG6 \"6$-I\"fGF)6#$!$6#!\"#I\"xGF)$\"&jN*!\"%" }{TEXT 210 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 203 34 "ici encore k=-1/10 devrait suffire" }} }{EXCHG {PARA 204 "" 1 "" {XPPMATH 20 "6#$\"+)Rf[)R!\"*" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 26 "Cauchy(-fk,-2.11,1 0^(-9));" }{MPLTEXT 1 204 0 "" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#7$\" \"($!+Tv!\\6#!\"*" }{TEXT 210 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 0 "" }}}{PARA 209 "" 0 "" {TEXT 214 0 "" }}{PARA 209 "" 0 "" {TEXT 214 0 "" }}{PARA 209 "" 0 "" {TEXT 214 0 "" }}{PARA 209 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }