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QMat^H;tBry::Zh;t>::W]Qmysy::nCt>ryvYt@W\\H;:t@i@vY`FW\\H;::Nxiy=::nCt>rgZhKabM::nKjy;::bUZH;::t@i@vYJJcMnC:::JJsyxIJqbMnCX EyaHLabM:vYxI:W=nC::JqryvY:Nt>W<::ZZlywY: :ZHKa:i_CMqbMnC::tByay=::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::lFa;nFJ:lFnFxyyyITP`?ZSxy:^S;:m^`C mSxyyqy;:JUEXyU;]vnwII]gyryvY:::::::::::::::::::::::::::::::::T<:yA::: ::::::::^JLB::E?fNLBC>:fNxIyA^RPDC^><:PDyay=PDC^>L<;BsvYLHCnryvYNxiy=:::ZC=Jq:W^qW]C=`Fi`ymysgZqlysy::::>N tBW]hKACbvYyA::::i_qNtBW]hKAsoZy=:::::::t@W\\HKAsyvY::::Nxiy=:::vPXEryvYjUc]nGt@W^Q=xI::::M`HLqbUZHl^afymy;:i`qxPi_HLabMnG `Fi_qlesyvYj^;i_ymy;BbnKtByay=:`FW^hKq:W^qNJqbUnKZyE:mvyyqyK`syyQmxi:^SlFmN:`CvYyA::b?Vf;qyPJjxUyyIyA::::::::::::: :::::::::::::::::::Z`:xIyA:::::::::::^JEB[::fNxIE_>L<;:E_ymy;:g`PM\\yT ?Uw\\_VxIyA:^VLxfYFjysy^Rjysy:^JLB:E?vY:JeCk^RTGyay=::::::i?xyy;::::::::::::::::::::::i_HKa:yay=::::::::::: ::::::::::::<:J\\syfNLBC>:fNxyywY::::::::: :::LHCnNJabUZy=::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::<:J\\;E_>L<;:E_ymy;:::::::::::BsZ>W=vY::::::::::::::::::::::::::::: ::t@W\\HKAsyvY:::XGW\\H;jUsyvY:::::::::::::::::::::::::::::::::::::::: ::::::::::::::tBW\\HKqryvY:::::::::::::::::JA;W\\h;xI::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::bC :vY:::::::::::J\\CkdB:jysy^VLFC@:nTjy;::::::::::::C`>=Je;yA::::::::::Z ::xI::::::::::::::bMnCJA;W\\Hlysy:::::::::::::::::::::::vPt>W:RlvYxI:::::::::::::::::::::::i?nCjysy:ZqryvY::::::::::::::::: ::tBbMnGxIyA:vPJa:yay=::::::::::::::::::::::::::t@W\\H;:::::JqryvY:::: :::::::::::::::::::::::::::::::::::::::::::::::::::tBbMnGjy;:::::::::: :::::::W>nCt@ry::vPJa:yay=:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::<:ry::::::::::::::ZZ@NbM:Zymy;:::::::::::::::::::::::i? nCjysy:ZqryvY:::::::::::::::::::XGi_ql^;yA:vPt>W:::::b]vYxI::::::::::::::::::::::::::::::::::::::::: ::::::::::::::roZql^;yA:::::::::::::::::vTjUSlZy=:jUcMnCjysy:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::Z::xI:::::::::::::::W \\H;JqryvY:::::::::::::::::::::::jU;W:1:" }{TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT 256 63 "Polyn \364me d'interpolation de Lagrange et polyn\364mes de Tchebychev" }} {PARA 19 "" 0 "" {TEXT -1 40 "\251 Apprendre Maple. Alain Le Stang. 20 06\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Soit " }{TEXT 257 2 "n " }{TEXT -1 23 "points de coordonn\351es \+ (" }{XPPEDIT 18 0 "(x[1],y[1]) " "6$&%\"xG6#\"\"\"&%\"yG6#F&" }{TEXT -1 7 "),...,(" }{XPPEDIT 18 0 "x[n],y[n] " "6$&%\"xG6#%\"nG&%\"yG6#F& " }{TEXT -1 6 " ).Le " }{TEXT 258 36 "polyn\364me d'interpolation de L agrange" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "relatif \340 c es points est l'unique polyn\364me " }{TEXT 259 1 "P" }{TEXT -1 10 " d e degr\351 " }{TEXT 260 3 "n -" }{TEXT -1 12 "1 tel que , " }{XPPEDIT 18 0 "P(x[i])=y[i]" "6#/-%\"PG6#&%\"xG6#%\"iG&%\"yG6#F*" }{TEXT -1 13 " , pour tout " }{TEXT 305 1 "i" }}{PARA 0 "" 0 "" {TEXT -1 13 "de \{1 ,2,. . ," }{TEXT 308 1 "n" }{TEXT -1 21 "\}. Il est d\351fini par " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&P(x)~G-%$SumG6$*&&%\"yG6#%\"kG\"\" \"-&%\"LGF+6#%\"xGF-/F,;F-%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-& %\"LG6#%\"kG6#%\"xG**-%(ProductG6$,&F*\"\"\"&F*6#%\"iG!\"\"/F3;F0,&F(F 0F4F0F0-F-6$F//F3;,&F(F0F0F0%\"nGF0-F-6$,&&F*F'F0F1F4F5F4-F-6$F@F:F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "1\260 E crire une proc\351dure " }{TEXT 261 8 "Lagrange" }{TEXT -1 45 " qui re \347oit comme param\350tres les deux listes " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 262 2 "x=" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "x[1" "6# &%\"xG6#\"\"\"" }{TEXT -1 5 ",...," }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#% \"nG" }{TEXT -1 4 "] , " }{TEXT 263 1 "y" }{TEXT -1 2 "=[" }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 5 ",...," }{XPPEDIT 18 0 "y[n ]" "6#&%\"yG6#%\"nG" }{TEXT -1 14 "] , et le nom " }{TEXT 264 1 "t" } {TEXT -1 34 " de la variable , et qui retourne " }{TEXT 265 4 "P(t)" } {TEXT -1 11 " d\351velopp\351." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "2\260 Soit " }{TEXT 266 1 "f" }{TEXT -1 17 " une fonction de " }{TEXT 267 2 "IR" }{TEXT -1 6 " dans " }{TEXT 268 2 "IR" }{TEXT -1 24 " . Ecrire une proc\351dure " }{TEXT 269 9 "La grangef" }{TEXT -1 12 " qui re\347oit " }}{PARA 0 "" 0 "" {TEXT -1 33 " comme param\350tres l'op\351rateur " }{TEXT 270 1 "f" }{TEXT -1 13 " , une liste " }{TEXT 271 2 "x=" }{TEXT -1 1 "[" }{XPPEDIT 18 0 "x [1" "6#&%\"xG6#\"\"\"" }{TEXT -1 5 ",...," }{XPPEDIT 18 0 "x[n]" "6#&% \"xG6#%\"nG" }{TEXT -1 14 "] , et le nom " }{TEXT 272 2 "t " }{TEXT -1 18 "de la variable, et" }}{PARA 0 "" 0 "" {TEXT -1 17 " qui reto urne " }{TEXT 273 4 "P(t)" }{TEXT -1 4 " o\371 " }{TEXT 274 1 "P" } {TEXT -1 56 " est le polyn\364me d'interpolation de Lagrange associ \351 aux" }}{PARA 0 "" 0 "" {TEXT -1 12 " points (" }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 2 ", " }{TEXT 275 1 "f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 10 ")), ... , (" }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 1 "," }{TEXT 276 2 " f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 15 ")) . On a donc " }{TEXT 277 1 "P" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 2 ")=" }{TEXT 306 1 "f" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT -1 14 ") , pou r tout " }{TEXT 278 1 "i" }{TEXT -1 14 " de \{1,2,. . ," }{TEXT 279 1 "n" }{TEXT -1 3 "\}. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "3\260 On d\351finit " }{TEXT 280 1 "f" }{TEXT -1 2 ": \+ " }{XPPEDIT 18 0 "x->1/(1+x^2) " "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6 \"*&\"\"\"F,,&F,F,*$F%\"\"#F,!\"\"F*F*F*" }{TEXT -1 14 ". Repr\351sent er " }{TEXT 281 1 "f" }{TEXT -1 43 " et le polyn\364me d'interpolation de Lagrange" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 307 2 "Lf" } {TEXT -1 42 " associ\351 aux points d'abscisses -1,0,1,2 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "4\260 On d\351fin it les polyn\364mes de Tchebychev " }{XPPEDIT 18 0 "T[n]" "6#&%\"TG6#% \"nG" }{TEXT -1 9 " d'ordre " }{TEXT 282 1 "n" }{TEXT -1 3 " , " } {TEXT 283 1 "n" }{TEXT -1 23 " entier naturel , par :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"TG6#%\"nG6#%\"xG*&-%$cosG6#*&F(\"\"\"-%'arcc osGF)F0F0)\"\"#,&F(F0!\"\"F0F6" }}{PARA 0 "" 0 "" {TEXT -1 39 " Ecr ire les polyn\364mes de Tchebychev " }{XPPEDIT 18 0 "T[1]" "6#&%\"TG6# \"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "T[2]" "6#&%\"TG6#\"\"#" } {TEXT -1 11 " , . . . , " }{XPPEDIT 18 0 "T[10]" "6#&%\"TG6#\"#5" } {TEXT -1 23 " sous forme d\351velopp\351e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "5\260 On travaille avec la fonc tion " }{TEXT 284 1 "f" }{TEXT -1 16 " d\351finie au 3\260 :" }}{PARA 0 "" 0 "" {TEXT -1 43 " Repr\351senter sur un m\352me graphique pou r " }{TEXT 285 1 "n" }{TEXT -1 22 "=2,5,10 , la fonction " }{TEXT 286 1 "f" }{TEXT -1 17 " et les polyn\364mes" }}{PARA 0 "" 0 "" {TEXT -1 23 " d'interpolation de " }{TEXT 287 1 "f" }{TEXT -1 54 " associ \351s lorsque l'on d\351coupe l'intervalle [-5,5] en " }{TEXT 288 1 "n " }{TEXT -1 15 " intervalles de" }}{PARA 0 "" 0 "" {TEXT -1 76 " m \352me longueur. Que remarque-t-on aux extr\351mit\351s de l'intervall e [-5,5] ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "6\260 On travaille avec la fonction " }{TEXT 289 1 "f" }{TEXT -1 16 " d\351finie au 3\260 :" }}{PARA 0 "" 0 "" {TEXT -1 43 " Repr \351senter sur un m\352me graphique pour " }{TEXT 290 1 "n" }{TEXT -1 21 "=2,5,10 , la fonction" }{TEXT 291 2 " f" }{TEXT -1 17 " et les pol yn\364mes" }}{PARA 0 "" 0 "" {TEXT -1 23 " d'interpolation de " } {TEXT 292 1 "f" }{TEXT -1 44 " associ\351s lorsque l'on prend pour abs cisses " }{XPPEDIT 18 0 "x[k]" "6#&%\"xG6#%\"kG" }{TEXT -1 5 " (o\371 \+ " }{TEXT 293 1 "k" }{TEXT -1 4 "=0.." }{TEXT 294 1 "n" }{TEXT -1 7 ") \+ , les" }}{PARA 0 "" 0 "" {TEXT -1 46 " racines du polyn\364me de Tc hebychev d'ordre " }{TEXT 295 1 "n" }{TEXT -1 22 "+1,multipli\351es pa r 5 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#%\"kG,$-%$cosG6#,$*( ,&F'\"\"#\"\"\"F0F0%#PiGF0,&%\"nGF0F0F0!\"\"#F0F/\"\"&" }}{PARA 0 "" 0 "" {TEXT -1 63 " Que constate-t-on par rapport aux r\351sultats o btenus au 5\260 ?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " " 0 "" {TEXT 296 8 "Corrig\351:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 297 11 "Question 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Lagrange:=proc(x::list,y::list,t::name)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " local k,L,n,P,Q;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " P:=0:n:=nops(x):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " for k to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " Q:=simplify(product(t-x[i],i=1..n)/(t-x[k])): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " L[k]:=Q/subs (t=x[k],Q):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " P :=P+y[k]*L[k]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " end do:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " expand(P);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)L agrangeGf*6%'%\"xG%%listG'%\"yGF)'%\"tG%%nameG6'%\"kG%\"LG%\"nG%\"PG% \"QG6\"F5C&>8'\"\"!>8&-%%nopsG6#9$?(8$\"\"\"FBF;%%trueGC%>8(-%)simplif yG6#*&-%(productG6$,&9&FB&F?6#%\"iG!\"\"/FR;FBF;FB,&FOFB&F?6#FAFSFS>&8 %FX*&FFFB-%%subsG6$/FOFWFFFS>F8,&F8FB*&&9%FXFBFZFBFB-%'expandG6#F8F5F5 F5" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Lagrange([4,0,-1,1,7],[7,2,2,8,1],x); # essai sur u n exemple particulier" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\" \"%\"\"\"#\"$R\"\"%35*&#\"%&)GF+F()F&\"\"#F(F(*&#\"$&=\"$E\"F(*$)F&\" \"$F(F(!\"\"*&#\"$j&F4F(F&F(F(F0F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 11 "Question 2:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Lagrangef:=proc(f ::operator,x::list,t::name)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " lo cal k,y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " y:=[seq(f(x[k]) ,k=1..nops(x))]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " Lagrang e(x,y,t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%*LagrangefGf*6%'%\"fG%)operatorG'%\"xG%%listG' %\"tG%%nameG6$%\"kG%\"yG6\"F3C$>8%7#-%$seqG6$-9$6#&9%6#8$/FA;\"\"\"-%% nopsG6#F?-%)LagrangeG6%F?F69&F3F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 11 "Question 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f:=x->1/(1+x^2):Lf:=unapply(Lagrangef(f,[-1,0,1,2] ,x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LfGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,**$)9$\"\"$\"\"\"#F1\"\"&*&#F1\"\"#F1*$)F/F6F1F1!\"\"*& #F1F3F1F/F1F9F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " plot(\{f(x),Lf(x)\},x=-1.2..2.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 254 224 224 {PLOTDATA 2 "6&-%'CURVESG6$7in7$$!3%**************>\"!#<$\"35, +++++Wo\"oEAG=k&F-7$$!3cJL$eN-*e\"*F-$\"3mJ@ s&y245'F-7$$!3=**\\ibUk&z)F-$\"3R\\F.4)R+`'F-7$$!3plmTbhQK%)F-$\"3iet) R[[?$pF-7$$!3Y***\\iR*f&4)F-$\"3Ql-r+G.\"G(F-7$$!37KL3PE\")exF-$\"3v?6 yLcl2wF-7$$!3E)**\\PFm81(F-$\"395\\0h#G\\@)F-7$$!3LJL3Zi1SjF-$\"35y=Lj ]\\[()F-7$$!3\\)**\\(Q%z5i&F-$\"3C6*o!Hf<*=*F-7$$!3!\\mm;88:)[F-$\"3G* 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