{VERSION 3 0 "APPLE_PPC_MAC" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 
{CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 
-1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 
1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE 
"Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }}
{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "TD 4 : Alg�bre lin�aire" 
}}{PARA 0 "" 0 "" {TEXT -1 27 "Jean-S�bastien ROY, 1997-98" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 1" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 51 "En pr�vision de ce qui va suivre, chargeons linalg." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "D�finissons A, B et la fonction u \+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A:=X^4-1;B:=X^4-X;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*$%\"XG\"\"%\"\"\"!\"\"F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,&*$%\"XG\"\"%\"\"\"F'!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u:=y->rem(A*y,B,X);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGR6#%\"yG6\"6$%)operatorG%&arrowG
F(-%$remG6%*&%\"AG\"\"\"9$F1%\"BG%\"XGF(F(6\"" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 45 "On prend deux polynomes g�n�riques P1 et P2 :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "P1:=add(a.i*X^i,i=0..3);P2:=
add(b.i*X^i,i=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,*%#a0G
\"\"\"*&%#a1GF'%\"XGF'F'*&%#a2GF'F*\"\"#F'*&%#a3GF'F*\"\"$F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#P2G,*%#b0G\"\"\"*&%#b1GF'%\"XGF'F'*&%#b2G
F'F*\"\"#F'*&%#b3GF'F*\"\"$F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "
Et on v�rifie que u est lin�aire :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "expand(u(l1*P1+l2*P2)-l1*u(P1)-l2*u(P2));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
51 "Calcul direct des coefficients de la matrice de u :" }}{PARA 0 "" 
0 "" {TEXT -1 76 "A la ligne i, colonne j, on trouve le coefficient de
 X^(i-1) dans u(X^(j-1))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"U:=matrix(4,4,(i,j) -> coeff(u(X^(j-1)),X,i-1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"UG-%'MATRIXG6#7&7&!\"\"\"\"!F+F+7&\"\"\"F*F+F-7&F+F
-F*F+7&F+F+F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Son noyau :" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "kernel(U);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<#-%'VECTORG6#7&\"\"!\"\"\"F)F)" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 80 "Que l'on peut visualiser de maniere plus agr�ab
le avec une fonction auxiliaire :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "vect2poly:= proc(v,x) local i; add(v[i]*x^(i-1),i=1..
op(2,op(2,eval(v)))) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "map(vect2poly,kernel(U),X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#,
(%\"XG\"\"\"*$F%\"\"#F&*$F%\"\"$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 39 "Une base de son image (avec colspace) :" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "map(vect2poly,colspace(U),X);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<%,&*$%\"XG\"\"#\"\"\"*$F&\"\"$!\"\",&F&F(F)F+,&F(
F(F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Ses valeur propre et ve
cteurs propres :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenve
cts(U);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7%,&#!\"$\"\"#\"\"\"*&%\"IG
F(\"\"$#F(F'F,F(<#-%'VECTORG6#7&\"\"!F(,&#!\"\"F'F(F)F4,&F4F(F)F,7%,&F
%F(F)F4F(<#-F/6#7&F2F(F6F37%F5F(<#-F/6#7&F5F2F2F(7%F2F(<#-F/6#7&F2F(F(
F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Ou, en plus lisible :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "map(vp -> print(valeur_prop
re=vp[1],multiplicit�=vp[2],base=map(vect2poly,vp[3],X)),[eigenvects(U
)]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%.valeur_propreG!\"\"/%-multi
