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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 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 14 "\n B:=evalm(A);" }{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(I MPOSSIBLE); fi;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 39 "\n B:=Dilat_ ligne(1/B[i,i],i,B); " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 27 "\n for k from i+1 to t[1] do" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 38 "\n B:=Transvect_ligne(k,-B[k,i],i,B); " }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 5 "\n od;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 10 "\ne valm(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 5" }{MPLTEXT 1 202 0 "" }}} {EXCHG {PARA 200 "> " 0 "" {TEXT 205 11 "question a)" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 25 "zero_sur_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 16 "\n B:=evalm(A); " }{TEXT 205 1 "B" }{TEXT 206 14 " contient une " }{TEXT 205 16 "copie exact de A" }{TEXT 206 13 ", mais B est " }{TEXT 205 10 "modifiable" }{TEXT 206 18 " contrairement \340 A" }{TEXT 206 0 "" }{TEXT 206 93 "\nla proc\35 1dure ne marche que si l'indice i ne d\351passe pas la taille de la ma trice et le pivot " }{TEXT 205 24 "a(ii) est DIFFERENT DE 1" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 75 "\n if is(i>t[2]) or is(i>t[1]) or is(B[i,i]<>1) then RETURN(IMPOSSIBLE); fi;" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 26 "\n for k from 1 to i-1 do " }{TEXT 205 1 "k" } {TEXT 206 14 " parcourt les " }{TEXT 205 7 "indice " }{TEXT 206 9 "de \+ ligne " }{TEXT 205 18 "au dessus 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 "" }}{PARA 200 "> " 0 "" {TEXT 205 11 "question b )" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 29 "Gauss_systeme_cramer:=proc(A)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 20 "\n local T,i,j,t,sol;" }{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 205 1 "T" }{TEXT 206 14 " contient une " }{TEXT 205 16 "copie exact de A" }{TEXT 206 13 ", mais T est " }{TEXT 205 10 "modifiable" }{TEXT 206 18 " contrairement \340 A" }{MPLTEXT 1 202 0 " " }{MPLTEXT 1 202 48 "\n if (t[1]<>t[2]-1) then RETURN(IMPOSSIBLE) fi; " }{TEXT 206 15 "Cette proc\351dure" }{TEXT 205 18 " ne fonctionne pa s" }{TEXT 206 7 " si le " }{TEXT 205 48 "systeme n'a pas autant de lig nes que d'inconnues" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 25 "\nfor j \+ from 1 to t[1] do " }{TEXT 206 93 "On annule les coefficients sous la \+ diagonale et on transforme les coefficients diagonaux en 1" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 32 "\n i:=CherchePivot_colonne(T,j);" } {MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 40 "\n if i<>0 then T:=Permut_lig ne(i,j,T); " }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 47 "\n \+ T:=zero_sous_pivot(j,T); " }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 6 "\n fi;" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 5 "\n od;" } {MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 63 "\n if (sum(T[k,k],k=1..t[1])<> t[1]) then RETURN(IMPOSSIBLE) fi; " }{TEXT 206 15 "Cette proc\351dure" }{TEXT 205 18 " ne fonctionne pas" }{TEXT 206 7 " si le " }{TEXT 205 27 "systeme n'est pas de CRAMER" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 2 "\n " }{MPLTEXT 1 208 10 "sol:=NULL;" }{MPLTEXT 1 202 0 "" } {MPLTEXT 1 202 2 "\n " }{MPLTEXT 1 209 32 "for j from t[1] to 1 by -1 do " }{MPLTEXT 1 207 1 " " }{TEXT 206 124 "On annule les coefficient s au dessus de la diagonale en commencant par la derni\350re colonne d e la matrice associ\351e au syst\350me" }{MPLTEXT 1 209 0 "" } {MPLTEXT 1 209 28 "\n T:=zero_sur_pivot(j,T);" }{MPLTEXT 1 207 5 " \+ " }{MPLTEXT 1 209 0 "" }{MPLTEXT 1 209 5 "\n " }{MPLTEXT 1 208 20 "sol:=T[j,t[2]],sol; " }{TEXT 206 130 "sol contient la s\351quence \+ des coefficients de la derni\350re colonne de B en commencant par le d ernier coefficient yn du second membre" }{MPLTEXT 1 209 0 "" } {MPLTEXT 1 209 5 "\n od;" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 2 "\n " }{MPLTEXT 1 210 6 "[sol];" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\n end:" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}{PARA 200 "> " 0 "" {TEXT 205 11 "question c)" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 87 "A:=array(1.. 4,1..4): for i from 1 to 4 do for j from 1 to 4 do A[i,j]:=min(i,j): o d: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&\"\"\"F+F+F+7&F+\"\"#F-F-7&F +F-\"\"$F/7&F+F-F/\"\"%" }{TEXT 211 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 87 "B:=array(1..4,1..5): for i from 1 to 4 do for j fr om 1 to 5 do B[i,j]:=min(i,j): od:od:" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 44 "B[1,5]:=y1:B[2,5]:=y2:B[3,5]:=y 3:B[4,5]:=y4:" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 11 "B=evalm(B);" }{MPLTEXT 1 202 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#/I\"BG6\"-I'matrixGF%6#7&7'\"\"\"F+F+F+I#y1GF%7'F+\" \"#F.F.I#y2GF%7'F+F.\"\"$F1I#y3GF%7'F+F.F1\"\"%I#y4GF%" }{TEXT 211 0 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 0 24 "Gauss_systeme_cramer( B);" }{MPLTEXT 1 0 0 "" }}{PARA 205 "" 1 "" {XPPMATH 20 "6#7&,&I#y1G6 \"\"\"#I#y2GF&!\"\",(F(F'F%F)I#y3GF&F),(F+F'F(F)I#y4GF&F),&F-\"\"\"F+F )" }{TEXT 211 0 "" }}}{PARA 206 "" 0 "" {TEXT 212 0 "" }}{PARA 206 "" 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 }