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{SECT 0 {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 8 "restart;" }
{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 1 "\n" }}}{SECT 1 {PARA 201 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 15 "taille
:=proc(A)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 16 "\n local t,u,n,p;"
 }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 24 "\n   t:=[op(2,evalm(A))];" }
{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 33 "\n   n:=op(2,t[1]);p:=op(2,t[2
]); " }{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 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(IMPOSSIBLE); fi;" }
{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 40 "\n B:=Dilat_ligne(1/B[i,i],i,B
);         " }{TEXT 204 17 "Li <--1/a(ii) *Li" }{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 39 "\n  B:=Transvect_ligne(k,-B[k,i],i,B);  " }{TEXT 
204 19 "Lk <-- Lk- a(ki)*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 204 "> " 
0 "" {TEXT 205 10 "Exercice 4" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 
205 "" 0 "" {TEXT 206 11 "question a)" }{TEXT 203 0 "" }}}{EXCHG 
{PARA 205 "" 0 "" {TEXT 203 148 "On modifie la proc\351dure Gauss_syst
eme en ajoutant la possibilit\351 de permuter les colonnes afin d'obte
nir un pivot non nul sur la diagonale concern\351e" }{TEXT 203 0 "" }}
}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 19 "Gauss_rang:=proc(A)" }
{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 18 "\n local T,i,j,k,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 "T" }{TEXT 204 14 " contient une " }{TEXT 207 16 "copie exact de A" 
}{TEXT 204 13 ", mais T est " }{TEXT 207 10 "modifiable" }{TEXT 204 
18 " contrairement \340 A" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 34 "\n
for j from 1 to min(t[1],t[2]) do" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 
202 32 "\n  i:=CherchePivot_colonne(T,j);" }{MPLTEXT 1 202 0 "" }
{MPLTEXT 1 202 2 "\n " }{MPLTEXT 1 208 9 " k:=j+1; " }{TEXT 207 1 "k" 
}{TEXT 204 19 " est lindice de la " }{TEXT 207 36 "colonne susceptible
 d'\352tre permut\351e " }{TEXT 204 25 "avec la colonne j \351tudi\351
e" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 29 "\n while(i=0 and k<t[2]+1)
 do " }{TEXT 204 121 "tant que l'on ne trouve pas de pivot non nul sou
s le coefficient a(jj) et tant qu'il reste des colonnes avec qui permu
ter" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 34 "\n     T:=Permut_colonne
(j,k,T);   " }{TEXT 204 64 "on permute la colonne \351tudi\351e et ave
c une des colonnes suivantes" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 36 
"\n     i:=CherchePivot_colonne(T,j); " }{TEXT 204 50 "on chercher un \+
pivot dans notre nouvelle colonne j" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 
208 12 "\n     k:=k+1" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 4 "\nod;" 
}{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 41 "\n  if i<>0 then T:=Permut_li
gne(i,j,T);  " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 40 "\n            \+
   T:=zero_sous_pivot(j,T);" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 16 "
\n               " }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 6 "\n  fi;" }
{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 5 "\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 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 
200 "> " 0 "" {MPLTEXT 1 202 54 "B:=array(1..4,1..3,[[0,0,4],[0,0,5],[
0,1,1],[0,1,1]]):" }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" 
{MPLTEXT 1 202 11 "B=evalm(B);" }{MPLTEXT 1 202 0 "" }}{PARA 206 "" 1 
"" {XPPMATH 20 "6#/I\"BG6\"-I'matrixGF%6#7&7%\"\"!F+\"\"%7%F+F+\"\"&7%
F+\"\"\"F0F/" }{TEXT 209 0 "" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 
1 0 14 "Gauss_rang(B);" }{MPLTEXT 1 0 0 "" }}{PARA 206 "" 1 "" 
{XPPMATH 20 "6#-I'matrixG6\"6#7&7%\"\"\"F)\"\"!7%F*F)F*7%F*F*F*F," }
{TEXT 209 0 "" }}}{EXCHG {PARA 205 "" 0 "" {TEXT 206 11 "question b)" 
}{TEXT 203 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 13 "rang:
=proc(A)" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 14 "\n local t,k,B;" }
{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 15 "\n t:=taille(A);" }{MPLTEXT 1 
202 0 "" }{MPLTEXT 1 202 19 "\n B:=Gauss_rang(A);" }{MPLTEXT 1 202 0 "
" }{MPLTEXT 1 202 35 "\n sum(B[k,k],k=1..min(t[1],t[2])); " }{TEXT 
204 107 "le rang est \351gal au nombre (donc la somme) des coefficient
s valant 1 sur la diagonale apr\350s Gaussianisation" }{MPLTEXT 1 202 
0 "" }{MPLTEXT 1 202 5 "\nend:" }{MPLTEXT 1 202 0 "" }{MPLTEXT 1 202 
2 "\n " }{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 
202 8 "rang(B);" }{MPLTEXT 1 202 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 
"6#\"\"#" }{TEXT 209 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 
202 0 "" }}}{PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 207 "" 0 "" 
{TEXT 210 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 207 "" 0 "" 
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