plicit|isG\"\"\"/%%baseG<#,&F%F(*$%\"XG\"\"$F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%/%.valeur_propreG\"\"!/%-multiplicit|isG\"\"\"/%%baseG<
#,(%\"XGF(*$F-\"\"#F(*$F-\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%
.valeur_propreG,&#!\"$\"\"#\"\"\"*&%\"IGF)\"\"$#F)F(F-/%-multiplicit|i
sGF)/%%baseG<#,(*&,&#!\"\"F(F)F*F6F)%\"XGF)F)*&,&F6F)F*F-F)F8F(F)*$F8F
,F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%.valeur_propreG,&#!\"$\"\"#\"
\"\"*&%\"IGF)\"\"$#F)F(#!\"\"F(/%-multiplicit|isGF)/%%baseG<#,(*&,&F.F
)F*F-F)%\"XGF)F)*&,&F.F)F*F.F)F8F(F)*$F8F,F)" }}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 10 "Exercice 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "restart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new
 definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new defi
nition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "M:=eval
m(matrix([[0,1,3],[3,0,1],[1,3,0]])/4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"MG-%'MATRIXG6#7%7%\"\"!#\"\"\"\"\"%#\"\"$F-7%F.F*F+7%F+F.F*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Apres avoir rentr� la matrice
 M, on regarde num�riquement la valeur aproch�e de M^60" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(evalm(M^60));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%$\"+LLLLL!#5F(F(F'F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "On en d�duit donc que M^n doit converger
, quand n tends vers l'infini, vers une matrice dont tous les �l�ments
 valent 1/3." }}{PARA 0 "" 0 "" {TEXT -1 57 "On va la diagonaliser et \+
calculer une matrice de passage." }}{PARA 0 "" 0 "" {TEXT -1 53 "Calcu
lons les valeurs propres/vecteurs propres de M :" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "E:=[eigenvects(M)];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"EG7%7%\"\"\"F'<#-%'VECTORG6#7%F'F'F'7%,&#!\"\"\"\"#
F'*&%\"IGF'\"\"$#F'F1#F'\"\"%F'<#-F*6#7%F',&F/F'F2F/,&F/F'F2F57%,&F/F'
F2#F0F7F'<#-F*6#7%F'F=F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "On e
xtrait de ce r�sultat les valeur propres avec leur multiplicit� DANS L
E MEME ORDRE que les vecteurs propres calcul�s pr�c�dement, (c'est tr�
s important)." }}{PARA 0 "" 0 "" {TEXT -1 106 "Dans l'expression ci-de
ssous, e[1] est la valeur propre, et e[2] sa multiplicit�. a$b = a,a,.
..,a (b fois)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "V:=map(e->
e[1]$e[2],E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG7%\"\"\",&#!\"
\"\"\"#F&*&%\"IGF&\"\"$#F&F*#F&\"\"%,&F(F&F+#F)F0" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 29 "D'ou une matrice de passage :" }}{PARA 0 "" 0 "" 
{TEXT -1 118 "(op(e[3]) est la sequence des vecteurs propres formant u
ne base du sous espace propre associ� � la valeur propre e[1]." }}
{PARA 0 "" 0 "" {TEXT -1 100 "e est un �l�ment de E, c'est a dire un t
riplet valeur propre/multiplicit�/liste de vecteurs propres)" }}{PARA 
0 "" 0 "" {TEXT -1 64 "'op' permet d'enlever les accolades/crochets : \+
op([1,2,3])=1,2,3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "P:=tra
nspose(matrix(map(e -> op(e[3]),E)));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"PG-%'MATRIXG6#7%7%\"\"\"F*F*7%F*,&#!\"\"\"\"#F**&%\"IGF*\"\"$#F
*F/F-,&F-F*F0F37%F*F4F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "La mat
rice diagonalis�e (avec les valeurs propres dans le BON ORDRE), mise �
 la puissance n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Nn:=dia
g(op(map(e->e^n,V)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NnG-%'MATR
IXG6#7%7%\"\"\"\"\"!F+7%F+),&#!\"\"\"\"#F**&%\"IGF*\"\"$#F*F1#F*\"\"%%
\"nGF+7%F+F+),&F/F*F2#F0F7F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "O
n calcule la limite de cette matrice quand n-> infini." }}{PARA 0 "" 
0 "" {TEXT -1 77 "Noter la pr�sence de l'evalc qui est indispensable p
our que limit fonctionne." }}{PARA 0 "" 0 "" {TEXT -1 71 "evalc mets (
c+I*d)^n sous la forme a(n)+I*b(n) avec a(n) et b(n) r�els." }}{PARA 
0 "" 0 "" {TEXT -1 67 "Noter aussi qu'il faut appliquer limit sur chaq
ue �l�ment avec map." }}{PARA 0 "" 0 "" {TEXT -1 53 "(limit ne fonctio
nne pas directement sur une matrice)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "L:=map(e->limit(evalc(e),n=infinity),Nn);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%'MATRIXG6#7%7%\"\"\"\"\"!F+7%F+F+F+F
," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "D'ou a limite de M^n:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(P &* L &* P^(-1));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#\"\"\"\"\"$F(F(F'F'" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "La fonction jordan permet d'ob
tenir directement la matrice diagonalis�e (J) et la matrice de passage
 (Q) :" }}{PARA 0 "" 0 "" {TEXT -1 44 "(le reste est identique). L'exp
ression est :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "J:=jordan(
M,'Q'):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 3" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "On entre la matrice A :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=matrix([[2,1,1],[1,2,1],[
0,0,3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"
\"#\"\"\"F+7%F+F*F+7%\"\"!F.\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
26 "Une matrice C quelconque :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "C:=array(1..3,1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG
-%&arrayG6%;\"\"\"\"\"$F(7\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "L
a matrice suivante doit etre nulle si C commute avec A :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "N:=evalm(C &* A - A &* C);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'MATRIXG6#7%7%,(&%\"CG6$\"\"\"
\"\"#F.&F,6$F/F.!\"\"&F,6$\"\"$F.F2,(&F,6$F.F.F.&F,6$F/F/F2&F,6$F5F/F2
,,F7F.F+F.&F,6$F.F5F.&F,6$F/F5F2&F,6$F5F5F27%,(F9F.F7F2F3F2,(F0F.F+F2F
;F2,,F0F.F9F.F@F.F>F2FBF27%,&F3F2F;F.,&F3F.F;F2,&F3F.F;F." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 77 "On resoud donc apres avoir converti la ma
trice en ensemble (set) d'equation :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "solve(convert(N,set));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#<+/&%\"CG6$\"\"$\"\"#\"\"!/&F&6$F)F(&F&6$\"\"\"F(/&F&6$F(F(,&&F&
6$F0F0F0&F&6$F0F)F0/&F&6$F)F)F5/&F&6$F)F0F7/&F&6$F(F0F*/F7F7/F5F5/F.F.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "On affecte � la matrice C les
 valeurs trouv�es :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assi
gn(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(C);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%&%\"?G6$\"\"\"F+&F)6$F+\"\"#
&F)6$F+\"\"$7%&%\"CGF-&F4F*&F4F07%\"\"!F8,&F5F+F3F+" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 91 "La ou il y a des ' ? ' la valeur est quelconque
. On verifie qu'une telle matrice convient :" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "map(eval,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'MATRIXG6#7%7%\"\"!F(F(F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "
Soit P un polynome quelconque en A (degr� 2 suffit car A est de dimens
ion 3) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P:=add(a.i*A^i,
i=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,(%#a0G\"\"\"*&%#a1G
F'%\"AGF'F'*&%#a2GF'F*\"\"#F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 
"On va essayer de trouver des valeurs pour a0, a1, a2 telles que P=C, \+
c'est � dire, telles que la matrice suivante est nulle :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(P-C);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'MATRIXG6#7%7%,*%#a1G\"\"#%#a2G\"\"&&%\"CG6$\"\"\"F0!
\"\"%#a0GF0,(F)F0F+\"\"%&F.6$F0F*F1,(F)F0F+\"\"'&F.6$F0\"\"$F17%F3F(F7
7%\"\"!F>,,F)F;F+\"\"*F-F1F5F1F2F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 44 "Meme methode que pr�c�dement pour r�soudre :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "solve(convert(%,set),\{a0,a1,a2\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%#a2G,&&%\"CG6$\"\"\"\"\"##!\"\"F+&
F(6$F*\"\"$#F*F+/%#a1G,&F'F0F.!\"#/%#a0G,(F'#!\"(F+F.#F0F+&F(6$F*F*F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(P);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'MATRIXG6#7%7%&%\"CG6$\"\"\"F+&F)6$F+\"\"#&F)6$F+\"\"
$7%F,F(F/7%\"\"!F4,&F(F+F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "evalm(P-C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%
\"\"!F(F(F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Tout va bien." }
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Exercice 4" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 "" 1 "" 
{TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 30 "Les fonctions interm�diaires :" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 73 "poly2vect:= proc(p,d) local i: [seq(coeff(p,i
ndets(p)[1],i),i=0..d)] end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 78 "vect2poly:= proc(v,x) local i: add(v[i]*x^(i-1),i=1..op(2,op(2,e
val(v)))) end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "La fonction pri
ncipale :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "sompoly:= p -
> vect2poly(linsolve(vandermonde([seq(i,i=0..degree(p)+1)]),[seq(add(s
ubs(indets(p)[1]=j,p),j=0..i),i=0..degree(p)+1)]),indets(p)[1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(sompolyGR6#%\"pG6\"6$%)operatorG%&a
rrowGF(-%*vect2polyG6$-%)linsolveG6$-%,vandermondeG6#7#-%$seqG6$%\"iG/
F9;\"\"!,&-%'degreeG6#9$\"\"\"FBFB7#-F76$-%$addG6$-%%subsG6$/&-%'indet
sGF@6#FB%\"jGFA/FQ;F<F9F:FMF(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
40 "Prennons un exemple avant de d�tailler :" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "sompoly(X^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,(%\"XG#\"\"\"\"\"'*$)F$\"\"#\"\"\"#F&F**$)F$\"\"$F+#F&F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 35 "Soit sous la forme plus classique :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$*(%\"XG\"\"\",&F%F&F&F&F&,&F%\"\"#F&F&F&#F&\"\"'" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "D�taillons :" }}{PARA 0 "" 0 "
" {TEXT -1 84 "On cherche � r�soudre le syst�me suivant (en a(0)..a(n+
1), ou n est le degr� de P) :" }}{PARA 0 "" 0 "" {TEXT -1 65 "a(0)+a(1
)*i+...+a(n+1)*i^(n+1)=sum(P(j),j=0..i) pour i de 0 � n+1" }}{PARA 0 "
" 0 "" {TEXT -1 62 "La matrice de ce syst�me est donc une matrice de v
andermonde :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:=X^3:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "vandermonde([seq(i,i=0..degr
ee(p)+1)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"\"
\"\"!F)F)F)7'F(F(F(F(F(7'F(\"\"#\"\"%\"\")\"#;7'F(\"\"$\"\"*\"#F\"#\")
7'F(F-F/\"#k\"$c#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "L'indetermi
n�e de P est indets(P)[1] (ici X) et l'on doit calculer la valeur de s
um(P(j),j=0..i) pour i de 0 � n+1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "subs(indets(p)[1]=j,p);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*$)%\"jG\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "add(%,j=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "[seq(add(subs(indets(p)[1]=j,p),j=0
..i),i=0..degree(p)+1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!\"
\"\"\"\"*\"#O\"$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "On aurait \+
pu �crire, avec unapply :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "[seq(add(unapply(p,indets(p)[1])(j),j=0..i),i=0..degree(p)+1)];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!\"\"\"\"\"*\"#O\"$+\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "On r�soud alors le syst�me avec li
nsolve, la solution etant les coefficients a(0)..a(n+1)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "linsolve(vandermonde([seq(i,i=0..d
egree(p)+1)]),[seq(add(subs(indets(p)[1]=j,p),j=0..i),i=0..degree(p)+1
)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7'\"\"!F'#\"\"\"\"
\"%#F)\"\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Que l'on convert
it en polynome :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "vect2po
ly(%,indets(p)[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"XG\"\"#
\"\"\"#\"\"\"\"\"%*$)F&\"\"$F(#F*F'*$)F&F+F(F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,$*&)%\"XG\"\"#\"\"\"),&F&\"\"\"F+F+F'F(#F+\"\"%" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4 25 0 0" 0 }{VIEWOPTS 1 1 0 1 
1 1803 }