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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT 210 12 "Introduction" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 200 413 "Au cours de ce bref article, nous allon
s approximer deux fonctions classiques, respectivement un signal trian
gulaire et un cr\351neau, sous forme de somme de cosinus. On n'explici
tera pas la th\351orie de Fourrier qui appuie les r\351sultats : l'int
\351ret ici est de consid\351rer plusieurs approches pour afficher les
 approximations successives, d'abord sous forme de graphe statique, pu
is sous forme anim\351e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 
"restart:\n" }{MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT 
211 49 "Warning, the name changecoords has been redefined" }}}}{SECT 0
 {PARA 3 "" 0 "" {TEXT 210 30 "D\351finition des coefficients" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 200 87 "On commence par d\351finir dans u
n tableau les coefficients des s\351ries de cosinus : " }{TEXT 203 1 "
a" }{TEXT 200 34 " pour la fonction triangulaire et " }{TEXT 204 1 "b"
 }{TEXT 200 20 " pour le cr\351neau." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "a := array(0..50):\n" }{MPLTEXT 1 0 20 "b := array(0.
.50):\n" }{MPLTEXT 1 0 23 "for i from 0 to 50 do\n" }{MPLTEXT 1 0 39 "
    a[i] := 4 / (((2*i+1)^2)*(Pi^2));\n" }{MPLTEXT 1 0 44 "    b[i] :=
 ((-1)^i) * 4 / (((2*i+1)*Pi));\n" }{MPLTEXT 1 0 3 "od:" }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 210 35 "Approximation avec une proc\351dure" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 200 76 "La fonction que l'on cherche \340
 obtenir pour le triangle est la suivante :" }}{PARA 0 "" 0 "" {TEXT 
200 0 "" }}{PARA 203 "" 0 "" {XPPEDIT 2 0 "f(x) = 1/2+(sum(a[k]*cos(t*
(2*k+1)), k = 0 .. n));" "/-I\"fG6\"6#I\"xGF%,&#\"\"\"\"\"#F*-I$sumG6$
%*protectedGI(_syslibGF%6$*&&I\"aGF%6#I\"kGF%F*-I$cosGF.6#*&I\"tGF%F*,
&*&F+F*F6F*F*F*F*F*F*/F6;\"\"!I\"nGF%F*" }}{PARA 0 "" 0 "" {TEXT 200 
0 "" }}{PARA 0 "" 0 "" {TEXT 200 135 "Une premi\350re id\351e pour la \+
repr\351senter est d'\351crire une proc\351dure qui calcule la valeur \+
en t, puis d'appeler la fonction " }{TEXT 205 4 "plot" }{TEXT 200 54 "
 directement au coeur de la proc\351dure, comme suit :" }{TEXT 212 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "triangle := proc(n,coul
eur,afficher)\n" }{MPLTEXT 1 0 17 "    local i, s;\n" }{MPLTEXT 1 0 
21 "    global a, b, f;\n" }{MPLTEXT 1 0 15 "    s := 1/2;\n" }
{MPLTEXT 1 0 26 "    for i from 0 to n do\n" }{MPLTEXT 1 0 43 "       \+
 s := s + a[i] * cos(t * (2*i+1));\n" }{MPLTEXT 1 0 9 "    od:\n" }
{MPLTEXT 1 0 16 "    f[n] := s;\n" }{MPLTEXT 1 0 31 "    if (evalb(aff
icher)) then\n" }{MPLTEXT 1 0 47 "        plot(s,t=-2*Pi..2*Pi, color=
couleur);\n" }{MPLTEXT 1 0 9 "    fi;\n" }{MPLTEXT 1 0 4 "end:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 212 0 "" }{TEXT 200 95 "L'affichage du r
\351sultat pour la somme des 42 premiers coefficients se r\351sume alo
rs \340 :" }{TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "triangle(42,red,true);" }}{PARA 13 "" 1 "" {TEXT 213 0 "" }
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{TEXT 210 28 "Animation de la construction" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT 200 96 "Cependant, l'expression sous forme de somme peut tout au
ssi bien s'\351crire avec l'instruction " }{TEXT 206 3 "sum" }{TEXT 
200 4 " de " }{TEXT 207 5 "Maple" }{TEXT 200 39 ". On peut donc d\351f
inir une fonction " }{XPPEDIT 18 0 "f_a(t, n);" "-I$f_aG6\"6$I\"tGF$I
\"nGF$" }{TEXT 200 45 " qui calcule la somme des n premiers termes :" 
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200 84 "L'avantage de cette \351criture est qu'elle permet l'utilisati
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iciel. Cette derni\350re s'utilise comme plot, si ce n'est que l'on pr
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45.0 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "
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Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curv
e 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32
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rve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve \+
64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" 
"Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Cur
ve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 8
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F77$Fdt$\"3bs(>tY+!4:F77$Fgt$\"3Qgtw57<1:F77$F\\u$\"3k-,-+#[p[\"F77$Fa
u$\"3;$f,+++++&FV7$Ffu$!3g<4?+?[p[FV7$F[v$!3]]Nn2@rh]FV7$F`v$!3m;x>tY+
!4&FV7$Fcv$!3/;>&eV;r.&FV7$Fhv$!3q5AReF?n\\FV7$F]wF^jel7$F`w$!3'H<#ReF
?n\\FV7$Few$!3Tp=&eV;r.&FV7$Fjw$!3y2x>tY+!4&FV7$F_x$!3wVOn2@rh]FV7$Fdx
$!3#>3,-+#[p[FV7$Fix$\"39'['**********\\FV7$F^y$\"3W#3?+?[p[\"F77$Fcy$
\"3s^tw57<1:F77$FhyFg[fl7$F]z$\"3r'>&eV;r.:F77$FbzF]hel7$FgzFjgelFizFe
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\"2w=r]iQ%[<F\\afl" 1 2 2 1 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1"
 "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8"
 "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Cur
ve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 2
1" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "
Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curv
e 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40
" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "C
urve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve
 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59"
 "Curve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Cu
rve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve \+
72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" 
"Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Cur
ve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 9
1" "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97" "
Curve 98" "Curve 99" "Curve 100" }}{TEXT 213 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
210 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT 200 214 "Bien que la premi
\350re approche soit souvent plus \"instinctive\" pour un programmeur,
 la derni\350re expression est beaucoup plus concise, tout aussi effic
iente, et permet l'utilisation de nouvelles instructions " }{TEXT 209 
5 "Maple" }{TEXT 200 45 " sympathiques tels que les graphes anim\351s.
" }{TEXT 212 0 "" }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }