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Résumé&du&cours&d’Analyse&Numérique&
1) Convergence+
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:!Quadratique>!)-!?.?-1!/1!')%F/1!'1!B*#(3$3)-!C!(D.4%1!3A#.A3)-E!
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:!1A(E!
2) Résolution+d’une+équation+f(x)=0+
.G H$A3&1*!?*.BD34%1&1-A!/.!*.(3-1E!I*.(1*!/1!?*.BD1!'1!/.!J)-(A3)-!1A!K))&1*!$%*!
/1$!*.(3-1$E!
FG Méthode&de&la&dichotomie>!L$%3A1!'+3-A1*M.//1$!1-(.'*.-A!/.!*.(3-1N!
3G
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f(an). f(w)<0[an+1,bn+1]=[an,w]
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(G Méthode&de&la&sécante>!L$%3A1!'1!M./1%*$!A1-'.-A!M1*$!*.(3-1N!
3G S3T1*!T8!1A!T9!
33G 6./(%/1*!!
xn+2=xn+1f(xn+1)xn+1xn
f(xn+1)f(xn)
!O-U8P9P7PEEEG!
333G ,M.-A.?1$>!:!Q)%*!%-1!J)-(A3)-!C!%-!$1%/!K#*)!B1%!'+D0B)ADR$1$!'1!'#B.*AE!
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3MG 5-()-M#-31-A$>!:!<1-A1!()-M1*?1-(1E!
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!!!!!!!!!!:!"3$4%1!'+%-1!'3M3$3)-!B.*!K#*)E!!
'G Méthode&du&point&fixe/d’itération>!
3G 6D)3$3*!T9!
33G 6./(%/1*!
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333G 6D)3$3*!?OTG!A1/!4%1!TU?OTGE!
3MG ,M.-A.?1$>!:!S.(3/1!C!&1AA*1!1-!)1%M*1P!?OTG!4%3!()-M1*?1!*.B3'1&1-A!-+1$A!
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!!OM#*3J31*!?*.BD34%1&1-A!$3!?*.BD1!
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!
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5"657! 7899:7897! ;<,26!<)=(! !
!
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1G Méthode&de&Newton(>Raphson)/de&la&tangente>!O*1&B/.(1*!/.!J)-(A3)-!B.*!$.!
A.-?1-A1>!.BB*)T3&.A3)-E!X#*)!'1!/.!A.-?1-A1!1$A!B*)(D1!'1!(1/%3!'1!/.!J)-(A3)-!$3!
/.!A.-?1-A1!1$A!A*R$!$1&F/.F/1!C!/.!J)-(A3)-G!L?#-R*1!%-1!$%3A1!'1!A.-?1-A1$!.%!
?*.BD1!'1!SOTG!1A!%-1!$%3A1!'1!M./1%*$!T-N!
3G 6D)3$3*!T9!B*)(D1!'%!K#*)!(D1*(D#E!
33G 6./(%/1*>!
xn+1=xnF(xn)
F'(xn)
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Y%!.BB*)T3&.A3)->!
xn+1=xnhF(xn)
F(xn+h)F(xn)
!O-U9P7PEEE!1A!D!A*R$!B1A3AGE!
333G ,M.-A.?1$>!:!6)-M1*?1!4%.'*.A34%1&1-A!$3!*.(3-1!$3&B/1E!
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3MG 5-()-M#-31-A$>!:!6)-M1*?1!/3-#.3*1&1-A!$3!*.(3-1!&%/A3B/1E!
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O'1!/+)*'*1!'1!98:]P!.M1(!]!4%1!/+)-!M1%AGE!
3) Résolution+de+systèmes+algébriques+linéaires+(problème+aux+CL,+etc.)+:+méthodes+
directes+
aij xj=bi
j=1
n
!O3U9P7PEEEP-G!)%!,TUF!.M1(!,>!&.A*3(1!(.**#1!'1$!.3^!O-T-GP!T!1A!F>!M1(A1%*$!
()/)--1$!T^!1A!F^!O-T9GE!_-`!#4%.A3)-$!C!_-`!3-()--%1$E!
