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{TEXT -1 22 "Stage Maple Agr\351gation" }}{PARA 257 "" 0 "" {TEXT -1 
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c1:=(a1+b1)/2;\n   Pa:=subs(x=a1,f); Pb:=subs(x=b1,f); Pc:=subs(x=c1,f
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0 "" {MPLTEXT 1 0 233 "newtongraph := proc(f,b,n)\nlocal a,i,s,t,y :\n
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bs(\{x=a\},f) : t := subs(\{x=a\},diff(f,x)) :\ns := s,y+t*(x - a) :\n
a := a - y/t\nod :\nplot([f,s],x=b,color=[red,blue$n])\nend;" }}{PARA 
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7\")*yKvU-'F[t7$Few$!3vEq$zx()p@(F[t7$Fjw$!3qI4Hyrkl%)F[t7$F_x$!3!Ht@v
ea#o'*F[t7$Fdx$!3c,(*)fV4)*3\"F37$Fix$!3OJC!Q/v<@\"F37$F^y$!3v=V_q([QK
\"F37$Fcy$!3st6')3zH_9F37$Fhy$!3a^HW\\V=n:F37$F]z$!3gA2Adsn*o\"F37$Fbz
$!3-.pq*f=p!=F37$Fgz$!3\")z0rL/\"H$>F3Fidl-%+AXESLABELSG6$Q\"x6\"Q!F\\
hm-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Sch\351ma de Horner" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,,*$)%\"xG\"\"%\"\"\"F(*$)F&\"\"$F(F(*&\"#BF()F&\"\"#F(!\"\"*&F+F(F&
F(F(\"#!*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "convert(Q,ho
rner,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"#!*\"\"\"*&,&\"\"$F%*&
,&\"#B!\"\"*&,&%\"xGF%F%F%F%F/F%F%F%F/F%F%F%F/F%F%" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,,*$)%\"xG\"\"%\"\"\"F(*$)F&\"\"$F(F(*&\"#BF()F&\"\"#F(!\"\"*&F+F(F
&F(F(\"#!*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "horner:=pr
oc(P,b)\nlocal r,d,i;\nd:=degree(P,x);\nr:=coeff(P,x,d);\nfor i from d
-1 to 0 by -1 do\n   r:=r*b+coeff(P,x,i);\n   od;\nRETURN(r);\nend;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'hornerGf*6$%\"PG%\"bG6%%\"rG%\"dG%
\"iG6\"F-C&>8%-%'degreeG6$9$%\"xG>8$-%&coeffG6%F4F5F0?(8&,&F0\"\"\"F>!
\"\"F?\"\"!%%trueG>F7,&*&F7F>9%F>F>-F96%F4F5F<F>-%'RETURNG6#F7F-F-F-" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "horner(Q,-5); subs(x=-5,Q
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "st:
=time():horner(Q,-9*10**99999):time()-st;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"%CD!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "st:=time():subs(\{x=-9*10**99999\},Q):time()-st;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"%&H$!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 114 "Il faut 2d-1 multiplications pour un
e m\351thode na\357ve et d pour la m\351thode de Horner, o\371 d est l
e degr\351 du polyn\364me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 18 "R\350gle de Descartes" }}{SECT 1 {PARA 
3 "" 0 "" {TEXT 256 18 "R\350gle de Descartes" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "R:=x^4-x^3-3*x+90;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"RG,**$)%\"xG\"\"%\"\"\"F**$)F(\"\"$F*!\"\"*&F-F*F(F*F.\"#!*F
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "coeffs(R);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&\"#!*!\"$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 2 "Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"x
G\"\"%\"\"\"F(*$)F&\"\"$F(F(*&\"#BF()F&\"\"#F(!\"\"*&F+F(F&F(F(\"#!*F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "coeffs(Q);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6'\"#!*\"\"$\"\"\"!#BF%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "v
ariation:=proc(P)\nlocal l,L,res:\nres:=[]:\nL:=[seq(coeff(P,x,i),i=0.
