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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning
, the protected names norm and trace have been redefined and unprotect
ed\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Le but de ce TP est de p
rogrammer l'algorithme du pivot de Gauss pour \351chelonner une famill
e u_1,...,u_p de vecteurs de R^n." }}{PARA 0 "" 0 "" {TEXT -1 127 "On \+
codera les vecteurs par des listes ; ainsi, une famille de vecteurs se
ra une liste de listes, comme dans l'exemple suivant :" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "u[1]:=
[0,2,3];u[2]:=[0,5,6];u[3]:=[7,8,9];u[4]:=[10,11,12];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"\"7%\"\"!\"\"#\"\"$" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%\"uG6#\"\"#7%\"\"!\"\"&\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"uG6#\"\"$7%\"\"(\"\")\"\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"uG6#\"\"%7%\"#5\"#6\"#7" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 31 "famille:=[u[1],u[2],u[3],u[4]];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%(familleG7&7%\"\"!\"\"#\"\"$7%F'\"\"&\"\"'7%\"\"(\"
\")\"\"*7%\"#5\"#6\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 50 "On manipule cette famille de la mani\350r
e suivante :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "famille[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7
%\"\"!\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "famille
[2][3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Choi
x d'un pivot non nul" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 241 "Ecrire une proc\351dure qui prend en entr\351e
 une famille de vecteurs, et qui donne en sortie, l'indice du premier \+
vecteur dont la premi\350re coordonn\351e est non nulle (la proc\351du
re donnera 0 si tous les vecteurs ont leur premi\350re coordonn\351e n
ulle)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Permutation des colonnes" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "O
n suppose qu'il existe au moins un vecteur dont la premi\350re composa
nte est non nulle. La deuxi\350me \351tape de l'algorithme est de le p
lacer en t\352te de liste ; cette composante est appel\351e premier pi
vot." }}{PARA 0 "" 0 "" {TEXT -1 46 "La proc\351dure suivante r\351pon
d \340 cette attente.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "permute:=proc(L)\nlocal
 i,res1,res2:\ni:=indice(L):\nconvert(swaprow(L,1,i),listlist);\nend;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(permuteGf*6#%\"LG6%%\"iG%%res1G
%%res2G6\"F,C$>8$-%'indiceG6#9$-%(convertG6$-%(swaprowG6%F3\"\"\"F/%)l
istlistGF,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "permute(f
amille);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7%\"\"(\"\")\"\"*7%\"\"!
\"\"&\"\"'7%F)\"\"#\"\"$7%\"#5\"#6\"#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "L'algorithme du pivot pour la pre
mi\350re ligne :" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 207 "On suppose encore qu'il existe au moins un vecteur \+
dont la premi\350re composante est non nulle. On effectue maintenant l
'algorithme du pivot afin d'\351liminer les premi\350res composantes d
es vecteurs u_2,...,u_p :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "L'algorithme g\351n\351ral" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Q
uestion : programmer l'algorithme g\351n\351ral du pivot de Gauss. On \+
pourra utiliser pour cela les proc\351dures suivantes." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "collezero:=proc(L)\n[seq([0,
op(j)],j=L)];\nend;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*collezeroGf*
6#%\"LG6\"F(F(7#-%$seqG6$7$\"\"!-%#opG6#%\"jG/F29$F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "famille;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&7%\"\"!\"\"#\"\"$7%F%\"\"&\"\"'7%\"\"(\"\")\"\"*7%\"#
5\"#6\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "collezero(fami
lle);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&7&\"\"!F%\"\"#\"\"$7&F%F%\"
\"&\"\"'7&F%\"\"(\"\")\"\"*7&F%\"#5\"#6\"#7" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "coll
epivot:=proc(v,L)\nlocal i,j,k,l,m:\n[v,op(L)];\nend;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%+collepivotGf*6$%\"vG%\"LG6'%\"iG%\"jG%\"kG%\"lG
%\"mG6\"F/7$9$-%#opG6#9%F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "collepivot([alpha,beta,gamma,delta],famille);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7'7&%&alphaG%%betaG%&gammaG%&deltaG7%\"\"!\"\"#\"
\"$7%F*\"\"&\"\"'7%\"\"(\"\")\"\"*7%\"#5\"#6\"#7" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
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