.G Méthode&du&déterminent>!
Ax =bx=A1b=(det A)1(A*)tb
!OV.A/.F>!TU,aFG!
FG Z0$AR&1!C!&.A*3(1!A*3.-?%/.3*1!3-J#*31%*1!b!$%B#*31%*1!
xk=
bkakj xj
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k1
akk
!O$3!.]]c8P!'1A,c8G!L3-J#*31%*1>!]U9P7PdPEEE-!e!$%B#*31%*1>!]U-P-:
9P-:7PEEEP9N!
c) Méthode&de&Gauss>&
Of!Z3!B3M)A!$+.--%/1>!B1*&%A1*!/.!/3?-1!'1!(1!B3M)A!.M1(!%-1!/3?-1!'+%-!3-'3(1!
$%B#*31%*G!
3G g#J3-3*!/.!&.A*3(1!,UO.3^G!'%!$0$AR&1!C!*#$)%'*1P!/1!M1(A1%*!FULEEEN!O9T-GE!
33G Q)%*!]U9PEEEP-:9>!
m=Ai,k
Ak,k
bi=bim.bk
!O3U]\9PEEEP-G!P!
Ai,j=Ai,jm.Ak,j
!O^U]PhP-GE!
333G
xn=bn
An,n
P!ZU8P!B)%*!3U-:9PhP9!>!!
S=S+(Ai,j.xj)
!O^U3\9PhP-GP!
xi=(biS)
Ai,i
!
d) Méthode&de&Newton>Raphson>&
3G HM./%1*!?*.BD34%1&1-A!OA*.(1*!J!1A!?G!/+3-A1*$1(A3)-!'1$!()%*F1$!O$)/%A3)-!'%!
$0$AR&1GE!
33G 6D)3$3*!%-!()%B/1!OT9P09G!B*)(D1!'+%-1!$)/%A3)-!'%!$0$AR&1E!
333G 6./(%/1*!
xn+1=xn+Δxn
!O-U9P7PdPhG!
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5"657! 7899:7897! ;<,26!<)=(! !
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xn=xn
yn
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J(xn)=
f
x(xn)
f
y(xn)
g
x(xn)
g
y(xn)
P!
F(xn)=f(xn)
g(xn)
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4) Systèmes+algébriques:+méthodes+indirectes+
.G Méthode&Jacobi!O&#AD)'1!3A#*.A3M1G>!
3G g#J3-3*!2P!/1!-)&F*1!'+3-A1*M.//1$!O->!-)&F*1!'+#4%.A3)-!U!2:9GP!kP!/1!-)&F*1!
'+3A#*.A3)-$E!
33G 6)-$A*%3*1!,UO.3^G!&.A*3(1!'%!$0$AR&1!,TUF!C!*#$)%'*1!O'3.)-./1!)%!
A*3'3.?)-./1!B.*!1T1&B/1GP!FULEEEN!M1(A1%*!()/)--1!'1$!A1*&1$!3-'#B1-'.-A$E!
333G Z)3A!/1!$0$AR&1!#4%3M./1-A>!TUST\(E!
3MG g#()&B)$1*!,!1-!,U<\g\l!Og>!'3.?)-./1!B*3-(3B./1P!l>!A*3.-?%/.3*1!
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M3G 5A#*.A3)->!
x0=xn,xn=F.x0+c
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5) Systèmes+mal+conditionnés+et+incompatibles+
:!EEE!
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6) Interpolation+et+lissage+
.G 5-A1*B)/.A3)-!B)/0-)&3./1!'1!m.-'1*&)-'1>!
3G I*)%M1*!%-!B)/0-n&1!B!'1!'1?*#!
n0
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3MG Q.$!()-$13//#1P!(.*!1//1!B)$1!'1$!B*)F/R&1$!-%&#*34%1$E!
b) Polyfit&sur&Matlab&:&p=polyfit(x,y,n)&
3G BUB)/0J3AOTP0P-G!>!')--1!'.-$!B!/1$!()1JJ3(31-A$!'+%-!B)/0-n&1!P(x)!'1!'1?*#!-!
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(G Interpolation&linéaire>!
3G g1%T!B)3-A$!$%((1$$3J$!$)-A!*1/3#$!B.*!%-!$1?&1-A!'1!'*)3A1E!
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3G S.3*1!%-1!F)%(/1!4%3!.M.-(1!B.*!B.$!'1!7!Ofor-i=a:2:bG>!Y-!B*1-'!/1$!.F$(3$$1$!
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1G Splines&cubiques>!
3G 6*#1*!'1$!.F$(3$$1$!O4%1/4%1$!B)3-A$G>!x=a:1:bE!
ii) HM./%1*!y=f(x)E-
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7) +Intégration+numérique+de+
f(x)dx
a
b
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3G Z%F'3M3$1*!L.PFN!1-!$)%$:3-A1*M.//1$!%-3J)*&1$>!T3U.\3D!.M1(!
h=ba
n
P!JOT3GUJ3P!
3U8P9P7PEEEP-E!
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h(fi+fi+1)
2
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333G S)*&%/1!'1$!A*.BRK1$>!
f(x)dx
a
b
h
2(f0+2f1+2f2+...+2fn1+fn)
!
3MG V#AD)'1!'+)*'*1!7!O1**1%*!B*)B)*A3)--1//1!C!D7GE!S)*&%/1!1T.(A1!B)%*!/1$!
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1
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f(x)dx
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12 [f'(a)f'(b)]
O1**1%*!
h4
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3G Z%F'3M3$1*!L.PFN!1-!-!$)%$:3-A1*M.//1$!%-3J)*&1$!LT3:9PT3N>!T3U.\3D!.M1(!
h=ba
n
P!JOT3GUJ3P!3U8P9P7PEEEP-E!
33G Z%*!(D.4%1!3-A1*M.//1!LT3:9PT3N>!/+.3*1!C!$)&&1*!M.%A!
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h
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3MG V#AD)'1!'+)*'*1!qP!J)*&%/1!1T.(A1!B)%*!/1$!B)/0-n&1$!JOTG!'1!'1?*#!
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c) V#AD)'1!'1&Romberg&
3G Q)%*!-U9P7PqPsPEEE!(./(%/1*>!
S0(n)=h
2(f0+2f1+2f2+...+2fn1+fn),h=(ba)
n
!
33G Q)%*!-U7&:9P7&P7&\9EEEt!&U9P7PdPEEE!(./(%/1*>!
Sm(2n)=4mSm1(2n)Sm1(n)
4m1
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d) Fonction&quad&
3G m)3*!'.-$!V.A/.F!_help-quad`E!
8) Equations+différentielles+ordinaires+
Résoudre:-
dy
dt (t)
y(a)=y0
=y'(t)=f(t,y(t)) t[a,b]
-
Données:-D!OB.$GP!T8PTJP
N=(xfx0)
h
P08P!JOTP0GUEEEP!'J'0UEEE!O'#*3M#1!'1!JOTP0G!Bb*!C!0GE!
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3G 08U()-$A!O')--#1!B.*!QF!6.%(D0GE!
33G Q)%*!]U8P9P7PEEEP2:9!>!
yk+1=yk+h.f(tk,yk)
!P!
tk+1=tk+h
E!
FG V#AD)'1!'+Euler&implicite!L*R?/1!'%!*1(A.-?/1N!
3G 08U()-$A!O')--#1!B.*!QF!6.%(D0GE!
33G Q)%*!-U9PEEEP2!>!:!Q)$1*!KU0-E!
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g=zynh.f(xn+1,z)
!
! ! ! !!!!u!
g'=dg
dz =1h.df
dy (xn+1,z)
!
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z=zg
g'
!
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g=zynh
2.[ f(xn,yn)+f(xn+1,z)]
!
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g'=dg
dz =1h
2.df
dy (xn+1,z)
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z=zg
g'
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k2=f(xn+1,(yn+h.f(xn,yn)))
!
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yn+1=yn+h
2.(k1+k2)
!
1G 2)A1$>!
1 / 7 100%
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