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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 263 48 "TP MAPLE  9  - ALGORITH
MES CLASSIQUES -  CORRIG\311" }{TEXT 264 1 " " }}}{EXCHG {PARA 259 "" 
0 "" {TEXT 262 61 "L'ALGORITHME D'EUCLIDE AVEC CALCUL DES COEFFICIENTS
 DE BEZOUT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 19 "Version it\351rative :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 182 "bezout:=proc(a,b)\nlocal u,v,q,t;\nu:=[1,0,a];v:=[0,1,b];\nwhil
e v[3]<>0 do  \n   q:=iquo(u[3],v[3]):  \n   (u,v):=(v,u-q*v) od: \npr
intf(`%a = (%a)*%a + (%a)*%a`,u[3],u[1],a,u[2],b)\nend:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "bezout(17,71);bezout(45*7*13,11*7*1
3);" }}{PARA 6 "" 1 "" {TEXT -1 21 "1 = (-25)*17 + (6)*71" }}{PARA 6 "
" 1 "" {TEXT -1 25 "91 = (1)*4095 + (-4)*1001" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 19 "version r\351cursive :" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "bezoutr:=proc(u,v,a,b)
\nlocal q;print(u,v):\nif v[3]=0 then printf(`%a = (%a)*%a + (%a)*%a`,
u[3],u[1],a,u[2],b) else  \n   q:=iquo(u[3],v[3]): \n   bezoutr(v,u-q*
v,a,b); \nfi: \nend:\nbezout:=proc(a,b)\nbezoutr([1,0,a],[0,1,b],a,b)
\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "bezout(17,71);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"\"\"\"!\"#<7%F%F$\"#r" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"!\"\"\"\"#r7%F%F$\"#<" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$7%\"\"\"\"\"!\"#<7%!\"%F$\"\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$7%!\"%\"\"\"\"\"$7%\"#@!\"&\"\"#" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$7%\"#@!\"&\"\"#7%!#D\"\"'\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$7%!#D\"\"'\"\"\"7%\"#r!#<\"\"!" }}{PARA 6 "" 1 "" 
{TEXT -1 21 "1 = (-25)*17 + (6)*71" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 259 "" 0 "" 
{TEXT 257 21 "PRIMALIT\311 D'UN ENTIER" }}{PARA 11 "" 1 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "estpremier:=proc(n)\nl
ocal k ;\nfor k from 2 while n mod k <>0 and k^2<=n do od:\nevalb(k^2>
n) :\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 28 "estpremier(456546545464689);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 
"une l\351g\350re modification donne le plus petit diviseur premier" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "pluspetitdiviseur:=proc(n
)\nlocal k;\nfor k from 2 while n mod k <>0 and k^2<=n do od;\nif k^2>
n then k:=n fi:\nk\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "pluspet
itdiviseur(46546469);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#J" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "liste des premiers : version utili
sant des ensembles" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "listepremiers := proc(n)\n \+
             local P,p,m,k;\n                  P := $ 2..n ; p:=2 : m:
=n :\n                  for k while p^2<= m do p:=P[k]:m:=max(P):\n   \+
                    P := op(\{P\} minus \{seq(k*p,k=p..m/p)\}) od;\n  \+
                [P]\n              end:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 46 "Une deuxi\350me, utilisant des listes et remove :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "listePremiers:=proc(n) \nlo
cal l,k,p:\nl:=[$ 2..n]:p:=2:\nfor k while p^2<=max(op(l)) do\n   p:=l
[k]:\n   l:=remove(x-> x >= p^2 and x mod p = 0, l) od :\nl\nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "listepremiers(400):listePrem
iers(400);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7jo\"\"#\"\"$\"\"&\"\"(
\"#6\"#8\"#<\"#>\"#B\"#H\"#J\"#P\"#T\"#V\"#Z\"#`\"#f\"#h\"#n\"#r\"#t\"
#z\"#$)\"#*)\"#(*\"$,\"\"$.\"\"$2\"\"$4\"\"$8\"\"$F\"\"$J\"\"$P\"\"$R
\"\"$\\\"\"$^\"\"$d\"\"$j\"\"$n\"\"$t\"\"$z\"\"$\"=\"$\">\"$$>\"$(>\"$
*>\"$6#\"$B#\"$F#\"$H#\"$L#\"$R#\"$T#\"$^#\"$d#\"$j#\"$p#\"$r#\"$x#\"$
\"G\"$$G\"$$H\"$2$\"$6$\"$8$\"$<$\"$J$\"$P$\"$Z$\"$\\$\"$`$\"$f$\"$n$
\"$t$\"$z$\"$$Q\"$*Q\"$(R" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "time(listepremiers(10000));time(listePremiers(10000));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"$J\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"%`?!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "D\351composition
 dans une base" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Version r\351cu
rsive" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "dec:=proc(n,b)\nif
 n=0 then NULL else dec(iquo(n,b),b),irem(n,b) fi:\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "dec(2008,10);dec(2008,7);dec(2008,2
);convert(2008,base,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"!F
$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"&F#\"\"'F$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6-\"\"\"F#F#F#F#\"\"!F#F#F$F$F$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7-\"\"!F$F$\"\"\"F%F$F%F%F%F%F%" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 149 "(Maple met les chiffres dans l'ordre des puissanc
es de b croissantes, alors que notre fonction dec les met dans l'ordre
 des puissances d\351croissantes)" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 
260 23 "L'EXPONENTIATION RAPIDE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
13 "Version lente" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "expol \+
:= proc(a,n)\n     local p;\n\011\011\011\011 p := 1;\n     to n do p:
=p*a od ;\n\011\011\011\011 p\nend :\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 51 "Utilisation de l'algorithme d'exponentiation rapide" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "expor := proc(a,n)\n     lo
cal p,r,b,m;\n\011\011\011\011 b := a; m := n; p:= 1;\n     while 0 < \+
m do r := irem(m,2);\n                    if r <> 0 then p := p*b fi; \+
\n                    m := iquo(m,2); b := b^2 od ;\n\011\011\011\011 \+
p\nend :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Puissance rapide, ver
sion r\351cursive :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "exp
orrecu:=proc(n,p)\n if p=0 then 1\n elif is(p,even) then exporrecu(n,p
/2)^2\n                 else exporrecu(n,p-1)*n \nfi\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "expomaple:=proc(a,n)\na^n\nend:" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "time(expol(2,10000));time(expor(2,10000));time(exporr
ecu(2,10000));time(expomaple(2,10000));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"$p%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#I!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$8$!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 1 0 75 "Remarque : plus rapide en temps, demande en g\351n
\351ral plus long en \351criture !" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Exponentiation \+
rapide modulo un entier" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 
"expormodulo := proc(a,n,b)\n     local p,r,c,m;\n\011\011\011\011 c :
= a mod b; m := n ; p := 1;\n     while 0 < m do\n       r := irem(m,2
);\n       if r <> 0 then p := p*c mod b fi;\n\011\011\011\011\011  m \+
:= iquo(m,2);c := c^2 mod b\n     od ;\n\011\011\011\011 p\nend :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "expormodulo(2,20052005,101);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#K" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 30 "LA DIVISION EN S
YSTEME DECIMAL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 17 "Version it\351rative" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 111 "f:=proc(a,b,k)\nlocal l,aa,q:\nl:=NULL:\naa:=a:\nto \+
k do\n q:=iquo(aa,b):\n l:=l,q:\n aa:=10*irem(aa,b):\nod:\n[l]\nend:\n
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f(2008,7,10);evalf(2008
/7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"$'G\"\")\"\"&\"\"(\"\"\"\"
\"%\"\"#F%F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H9doG!\"(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Version r\351cursive :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "f:=proc(a,b,k)\nif k=0 then NULL el
se iquo(a,b),f(10*irem(a,b),b,k-1) fi end:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "f(2010,7,10);evalf(2010/7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6,\"$(G\"\"\"\"\"%\"\"#\"\")\"\"&\"\"(F$F%F&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#$\"+r&G9(G!\"(" }}}{EXCHG {PARA 259 "" 0 "" 
{TEXT 258 50 "L'EXTRACTION DE RACINES CARR\311ES EN SYST\310ME D\311CI
MAL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 154 "f:=proc(n,k)\nlocal l,b,r,m:\nl:=NULL:r:=0:m:=n:\nto
 k do\n\011for b while (20*r+b)*b<=m do od: b:=b-1;\n\011l:=l,b;\n\011
m:=100*(m-(20*r+b)*b);\n\011r:=10*r+b od:\n[l]\nend:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f(2010,10);evalf(sqrt(2010));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"#W\"\")\"\"$F&\"\"!\"\"#F&\"\"&\"
\"%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aBI$[%!\")" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "Version r\351cursive :" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 129 "f:=proc(n,r,k)\nlocal b:\nif k=0 then NU
LL else for b while (20*r+b)*b<=n do od: b:=b-1;\nb,f(100*(n-(20*r+b)*
b),10*r+b,k-1) fi end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f
(2,0,10);evalf(sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"
%F#F$\"\"#F#\"\"$\"\"&\"\"'F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i
N@99!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 34 "DIVISION EUCLIDIENNE DES POLYN\326MES" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=3*X^4+X+1:\nB:=2*X^2+1:R
:=A:Q:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "while degree(R) >= de
gree(B)  do q:=degree(R)-degree(B):a:=lcoeff(R)/lcoeff(B):Q1:=a*X^q:\n
R:=expand(R-Q1*B) :Q:=Q+Q1 od:\nprintf(`%a = (%a)(%a) + %a\\n`,A,B,Q,R
):\n" }}{PARA 6 "" 1 "" {TEXT -1 42 "3*X^4+X+1 = (2*X^2+1)(3/2*X^2-3/4
) + X+7/4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 20 "ALGORITHME DE HORNER" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "version it\351rative" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 79 "horner:=proc(a,x,n)\nlocal y,k;\ny:=a[n];\nf
or k to n do y:=x*y+a[n-k] od;\ny\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 17 "version r\351cursive" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "hornerrecu:=proc(a,x,n,q)\nif n=0 then a[q] else a[q]
+x*hornerrecu(a,x,n-1,q+1) fi:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "horner(a,x,5);hornerrecu(a,x,5,0);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&&%\"aG6#\"\"!\"\"\"*&%\"xGF(,&&F%6#F(F(*&F*F(,&&F%6
#\"\"#F(*&F*F(,&&F%6#\"\"$F(*&F*F(,&&F%6#\"\"%F(*&F*F(&F%6#\"\"&F(F(F(
F(F(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&&%\"aG6#\"\"!\"\"
\"*&%\"xGF(,&&F%6#F(F(*&F*F(,&&F%6#\"\"#F(*&F*F(,&&F%6#\"\"$F(*&F*F(,&
&F%6#\"\"%F(*&F*F(&F%6#\"\"&F(F(F(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Par le calcul direct :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "simple:=proc(a,x,n)\nlocal y,k;\ny:
=a[n];\nfor k to n do y:=y+a[k]*x^k od;\ny\nend:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 35 "for k from 0 to 1000 do a[k]:=k od:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 84 "time(horner(a,4.55,1000));time(hornerrecu(a,4.55,1000,0));time(s
imple(a,4.55,1000));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#;!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#:!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"$0%!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "util
isation d'autres fonctions maple :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "P:=add(a[k]*X^k,k=0..1000):\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "time(subs(X=4.55,P));time(rem(P,X-4.55,X));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#J!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"&?f$!\"$" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Q:=convert(P,horner):time(
subs(X=4.55,Q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#:!\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "TRI" }}{PARA 0 "" 0 "" {TEXT -1 33 
"une premi\350re version par \351changes" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "echange:=proc(L,i,j)\nlocal x,M:\nM:=L:\nx:=M[j]; M[j
]:=M[i]; M[i]:=x:\nM\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
141 "tri1:= proc(L)\nlocal M,n,i,j:\nM:=L:n:=nops(L):\nfor i to n-1 do
\nfor j from i+1 to n do if M[j] < M[i] then M:=echange(M,i,j) fi od\n
od:\nM;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Une deuxi\350me version par insert
ion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 118 "inser:=proc(L,x)\nlocal k:\nfor k to nops(L) while x
>= L[k]  do od : \n[seq(L[i],i=1..k-1),x,seq(L[i],i=k..nops(L))]\nend:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "inser([1,2,8],6);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"\"#\"\"'\"\")" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "tri2:=proc(L)\nlocal k,M:\nM:=[L[1]
]:\nfor k from 2 to nops(L) do M:=inser(M,L[k]) od\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 93 "Une troisi\350me version utlisant la fonction min de maple (id
\351e de Nicolas Polidano, PCSI 2009)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 99 "rangdumin:=proc(L)\nlocal k,x:\nx:=min(op(L)):\nfor k
 while L[k]>x do od:\nk\nend:\nrangdumin([1,5,2,0]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "
tri3:=proc(L)\nif nops(L)=1 then L else [min(op(L)),op(tri3(subsop(ran
gdumin(L)=NULL,L)))] fi end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Une quatri\350me version
 par fusion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 " fusion:=pr
oc(L,M)\nlocal n1,n2,i,j,N;\ni:=1:j:=1;\nn1:=nops(L):n2:=nops(M);\nN:=
NULL:\nwhile i<=n1 and j<=n2 do\nif L[i]>M[j] then N:=N,M[j]:j:=j+1\ne
lse N:=N,L[i]: i:=i+1 fi\nod:\n[N,seq(M[k],k=j..n2),seq(L[k],k=i..n1)]
;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "fusion([1,5,6],[
2,4,5]);#fusionne deux listes croissantes en une liste croissante." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"\"#\"\"%\"\"&F'\"\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "tri4:=proc(L)\nlocal n,m:\n
n:=nops(L):m:=floor(n/2):if n=1 then L\nelse fusion(tri3([seq(L[i],i=1
..m)]),tri3([seq(L[i],i=m+1..n)])) fi\nend:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 66 "L:= [seq(rand(0..100)(),i=1..50)];tri1(L);tri2(L);
tri3(L);tri4(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7T\"#:\"#;\"
#**\"\"!\"#x\"#i\"#L\"#d\"#$*\"#]\"#W\"#\\F1F,\"#w\"#'*\"\"&\"#zF&\"#V
\"#a\"\"'\"#6\"\"(\"#K\"#_\"#O\"#(*\"#s\"#*)F)F'F*\"#lF/\"#%*\"$+\"FA
\"#GF7F8\"#P\"#r\"#oFA\"#7\"#hF?FI\"#)*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7T\"\"!F$\"\"&\"\"'F&\"\"(\"#6\"#7\"#:F*\"#;F+\"#G\"#K\"#LF.\"#O
\"#P\"#V\"#W\"#\\F3\"#]F4\"#_\"#aF6\"#d\"#hF8\"#i\"#lF:F:\"#o\"#r\"#sF
=\"#w\"#xF?\"#z\"#*)\"#$*\"#%*\"#'*\"#(*\"#)*\"#**\"$+\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7T\"\"!F$\"\"&\"\"'F&\"\"(\"#6\"#7\"#:F*\"#;F+\"
#G\"#K\"#LF.\"#O\"#P\"#V\"#W\"#\\F3\"#]F4\"#_\"#aF6\"#d\"#hF8\"#i\"#lF
:F:\"#o\"#r\"#sF=\"#w\"#xF?\"#z\"#*)\"#$*\"#%*\"#'*\"#(*\"#)*\"#**\"$+
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7T\"\"!F$\"\"&\"\"'F&\"\"(\"#6\"
#7\"#:F*\"#;F+\"#G\"#K\"#LF.\"#O\"#P\"#V\"#W\"#\\F3\"#]F4\"#_\"#aF6\"#
d\"#hF8\"#i\"#lF:F:\"#o\"#r\"#sF=\"#w\"#xF?\"#z\"#*)\"#$*\"#%*\"#'*\"#
(*\"#)*\"#**\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7T\"\"!F$\"\"&\"
\"'F&\"\"(\"#6\"#7\"#:F*\"#;F+\"#G\"#K\"#LF.\"#O\"#P\"#V\"#W\"#\\F3\"#
]F4\"#_\"#aF6\"#d\"#hF8\"#i\"#lF:F:\"#o\"#r\"#sF=\"#w\"#xF?\"#z\"#*)\"
#$*\"#%*\"#'*\"#(*\"#)*\"#**\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "L:= [seq(3^17 mod 51,i=1..1000)]:time(tri1(L));time(
tri2(L));time(tri3(L));time(tri4(L));time(sort(L));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"%/7!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%'H
\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$]#!\"$" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"#$*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!