.degree(P))]:\nfor l in L do\nif l <> 0 then res:=[op(res),l]:\nfi:\no
d:\nend;\n" }{XPPMATH 20 "6#>%*variationGf*6#%\"PG6%%\"lG%\"LG%$resG6
\"F,C%>8&7\">8%7#-%$seqG6$-%&coeffG6%9$%\"xG%\"iG/F<;\"\"!-%'degreeG6#
F:?&8$F2%%trueG@$0FDF?>F/7$-%#opG6#F/FDF,F,F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*variationGf*6#%\"PG6%%\"lG%\"LG%$resG6\"F,C%>8&7\">8
%7#-%$seqG6$-%&coeffG6%9$%\"xG%\"iG/F<;\"\"!-%'degreeG6#F:?&8$F2%%true
G@$0FDF?>F/7$-%#opG6#F/FDF,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 180 "descartes:=proc(L)\nlocal n:\nn:=nops(L):\nif n = 1 then RETU
RN(0):\nelse\nif sign(L[n-1]*L[n]) = -1 then RETURN(descartes(L[1..n-1
])+1):\nelse RETURN(descartes(L[1..n-1])):\nfi:\nfi:\nend;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%*descartesGf*6#%\"LG6#%\"nG6\"F*C$>8$-%%no
psG6#9$@%/F-\"\"\"-%'RETURNG6#\"\"!@%/-%%signG6#*&&F16#,&F-F4F4!\"\"F4
&F16#F-F4FB-F66#,&-F$6#&F16#;F4FAF4F4F4-F66#FHF*F*F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$
)%\"xG\"\"%\"\"\"F(*$)F&\"\"$F(!\"\"*&F+F(F&F(F,\"#!*F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "variation(R);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&\"#!*!\"$!\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "descartes(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "Q;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,,*$)%\"xG\"\"%\"\"\"F(*$)F&\"\"$F(F(*&\"#BF()F&\"\"#F(
!\"\"*&F+F(F&F(F(\"#!*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"variation(Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"#!*\"\"$!#B\"\"
\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "descartes(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 41 "Application : matrices d\351finies positives" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdeco
mpG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(
augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)ch
arpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'co
ncatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delro
wsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenv
ectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffg
ausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG
%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG
%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%
'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG
%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'
mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG
%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%
'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smith
G%,stackmatrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG
%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%
(vectdimG%'vectorG%*wronskianG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 128 "dp:=proc(M)\nlocal P:\nP:=charpoly(M,x):\nif descartes(variat
ion(P)) = degree(P) then RETURN(true):\nelse RETURN(false):\nfi:\nend;
\n\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dpGf*6#%\"MG6#%\"PG6\"F*C
$>8$-%)charpolyG6$9$%\"xG@%/-%*descartesG6#-%*variationG6#F--%'degreeG
F:-%'RETURNG6#%%trueG-F>6#%&falseGF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "A:=randmatrix(4, 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%'matrixG6#7&7&!#&)!#b!#P!#N7&\"#(*\"#]\"#z\"#c7&\"#\\\"#j\"
#d!#f7&\"#X!\")!#$*\"##*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"M:=A+transpose(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG,&%\"AG\"
\"\"-%'matrixG6#7&7&!#&)\"#(*\"#\\\"#X7&!#b\"#]\"#j!\")7&!#P\"#z\"#d!#
$*7&!#N\"#c!#f\"##*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eva
lm(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&!$q\"\"#U\"#
7\"#57&F)\"$+\"\"$U\"\"#[7&F*F.\"$9\"!$_\"7&F+F/F2\"$%=" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dp(M);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(eig
envalues(M));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"+ch5z=!\"($\"++7E:
KF%$!+(=(HX5F%$!+p,2p<F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"Diag:=diag(1,2,3,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DiagG-%'ma
trixG6#7&7&\"\"\"\"\"!F+F+7&F+\"\"#F+F+7&F+F+\"\"$F+7&F+F+F+\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dp(Diag);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }