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" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 "\n [n,p];" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\nend:" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 203 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 25 "Permut_ligne:=proc(i,j,A)" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 13 "\nlocal B,k,t;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 14 "\nt:=taille(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 27 "\nB: =array(1..t[1],1..t[2]);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 13 "\nB :=evalm(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 24 "\nfor k from 1 t o t[2] do" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 33 "\n B[i,k]:=A[j,k]; B[j,k]:=A[i,k];" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 4 "\nod;" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 "\nevalm(B);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 6 "\nend: " }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 24 "Dilat_ligne:=proc(a,i,A)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 13 "\nlocal k,t,B;" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 14 "\nt:=taille(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 27 "\nB:=array(1..t[1],1..t[2]);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 13 "\nB:=evalm(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 24 "\nf or k from 1 to t[2] do" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 19 "\n B[ i,k]:=a*A[i,k];" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 4 "\nod;" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 "\nevalm(B);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\nend:" }{MPLTEXT 1 202 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 30 "Transvect_ligne:=proc(i,a,j,A)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 13 "\nlocal k,t,B;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 14 "\nt:= taille(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 27 "\nB:=array(1..t[1 ],1..t[2]);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 13 "\nB:=evalm(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 24 "\nfor k from 1 to t[2] do" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 26 "\n B[i,k]:=A[i,k]+a*A[j,k];" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 4 "\nod;" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 10 "\nevalm(B);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\nend:" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 31 "CherchePivot_colonne:=proc(A,i)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 12 "\n local t,k;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 16 "\n t:=taille(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 44 " \n if is(i>t[1])then RETURN(IMPOSSIBLE); fi;" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 26 "\n for k from i to t[1] do" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 40 "\n if is(A[k,i]<>0) then RETURN(k); fi;" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 6 "\n od;" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 13 "\n RETURN(0);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\nend:" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 27 "Permut_colonne:=proc(i,j,A)" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 13 "\nlocal B,k,t;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 14 "\nt:=taille(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 27 "\nB: =array(1..t[1],1..t[2]);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 13 "\nB :=evalm(A);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 24 "\nfor k from 1 t o t[1] do" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 33 "\n B[k,i]:=A[k,j]; B[k,j]:=A[k,i];" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 4 "\nod;" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 "\nevalm(B);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 6 "\nend: " }{MPLTEXT 1 202 0 "" }}}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 202 0 "" }}{PARA 204 "> " 0 "" {TEXT 204 10 "Exercice 3" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {TEXT 205 11 "question a)" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 " > " 0 "" {MPLTEXT 1 202 82 "A:=array(1..3,1..4): for i from 1 to 3 do \+ for j from 1 to 4 do A[i,j]:=i+j: od:od:" }{MPLTEXT 1 202 0 "" }}} {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 11 "A=evalm(A);" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/I\"AG6\"-I'matrixGF%6#7 %7&\"\"#\"\"$\"\"%\"\"&7&F,F-F.\"\"'7&F-F.F0\"\"(" }{TEXT 206 0 "" }}} {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 26 "A1:=Dilat_ligne(1/2,1,A) ; " }{TEXT 205 11 "L2 <-- L2/2" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I#A1G6\"-I'matrixGF%6#7%7&\"\"\"#\"\"$\"\"#F.#\"\"& F.7&F-\"\"%F0\"\"'7&F2F0F3\"\"(" }{TEXT 206 0 "" }}}{EXCHG {PARA 200 " > " 0 "" {MPLTEXT 1 202 32 "A2:=Transvect_ligne(2,-3,1,A1); " }{TEXT 205 15 "L2 <-- L2- 3*L1" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I#A2G6\"-I'matrixGF%6#7%7&\"\"\"#\"\"$\"\"#F.#\"\"&F.7 &\"\"!#!\"\"F.F4#!\"$F.7&\"\"%F0\"\"'\"\"(" }{TEXT 206 0 "" }}}{EXCHG {PARA 200 "" 0 "" {MPLTEXT 1 202 33 "A3:=Transvect_ligne(3,-4,1,A2); \+ " }{TEXT 205 15 "L3 <-- L3- 4*L1" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#>I#A3G6\"-I'matrixGF%6#7%7&\"\"\"#\"\"$\"\"#F.#\" \"&F.7&\"\"!#!\"\"F.F4#!\"$F.7&F2F4!\"#F6" }{TEXT 206 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {TEXT 205 11 "question b)" }{MPLTEXT 1 202 0 "" }} }{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 15 "Gauss1:=proc(A)" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 14 "\n local t,B,k;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 17 "\n t:=taille(A); " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 28 "\n B:=array(1..t[1],1..t[2]);" }{MPLTEXT 1 202 0 " " }{MPLTEXT 1 202 15 "\n B:=evalm(A); " }{TEXT 205 1 "B" }{TEXT 207 14 " contient une " }{TEXT 205 16 "copie exact de A" }{TEXT 207 13 ", \+ mais B est " }{TEXT 205 10 "modifiable" }{TEXT 207 18 " contrairement \+ \340 A" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 40 "\n B:=Dilat_ligne(1/B [1,1],1,B); " }{TEXT 205 17 "L1 <--1/a(11) *L1" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 25 "\n for k from 2 to t[1] do" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 39 "\n B:=Transvect_ligne(k,-B[k,1],1,B); " }{TEXT 205 19 "Lk <-- Lk- a(k1)*L1" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\n od;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 "\nevalm(B);" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 6 "\nend: " }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 2 "\n " }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 11 "Gauss1(A); " }{TEXT 207 24 "On retrouve comme p r\351vu:" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#-I'm atrixG6$I*protectedGF&I(_syslibG6\"6#7%7&\"\"\"#\"\"$\"\"#F/#\"\"&F/7& \"\"!#!\"\"F/F5#!\"$F/7&F3F5!\"#F7" }{TEXT 206 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {TEXT 205 11 "questi on c)" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 26 "zero_sous_pivot:=proc(i,A)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 14 "\n local t,B,k;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 17 "\n t :=taille(A); " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 28 "\n B:=array(1 ..t[1],1..t[2]);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 15 "\n B:=evalm (A); " }{TEXT 205 1 "B" }{TEXT 207 14 " contient une " }{TEXT 205 16 " copie exact de A" }{TEXT 207 13 ", mais B est " }{TEXT 205 10 "modifia ble" }{TEXT 207 18 " contrairement \340 A" }{TEXT 207 0 "" }{TEXT 207 94 "\n la proc\351dure ne marche que si l'indice i ne d\351passe pas l a taille de la matrice et le pivot " }{TEXT 205 17 "a(ii) est NON NUL" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 74 "\n if is(i>t[2]) or is(i>t[1 ]) or is(B[i,i]=0) then RETURN(IMPOSSIBLE); fi;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 40 "\n B:=Dilat_ligne(1/B[i,i],i,B); " }{TEXT 205 17 "Li <--1/a(ii) *Li" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 30 "\n for k from i+1 to t[1] do " }{TEXT 205 1 "k" }{TEXT 207 14 " parcou rt les " }{TEXT 205 7 "indice " }{TEXT 207 9 "de ligne " }{TEXT 205 19 "en dessous du pivot" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 39 "\n \+ B:=Transvect_ligne(k,-B[k,i],i,B); " }{TEXT 205 19 "Lk <-- Lk- a(ki)* Li" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\n od;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 "\nevalm(B);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 6 "\nend: " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 2 "\n " } {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {TEXT 205 11 "question d)" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 205 "" 1 "" {TEXT 206 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 22 "Gauss_systeme:=proc(A)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 16 "\n local T,i,j,t;" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 16 "\n t:=taille(A); " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 28 "\n T:=array(1..t[1],1..t[2]);" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 15 "\n T:=evalm(A); " }{TEXT 207 1 " " }{TEXT 205 1 "T" }{TEXT 207 14 " contient une " }{TEXT 205 16 "copie exact de A" } {TEXT 207 13 ", mais T est " }{TEXT 205 10 "modifiable" }{TEXT 207 18 " contrairement \340 A" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 35 "\nfor j from 1 to min(t[1],t[2]) do " }{TEXT 207 31 "on parcourt les \351l \351ments de la " }{TEXT 205 21 "diagonale d'indice jj" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 33 "\n i:=CherchePivot_colonne(T,j); " } {TEXT 207 15 "On cherche un " }{TEXT 205 31 "pivot non nul dans la co lonne j" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 42 "\n if i<>0 then T:= Permut_ligne(i,j,T); " }{TEXT 207 35 "Au cas o\371 l'on trouve ce pi vot, on " }{TEXT 205 21 "permute avec la ligne" }{TEXT 207 15 " actuel et on " }{TEXT 205 37 "annule les coefficients sous celui-ci" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 40 "\n T:=zero_sous_ pivot(j,T);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 7 "\n fi; " }{TEXT 205 36 "sinon on passe \340 la colonne suivante" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 6 "\n od; " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 " \nevalm(T);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\nend:" } {MPLTEXT 1 202 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {TEXT 205 11 "question e)" }{MPLTEXT 1 202 0 "" }} }{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 82 "B:=array(1..4,1..5): fo r i from 1 to 4 do for j from 1 to 5 do B[i,j]:=i+j: od:od:" } {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 11 "B =evalm(B);" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/I \"BG6\"-I'matrixG6$I*protectedGF)I(_syslibGF%6#7&7'\"\"#\"\"$\"\"%\"\" &\"\"'7'F/F0F1F2\"\"(7'F0F1F2F4\"\")7'F1F2F4F6\"\"*" }{TEXT 206 0 "" } }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 17 "Gauss_systeme(B);" } {MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#-I'matrixG6$I*pro tectedGF&I(_syslibG6\"6#7&7'\"\"\"#\"\"$\"\"#F/#\"\"&F/F.7'\"\"!F,F/F. \"\"%7'F3F3F3F3F3F5" }{TEXT 206 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 203 108 "Ainsi, puisque les deux derni\350res \351quations sont \+ compatibles, il ne nous reste plus qu'\340 \351tudier le syst\350me" }{TEXT 203 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 33 "x+3*y /2+2*z+5*t/2=3; y+2*z+3*t=4;" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/,*I\"xG6\"\"\"\"I\"yGF&#\"\"$\"\"#I\"zGF&F+I\"tGF&#\" \"&F+F*" }{TEXT 206 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/,(I\"yG6 \"\"\"\"I\"zGF&\"\"#I\"tGF&\"\"$\"\"%" }{TEXT 206 0 "" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 203 52 "o\371 les inconnues z et t jouent le r \364le de param\350tres" }{TEXT 203 0 "" }}}{PARA 207 "" 0 "" {TEXT 208 0 "" }}{PARA 207 "" 0 "" {TEXT 208 0 "" }}{PARA 207 "" 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 }