F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "CONCLUSION :  tri de maple
 > tri par fusion > tri par minimum > tri par insertion > tri par \351
changes." }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
259 "" 0 "" {TEXT 261 13 "LA DICHOTOMIE" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "Digits:=50:\nf:
=x->x^2-2:\na:=0.:b:=2.:d:=2*(b-a):\nif f(a)*f(b)<=0 then \n   while (
b-a)<d do\n      d:=b-a;\n      milieu:=(a+b)/2;\n      if f(milieu)>0
 then b:=milieu else a:=milieu fi\n   od \nfi;\n   printf(`solution ap
proch\351e:%a`,(a+b)/2);" }}{PARA 6 "" 1 "" {TEXT -1 70 "solution appr
och\351e:1.4142135623730950488016887242096980785696718753770" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(sqrt(2));;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"SpPv=np&y!)p4Us)o,)[]4tBc8UT\"!#\\" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Une version plus sophistiqu\351e :
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 378 "dichotomie:=proc(F,a,b
,p)\n  local g,d,k:\n  Digits:=p+floor(max(abs(a),abs(b)));\n  g:=eval
f(a): d:=evalf(b): k:=(g+d)/2:\n  while abs(evalf(subs(x=k,F)))>=10^(-
p) do\n  k:=(g+d)/2:\n  if evalf(subs(x=k,F))*evalf(subs(x=d,F))>0\n  \+
then d:=k: else g:=k:\n  fi: od:Digits:=p+floor(k);\n  printf(`Une sol
ution de l'\351quation %a=0 sur [%a;%a] est x=%a \340 10^-%a pr\350s.`
,F,a,b,evalf(k),p):\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "dichotomie(x^2-2,0,5,10);\n" }}{PARA 6 "" 1 "" {TEXT -1 78 "Une so
lution de l'\351quation x^2-2=0 sur [0;5] est x=1.4142135624 \340 10^-
10 pr\350s." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "ALGORITHME GLOUTON
 POUR LES FRACTIONS EGYPTIENNES" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 80 "glouton:=proc(x) local n;\nif x>0 then n:=ceil(1/x) ; 1/n,glou
ton(x-1/n) fi\nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "x:=
4/97:x=somme(glouton(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\"%\"
#(*-%&sommeG6&#\"\"\"\"#D#F+\"$4)#F+\"'84)*#F+\".DikzV#>" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "algorithme si l'on autorise les sommes ou
 diff\351rences de fractions \351gyptiennes :" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 84 "glouton2:=proc(x) local n;\nif x<>0 then n:=roun
d(1/x) ; 1/n,glouton2(x-1/n) fi\nend: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "x:=4/97:x=somme(glouton2(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/#\"\"%\"#(*-%&sommeG6$#\"\"\"\"#C#!\"\"\"%GB" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "48 0 0" 45 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }