Introduction à l’analyse numérique

Introduction lanalyse numrique

Introduction lanalyse numriqueJacques Rappaz Marco Picasso

Presses polytechniques et universitaires romandes

Les auteurs et lditeur remercient lEcole polytechnique fdrale de Lausanne dont le soutien financier a rendu possible la publication de cet ouvrage.PROFESSEUR

LA COLLECTION ENSEIGNEMENT DES MATHMATIQUES EST DITE SOUS LA DIRECTION DU ROBERT C. DALANG Recherche oprationnelle pour ingnieurs I Dominique de Werra, Thomas M. Liebling, Jean-Franois Hche Recherche oprationnelle pour ingnieurs II Jean-Franois Hche, Thomas M. Liebling, Dominique de Werra Calcul diffrentiel et intgral Jacques Douchet et Bruno Zwahlen 1 Fonctions relles dune variable relle 2 Fonctions relles de plusieurs variables relles 3 Fonctions relles dune variable relle Exercices rsolus 4 Fonctions relles de plusieurs variables relles Exercices rsolus Algbre linaire Aide-mmoire, exercices et applications Robert C. Dalang et Amel Chaabouni Analyse avance pour ingnieurs Bernard Dacorogna, Chiara Tanteri Initiation aux probabilits Sheldon M. Ross Cours dAnalyse Srishti D. Chatterji 1 Analyse vectorielle 2 Analyse complexe 3 Equations diffrentielles DANS LA COLLECTION MTHODES MATHMATIQUES POUR LINGNIEUR

Introduction la statistique Stephan Morgenthaler Aide-mmoire danalyse Heinrich Matzinger

Les Presses polytechniques et universitaires romandes sont une fondation scientifique dont le but est principalement la diffusion des travaux de lEcole polytechnique fdrale de Lausanne ainsi que dautres universits et coles dingnieurs francophones. Le catalogue de leurs publications peut tre obtenu par courrier aux Presses polytechniques et universitaires romandes, EPFL Centre Midi, CH-1015 Lausanne, par E-Mail [email protected], par tlphone au (0)21 693 41 40, ou par fax au (0)21 693 40 27. www.ppur.org ISBN 2-88074-363-X 1998, 2000, 2004, Presses polytechniques et universitaires romandes, CH 1015 Lausanne Imprim en Italie Tous droits rservs. Reproduction, mme partielle, sous quelque forme ou sur quelque support que ce soit, interdite sans laccord crit de lditeur.

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!" !k k = 0, 1, . . . , n Pn n

Pn

P2

1@u RT3Q"X57v)AEn 1dR'SU1iC0&%&X57DG 5B$'57AU6 1"AUC0&($'$'IU67A( t0DG5763Q5 5B18"(We v)8 WBC0C0A(A(6n 6i@7D# SU&% 57AEAU1d1h5BA(1dC00A&(U6 $'576 $t 57V 6i"(0WB5 5HWs1dC0)dR`57$62CECE)iAUDG v C d 1 vR ' 1 j 6 y R`57AWrCEA%AP&SU&%5 5H6q1#&%A~576i"(EWr5 y 57Wr1iC0)dR'5B$ 3% B 5 0 c d ) v @ ` R Q A 7 @ i ) ' R B 5 ( & ) E C & 7 @ E c v$ x w $ 7 5 2 6 G 1 ( a 3%5g 3% R`vRDG 5BAXSU62&%R'5C0A 5r18SU&%5w6dja( E6R`2575w W7vA3Q%57CE6v A%R'"SUC0&%$'5f 5762t018 3Q57C06i A(3Q A%5@B5f3Q"(57v)c0)d@ $'R'A%@70R`)d5BDG5BAU1 R' 5A( ` i 1 2 6 E C E A i 1 P I % A G D E1Y$T6h &%5g E3Q3QC0@7"%"X1i575BA()dC03%AX0A6 #3D Q@BCEl(AUA%1i)dR`1i54R'C0SUAb 62&(WrR 57y 6h vAU3Q1d57I 5 )iA%B R'5B)Y6CE)iDG57AE1v&(6d6iRX&%A(54a(E625f3Q5 UdeRÀX62RA%CE&(6 twbwvEx (ab0tdQ t & pfhp i UtdSt Yba P7# b(b5dX 9A58f e(7 t p!`Q X VYXaXaegiT`cvb b daegfhXpiqb r0ùt V $'576"XCE$`%IP a (57E66 25w3Q5f!vcE)d0A% c0@Bl( 5fA%3QRT56 f e(' & 6iCPC WB"8R`( !v&%Q sGm"X)iCE57RÀ%AUC01Y6A(6 V 5B1V 57621gCE)iV DG@B5e"(v)g 06 d 7 @ 5 % A E t G D Q 3 "Xv) H }0 SSI H }0 RpI H }0 VSI 66 cE)d$Tnl 0"%(9%c057&(6)i5G 3Q5 }0'}0 V V 6i&%)$et R'AE1d5B) y v$'$`5 &%A%RTSU&%5BDG57AE1e6iC0AU1)i57"%)d@76i5BAU1i@H6 3(vA57(Pn (n + 1) 0 , 1 , 2 , . . . , n 1, t, t2 , t3 , . . . , tn n Pn 0 , 1 , 2 , . . . , n t0 , t1 , t2 , . . . , t n n = 2 t0 = 1 t1 = 0 t2 = 1 1 0 1 0 , 1 , 2 (t t1 )(t t2 ) 1 1 1 = t(t 1) = t2 t; (t0 t1 )(t0 t2 ) 2 2 2 (t t0 )(t t2 ) 1 (t) = (t + 1)(t 1) = 1 t2 ; (t1 t0 )(t1 t2 ) (t t0 )(t t1 ) 1 1 1 2 (t) = (t + 1)t = t2 + t. (t2 t0 )(t2 t1 ) 2 2 2 0 (t) 0 1 2 [1, +1] 1 0.8 0.6 0.4 0.2 0 -0.2 0 2 1

I

5%4!% H06i5f3Q5f!vcE)d0A%c05f3Q5 06d62CQWBR`@75g0&Qsu"C0R'AU1d6 V 5r1 P2 1 0 1

-1

0

1

t0 t1 t2 . . . t n

1Y'yF 8)6 !$B #A'y%0!$1 2F %'#)(A8 #$10($F A%5f C0AX3%6g57v&6 #v$"%')d5BC0&%a()Y$`6F73% DC05 A%A% }0@7'57}6 IWBC0V A(6iRT6qV 1YvAUV1fWYV 9%57)dWY579%Ar 57)3Q&%5H6h Aq"X"CEC0R`$'AUIP1dA%6t03QDGRT625 1iR'pA(Ws3%1Y5e6h3Q3%5BC0cEA%)iA%@ @Hn6 H SU&%V R8 "%)d5V 5By A(57A% p p p ... p V V y0 1 2 n

0 , 1 , 2 , . . . , n t2 . . . t n

2 . . . n p Pn

t1 = 0 1

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Q!#"! !"

p(t) = p0 0 (t) + p1 1 (t) + + pn n (t) = p n p

pj j (t).

j =0

(n + 1)

0 1

n

j

k = 0, 1, 2, . . . , n

n

p(tk ) =

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j =0

p0 , p1 , . . . , pn a

p

T

p1 = 3

t2 = 1

t 0 = 1 p0 = 8 p2 = 6 p(t) = 80 (t) + 31 (t) + 62 (t) 0 1 2 1 t + t 2 2

2

p(t) = 8

1 2 1 t t + 3(1 t2 ) + 6 2 2

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0 1 2 n j j j j j n j =0 j j j n 0 1 2 n n j j j =0

$

= 4t2 t + 3.

f

f

e2 (t)

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p(t) = e1 0 (t) + e0 1 (t) +

2

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1 2 1 1 2 1 t t + (1 t2 ) + e t + t 2 2 2 2 e 1 e 2 1 t + t + 1. 1+ 2e 2 2 2e

[a, b] tj j = 0, 1, 2, . . . , n j = 0, 1, 2, . . . , n t0 , t1 , . . . , t n n

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-1

0

1

p

pn (t) =

f (tj )j (t), Pn

j =0

0 , 1 , . . . , n

t0 , t1 , . . . , t n

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S

Q!#"! !"

... n

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1 n 2(n + 1) lim

ba n

(n+1)

= 0.

[1, +1]

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"! " !I

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maxt[a,b] |f (t) pn (t)|

1i57A(3

R

57 aUfIQ6WY 9%5BwxhkAU 1i57)i"C0$TvAU1#3Q5 WY5%9% k

f (t) = 1/(1 + 25t2 )

3Q5b3Q5BcE)i@}7 y H5 Wu3Q5H6#"C0R'AE1Y6#3Q5

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[t0 , t1 ]

[t0 , t1 ]

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H }0'}rRQI

!" #! "! 11 0.8 0.6 0.4 0.2 0 -0.2 t0 0 1 t1

T

0

1

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{R'AUA ""CEC06d$'vA5BU) 1 v&%sG"C0R'V AU1d6 V y 57W VP"(v)h&%Au"C0$'IPA%t0VDG5g3Q5g3Q5BcE)i@ VA%CEWr&( 6DG "XCE54& A%y C0CE&( A(66 i 1 7 5 i ) E C D $t y C0AX6 0R1e3(vA(6$T#625HWs1dR`CEAy}E X fvAX6$'G6i&%R`1i5wA%CE&(63Q@rlXA%R'6d6iC0A(6xi,1 < xi,2 < xi,3 < . . . < xi,n1 . t0 = xi tj = xi,j 1 j n 1 tn = xi+1 f tj 0 j n n h=0iN 1

n1

[a, b]

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max

|xi+1 xi |,

fh

max |f (x) fh (x)| Chn+1 .

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deRÀ(6iRpA%CE&(6 y CEA(6t[xi ,xi+1 ]

max

|f (t) fh (t)|

1 2(n + 1)

xi+1 xi n

(n+1) t[xi ,xi+1 ]

max

|f (n+1) (t)|.

n+1

CSRr$TGWrC0AX6q1YvAU1i5 57621e3QC0A%A(@B5g"(v)t[xi ,xi+1 ]

max C

|f (t) fh (t)| C

0j N 1

max

|xj +1 xj |

H }0UGuIUI

5B1kCSR

C= i = 0, 1, 2, . . . , N 1

xi = a + ih

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1 max |f (n+1) (t)| 2(n + 1)n(n+1) t[a,b]

0.8 N = 4 x4 = 0.8

xi,1 =

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max |f (x) fh (x)| Chn+1 .

f (x) = x1.7 + 0.1e3x sin(13x) a = 0 b = x0 = 0 x1 = 0.2 x2 = 0.4 x3 = 0.6 [xi , xi+1 ]

[xi , xi+1 ]

xi

C1

[xi , xi+1 ]

f

[a, b]

[xi , xi+1 ]

f

f

[t0 , t1 ] f (xi )

xi

C1

f (x i )

[a, b]

!" #! "! 11.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 x1.7 + 0.1e3x sin(13x) interpolant

}0}

58lf ) wxAU1d5B)d"XCE$'v1iR'C0Au"Xv)R'AE1d5B) y v$'$`5H63Q5 "(0)83Q576"C0$'IPA%t0DG5H63Q5w3Q5BcE)i@G}0f 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 x1.7 + 0.1e3x sin(13x) interpolant

0 a = x0

0.2 x1

0.4 x2

0.6 x3

0.8 b = x4

58lf q wxAU1d5B)d"XCE$'v1iR'C0Au"Xv)R'AE1d5B) y v$'$`5H63Q5 "(0)83Q576"C0$'IPA%t0DG5H63Q5w3Q5BcE)i@nGQf 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 x1.7 + 0.1e3x sin(13x) interpolant

0 a = x0

0.1 0.2 x0,1 x1

0.3 0.4 x1,1 x2

0.5 0.6 x2,1 x3

0.7 0.8 x3,1 b = x4

a R'5%SU&% ( 5H6B! 5 wxAU1d5B)d"XCE$'v1iR'C0A "(0)nR'AU1i57) y v$'$'5763Q5 "Xv)G3Q576n"XCE$ÌPA%tEDG5763ut Qe5B)dDGR1d5vWB&Qxf

0 a = x0

0.2 x1

0.4 x2

0.6 x3

0.8 b = x4

y

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f (t)

t

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t ]k, k +1[

1 0 (x) := x(x 1)(x 2) 6 1 1 (x) := (x + 1)(x 1)(x 2) 2 1 2 (x) := (x + 1)x(x 2) 2 1 3 (x) := (x + 1)x(x 1) 6 t k := E [t] t=k

"(v)i1iR')83Q5 DE#"%('Deh f8 d8$'c0CE)iR`1i9%DG5fC0&()iA%RT6d6i0AE1 "%!!5d94&&)( $f&)&)('7@9FD5d('0 e@Bl(A%R`1iR'C0Ao3Q5H6hC0A(Wr1iR'C0A(6 V V Vp = p(t) t 0 1 2 3

pR')i5 PR PRÀ(C0Ap

hv$TWr&($p3Q5t

k

p := fk ;

QR 5H6q157AE1dR`57) PR'A%C0AvC0AbW7v$TWr&%$'5

p(t)

:= fk1 0 (t k ) +fk 1 (t k ) +fk+1 2 (t k ) +fk+2 3 (t k )

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x ]0, 1[

{gh5HWs1i5B&(18C0A%A(C06e&($`6 5j3%WY9X %RTcE6d5B62DG AU1e V m&%R'6d"(SUv)&%5 @rl(vAA% CEA(5B6 $`5H3Q6h5 C0y A(0)Wri1iRTR'vaC0%A($'65 V V (0 (x) = 0 (x + k ), 2 (x) = 2 (x + k ), 1 0 (x) = x(x 1)(x 2), 6 1 1 (x) = (x + 1)(x 1)(x 2), 2 1 2 (x) = (x + 1)x(x 2), 2 1 3 (x) = (x + 1)x(x 1). 6

V(0$`CE)d6

6iC0R`185BAXWrC0)d5

1 (x) = 1 (x + k ), 3 (x) = 3 (x + k ),

t @7cE0$`R`1i@ H E } GvUIh67t @HWr)dR1D#vR'AU1i5BAXvAU1p(t) tR

B

< t 1 t0

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p(t) =f (k 1)0 (t k ) + f (k )1 (t k )

00 h f

B

f : R R h f (x) = f (x + h) f (x) h h h > 0

h h

h

W CE&(6 y @B)dRl(CEA(6hD#vR'AE1d5BA(0AU1$'5g)i@H62&($1Y1862&(R y vAU1 B 8 a'$'&)(Xe8 &t0b X 0fhpv t` f(b dQt dae s0fhti(t ft0 e0ft tv b0Xpiuv x e i pegfht0b Yb a'&)!7#0F9DE9F5%!7 bCEAU1i)dC0A(6$'5)i@H62&($1Y1p"C0&%)p$t C0"@B)Y1i57&%) 7mC0&%)WB5 vR')i5EVBWY9%CER'6iR'6d6iC0A(63Q57&Qs4A%CEDja%)d576 )dWe@BC057&( $'66 SU&%@B57)d$'RWBl(C0CEA(A(SU6&%6i5H0A6 (63Q5BR`1 qWr&%5r$`14 3Q57SU&Q&%su CEA(Ws1dR`CEA(68WBC0AU1iR'AP&%576 SU&%57$'WBC0A(SU&%5H6B i 1 f @ 5 y h h h h f, g : R R

}AV

Q!# % !#

!#

[x0 , x0 + h0 ]

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h (f + g )(x) = f (x + h) + g (x + h) f (x) g (x)

x0 R C

( )| C

h h0

h f (x 0 )

h f (x0 )

f (x0 )

f

h f (x0 ) h

f (x0 + h/2) = f (x0 ) + f (x0 ) f (x0 h/2) = f (x0 ) f (x0 )

h f (x0 ) + 2 2! h f (x0 ) + 2 2!

h 2 h 2

2

+

f ( ) 3!

h 2

3

,

2

f ( ) 3!

h 2

3

,

" ! ) !#

! !#'

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}FT

QR 5H6q18&(AvA(C0Dja()i5f"XCU62R`1iR`kl%sQ@w5r1e6iRuA%C0&(63%@rl(A%RT6d62CEA(6h0 C=

f ( ) + f ( ) h2 h f (x0 ) = h 6 8 |f ( )| + |f ( )| h2 . 2 24

H G%`}E}I

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1 max |f (x)|, 24 x[x0 h0 /2,x0 +h0 /2]

x0 R C

f (x0 ) h

h2

h f (x0 )/h

h f (x0 )/h

f (x0 )

f (x0 )

h f (x0 ) h0 (

f (x0 )

h f (x0 ) h

"

#7

40

h 1 h h f (x0 )/h

f (x0 )

2

h

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f (x0 )

h f (x0 ) h

0G Q!# % !# !# A%6 %CE)k &(64 A%5 Cv1dv"5B%)d"%C0)dA(C6 sQR`D# de R`R'AXC062A R "0)C0! &%)g 3Q5BCEDGA%"%A%$'5B50)fV &%A%5)d5B"%)d3Q@75 6i5BAUR'5B1dA(v13%iR')dC0Ao3Q@HWrR'D#v$'5fl(A%R'SU5G 3Q 557621!a(SER`&(575n "( B 5 Q s % & Q R A % & % A v1 i y AP&%DG@B)dRTSE&(5g3%5 Pt( ce f twcb e (fhXat0 bbwv e wpb iTe Yba 7#5d9A58f7Xe x VXp`cbdpegfhXaiubr0` ! ` i iXa fht t0bwvdSXa iuPi t( pve b t0iWfht fwbtiuv av eXai"dQ(egpxt b ! egx t0b(v dQXaiTi afwv egfda` fht0 etf e fhtYiXpi iu` x f e('& C#"%( Xe8 576A%CEDja%)d576 B

c

c

c = 1/3 c

c = 0.333333

c

N

N

0.333333 = 0.333333 100 34.2456 = 0.342456 102 0.000345033 = 0.345033 103

6 |c c | = 1 5|c| 3 10

0.01

6iC0AU183QC0A(A%@76 y 57W SWY9(Rh)i5H662R'c0A(RlXW71dR 67 5d9A58f7Xe6 Xpegv ` i iXp fhtfh(t0xytv bXpegv b aax t0` fY Q fhX q(t af$` i (px Yb ` a xav 7# t` fti egf p` x t Xav v piuv tAv fh aaegxgx aiuv at( ce ftwb dQ(0ega`Q bwe piTe (av e b t0bwv S tx(tt0fwft0` fd ! afwfhXpidae bAbw` f &r0ùpiuv egv t0bwv Q t0x(t fe bweXai fhtxav ePt da` (px0` x av t0` f & 0 r u ` p u i v g e v R`DG5B"%AU$'1g5G$`5e WBv$A%TC0WrDn &($a%A%)i5C0&(6f DG0)C0gAU5r1dsQ)i5G SU"% &%$'5 50VU$'C0)Y6iSU&%5 62R5r$'1h 5#SEWB&(v$5e TWr$&(t CE$'v1Aqi5B3Q&(R'6i)g"0)CEi6i)d5eC03Q AX5g 3QRS 1 WBWY9% C0)dR`h)i85H)iWs5HAp 1i6576i62DG E m B 5 G D R'c0A%R`lXW71iR` 67V(v$'C0)Y"%6&%C0RT6dAb v&( )d 5B18$et 5B)d)i57&%)83ut v)d)iCEA(3QRT662&() 6i5B)Y E S ( & 5 6D# 3QRT6{ $'C01d)Y&(6i3Q SUR'&pC0t A( C0Av @ y 0Rv$ÀU '&%1i557A(vAU1&%A5rsQ5BDG""%C0$'&(5)8R`$'v"$'&( %6q"%1d)d)dCQ0AWYU9%15B$)t RÀPDG&%"DGC0@B)i1d)dvARTSEX&(Wr5B5DG3Q5B57AU61 57)i)d5B&()d6k 3ut v)d)iCEAQx CEA(WB6 0$'WB&%$T1i57&%)8SU&%$`R CE)dA(6d5j SU&%3Q5RT62"CE6i5jSU&%5j3QV 5I#WY9%Rh V )d576e6iR'c0A%R`lXC0WB& v1iR` 6BV f e('& PC# "%A%( C0&(Xe6e X hC00$AX'6WB&%&%$À ! R A%CE&(6C0a%1i5BA(C0A(6y "C0&%) 5r1 $`5H6 y 0$`57&%)Y662&%R y vAU1i5H6 B3.42550 1018 = 0.342550 1019 c c N |c c | c = 10N

c

|c c | 5|c| c = 1/3 c = 0.333333 = 106

c

h f (x0 )/h

f (x0 )

h f (x0 )/h

f (x) = x2 x0 = 7 h = 0.06

h=

h = 0.06

h = 0.01

m3QR`&(hR'@B6d)dSU5B&%A(5 WB576 g A%C05H6q&(1n 6"%Wr$`CE&XA(6jWr$'cE&%)dCE0AA((6n SU&%"5r $et) 5B)d)d5B&%)nC0aQ1d5BAPSU&%&%5v "("Xv)CE $'oC0)dDj&%$'5r0& Qs 5 Q 3 q 5 0 C % & r 5 % & ) "% 9( @BvAA%UCE1 D3QF75 A%Sf 5fWY5H9%6q1f $`)dR'@j v&6i%R`cEsvA%57R`lX)i)dWB5Bv1&%i)YR` 68 3u"t C0v)&(d)d)kC0CEA(aQ3Q1iRT5767A%QR'wx)k$k$T6ifW&QrG 1fWr3Q$'&( 5j62"%R'C0)iAj 57A(R`3Q )d5f &%6i50A%H5n{gD# 0W5rYh9%5rR'1HA%V5jA%3QC0RT&X626x " E C d 6 ` R h 7 5 6 ! 6 0 C X A A 7 5 d ) G A y CEaQ1i57A%C0A(63(vA(6WB5gW706$'576 y 0$`57&%)d66i&%R y 0AU1i576 Bf (x0 ) = 14 h f (x0 )/h h = 0.01 h = 0.06 h = 0.06 : h f (x0 ) (7.06000)2 (7.00000)2 49.8436 49.0000 = = = 14.0600, h 0.0600000 0.0600000 h f (x0 ) (7.01000)2 (7.00000)2 49.1401 49.0000 h = 0.01 : = = = 14.0100. h 0.0100000 0.0100000

(7.06)2 (7.00)2 h f (x0 ) = h 0.0600 h f (x0 ) (7.01)2 (7.00)2 h = 0.01 : = h 0.0100 h = 0.06 :

49.8 49.0 0.0600 49.1 49.0 0.0100

13.3, 10.0.

|c c | 5|c|

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108 . . .

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f (x0 + h) f (x0 ) h f (x0 ) = def h h

h = 107

V

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f (x0 ) = 49 f (x0 ) = 14

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er

2 |f (x0 )| 2f (x0 )/h = h f (x0 )/h |f (x0 + h) f (x0 )|

2 |f (x0 )| ; |f (x0 )|h

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f : RR

f (x 0 )

x0 R h f (x0 ) h f (x0 )/h

1 h Et = 2 |f (x0 )|h

h Ea = 2 |f (x0 )|/h f (x0 )

h f (x0 )/h

h h E h = Ea + Et h E

Eh =

|f (x0 )| 1 |f (x0 )|h + 2 . 2 h h g

Eh

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b x

p

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b x2 x = b/a g ( x) > 0 x>0 h

x

h f (x0 )/h

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3 |f (x0 )| |f (x0 )|

1/3

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[f (x h/2 + h/2) f (x h/2 h/2)] =f (x + h) 2f (x) + f (x h).

=h f (x + h/2) h f (x h/2) =f (x + h/2 + h/2) f (x + h/2 h/2)

H G%`}AVSI

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h h0 , h h0 .

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pm (x) = f (x0 ) +

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h f (x0 ) 2 f (x0 ) (x x0 ) + h 2 (x x0 )(x x1 ) h 2!h 3 f ( x ) 0 + h 3 (x x0 )(x x1 )(x x2 ) + 3!h m f (x0 ) + h m (x x0 )(x x1 ) (x xm1 ). m!h pm B m pm (x0 ), pm (x1 ), . . .

H GQUGcI

pm

tG P {AG5HvR6q`1h17Vv&%A%A#CE"X&(CE6$`" C0&tEDG A(3Q 65e DG3QC057AUc01d)d)i@ 57)SUVU&%v$5 'C0)Y6 57621$et &%A%RTSUV &%5e"C0$'IPA%t0DG583Q5f3Q5B5BcE1g)i"%@ &%RT6iSUSU&%&%5 R y C05e P I % A R'AUC01iR'57)8)iSU "C0$'5 3%0A(64$`5H6 "XCERÀU1d6 H WYC09XaQv"1dp5BkA%}CEIrA(X6 wx$57621e EWrR'$`53Q5 y &%5w62RpA(C0&(63Q@7)iR y CEA(6 C0RT6$Tn)d5B$T1dR`CEA H GQUGcIsVPA%C0&X6 !#" %!# !# m f (m + 1) m pm (xj ) = f (xj ) j = 0, 1, 2, . . . , m pm x 0 , x1 , x2 , . . . , x m m

xj = x0 + jh (i)

WeC0&(6e y C0AX60RÀ(6iRp"(0)21dR`57$`$'5BDG57AE1DGC0AU1d)i@g$'5f)d@76i&%$`1dv16i&%R y vAU1 B &a'!r$'&)(iXe ) Te twbwv xt XpxaiptsdQtEdSt Qfh r0` eegiuv tf Xpxt dSaiub xt0b XaegiTvb Pt( pxXaf(bYXpi pm j = 0, 1, 2, , m m f pm (x) = f (x0 ) + (ii) h f (x0 ) 2 f (x0 ) (x x0 ) + h m (x x0 )(x x1 ) h 2!h m h f (x0 ) (x x0 )(x x1 ) (x xm1 ); + + m!hm dm m h f (x0 ) p ( x ) = . m 0 dxm hm

dm m h f (x0 ) p ( x ) = . m dxm hm

f

W C0&(6h1d5B)dDGRÀ%CEA(6Wr5B121d5f625HWs1iR'C0Av"(0)83Q5B&Qsv)i57D#v)YSE&(5767 e PR$TrC0A(Wr1iR'C0A 57621 6qCE1dR'R`6w WB1C0dAUR`CE1iA R'A3p %DG5BAU1f3Q@B)dR y 0a%$`5EVA%C0&(6 C0('& &)C0DEAX6pi&Qh1i(i Xe 8) $'5 " ' R ` $ T R i 6 B 5 i 1 % 9 7 @ 0 C d ) B F G D 4 5 E } ` } " 0 C % & ) B @ d 1 0a % ` $ ' R ) $ t H 5 # D t 57)i)d5B&%)6i&%R y 0AU1i55BAU1i)d5 5r1 y Bf (m + 1) pmx[x0 ,x0 +mh]

H G% GvPQI H G% GtSSI

f

YSU"X&%v)5B8 )f3Q$t@ vA(5B0$$'C0`CE"(c0"XR'5 57AU57AU1i)d15#3%$`5g 5qk"IC0P$'IP$`CEA%)t03Q DG5 5q3Q50&QWe 5 &%E)81dC03QA 5 H G% Gv UIf5r1n$`5#"C0$'IPA%t0DG5 F C8aQ5B1iD# E 57APv)&v 7 5 D i 1 E C y "CE0R6d'626fR'a%$'5573u t @B$`5H1d6w 0a%C0$`" R')g@B3Q 5Hd65B)i&%@H)Y626 &%$`1dv1d65r6i1 5BDna%$'p0am %$'v)57w 65BgsQWr5B5BDG &%sn @B50A(VpC0R'$g A(WB57@76216 DE pi1i9(h@B(i XeFB DG X 5 wx$QGQ5H S%6q1g 3(EWvAr('(R'&) 6$`w ' $ G 5 0 C d ) u V # D n W Y ) 1 % " ' $ y $'5f"XCE$ÌPA%tEDG5f3Q5f3Q5BcE)i@2GGSU&%R!RÀU1d5B)d"XCE$`5w$TC0A(Wr1iR'C0A n 5 Q 3 w 5 G D 0 C U A i 1 d ) B 5 e ) U S % & f 5 i 6 R 7 5 2 6 8 1 5BAv$'576"C0R'AU1d6 V 5B1 VXv$'C0)Y6f pm x = x0m h

max

|f (x) pm (x)|

1 hm+1 max |f (m+1) (x)|. 2(m + 1) x[x0 ,x0 +mh]

q2 x0 h x0

m h

x0 + h

vAWrX5B6!$'&%$Tl X3Qc05 &%)d5 eG%W8 `}gCEA(&(C06f&(C06ka(6iy 5BCE) A(CE6pA()d6e 5B"%SU)d&%@75 6i5B$ÀU5H1d6w@0VBWrCE"&%C0)i&(a()!&%&%)iA(5H5 3%CE57A(6fWs1dc0R`)YCEv"A %9(573Q6fC03QA(5 A%@B5EVH5r6i1f C0Af cE)d0"%v&9%5 B" 5f1f g R 4 6 Q 3 5 C0R'AU1 6iC0AU1"%)dCQWYy 9%576h$'C0)Y6iSU&%5 576218"X5B1iR`17f q2 (x0 , f (x0 ))

2 h f (x0 ) f (x0 + h) 2f (x0 ) + f (x0 h) d2 q ( x ) = = . 2 0 dx2 h2 h2

f

q2

h

'y8#A6 !$B #')%0!$1 2F % 0#A892#y!$1 "XCU R'a%A%$`5f 541i)dC0C0&AXy6e5BWr)CE3%A(576i6 @7C0)i)d@7Dj &%$'zq57&(6h6dSE3Q& 5gt r 3Q@B"%)dR)dy @7v16ii5BR'AUC017Aq AP &% DG @BCU)d62RTSECE&(A(5e "%"X$'v)&(g 6h5r"%sQ)d5B@7DG WrRT"%6i57$'56 SU&%5#wx$ WB5B5H$'6q$`1 5H6f SU6i&%6i5 CE&(3Q6g ' R Q 3 7 5 6 P % & ( " X " e 6 y

GtS

Q!# % !#

!#

f q2

x0 x0 + h

5% V lXe8 xw AU1i57)i"C0$T1dR`CEAv3%5 "(v)8&(Ab"XCE$ÌPA%tED5 3Q5j3%5Bc0)d@2GGv&%sv"C0R'AU1d6 V $T$'R`r 1iR'C0$t ACE)d3Q)d5n WrR'A(PQSo DGR`C01iA(@fWr eWeC0C0RT&(6w6WrCEC0aQAU1i1d57RÀA%%C0DG A(6 5BAUB 1f3Q@7)iR y va%$'5G5r1fWrCEA(62RT3Q@7)iCEA(6f$`5q3Q@ y 57$`CE"%"X57DG5BAU1f q2 x0 h f f (x0 + h h f (x0 ) ) = f (x0 ) + f (x0 ) + 2 2 2! + f (x0 f IV (x0 ) 4! h 2 h 2 h 2 h 22 2

x0 h x0 x 0 + h

+4

f (x0 ) 3! f V ( ) 5!

h 2 h 2 h 2 h 2

3

5

+ 4

,3

h f (x0 ) h ) = f (x0 ) f (x0 ) + 2 2 2! + f IV (x0 ) 4!

f (x0 ) 3! f V ( ) 5!

5

AU1f y 0$3Q`CE$'5A(WB mCSR v)86iC05762&X1g 6q1d&()dA EWs" 1iR'C0C0R'Ar A%C03Q&X5G6$et RÀUy C01d5BAX) 6

[x0 , x0 + h 2]

5r1 57621g&(A"C0R'AU1f3Q5

,

[x 0 h 2 , x0 ]

H GQUGvRQI CS621iRb 62R'c0A%3%5w0A&((Ao )i5HG%6q GB1d5f 3uVQt A%C0C0)Y3Q&()i65 CEhaQ1i57$'A%C0C0)Y6dAXSE6&(v&5 (h6d62Ri1 57B A(3 y 57)d631B@B)dC( PR!A(C0&(6462&%a%x R`1iR'&%WBCER A(O6 (hh "()0)3Q@Hh/ 6 p R r I 2 H f (x ) f (x ) h H GQUGuVUI = f (x ) + + O(h ). h/2 24 4 {gAb6iC0&(621i)YIE0AU1hSU&(1d)i5fCER'6 H GQUGuVSIh H G% GBRpIA%CE&(6 y CEA(63QC0A(I WBf (x0 ) 2 h f (x0 ) = f (x0 ) + h + O(h4 ), h 244 4 h/2 0 0 0 2 4

6iC0R`17VQ"(0)83Q@rlXA%R1dR`CEAb3Q5

f (x0 ) 2 f V ( ) + f V ( ) 4 f (x0 + h/2) f (x0 h/2) = f (x0 ) + h + h h 24 5!25 h f (x0 )

B

h f (x0 ) 8h/2 f (x0 ) = 3f (x0 ) + O(h4 ) h h

!

5B1l(A(0$`57D57AU1 m0)83Q@rl(A(R1dR`CEAi3%5w$et C0"@B)Y1d5B&%) VQCE)iDn&%$`5 H Q G UPQIsVQA(C0&(6CEaQ1i57A%C0A(6h f (x0 ) = 8h/2 f (x0 ) h f (x0 ) + O(h4 ). 3h

GvRH G% GtTSIB

8h/2 f (x0 ) h f (x0 )

5B1e3QC0AXW H Q G I0UI3(vA(6 H GQUGuTSIA%CE&(606d6i&%)i5wSU&%5f (x0 )B

= 8f (x0 + h/4) 8f (x0 h/4) f (x0 + h/2) + f (x0 h/2),

H GQ I0UI

6 8 ... h h/4 f (x0 ) h/8 f (x0 ) . . .

5HWB@76q18 &(A%"X5f %"(CE)iaQC1isQ57R'A%DG v1iWrR'5BC012Av 3Q5 C0)dDj&%$'5f5762$14 t 570")i% )d5B"X&(57) 3%5g 1i)dC0A(W71i&()i5g57 621e3ut C0)Y3Q)d5 (5w "%)dCvx 3 Q@j CE&%v")e ` R f ) i 1 f 5 ` $ @ 5#CEc0&%@B)A(0"@B)Y%v$"%')dR'6iCQ5BWY)w CP6i3%&Q5q#l)d(j3 ApQ3u58t C01d5BaQA%1dR'5B)A%WBR')j 3Q5H"Q6g C03Q)i5 x Dn &%$'576h3ut CE)dV 3%!)iwx5 $g5762V 1fV "XCU6iV 6iVUR'a% 5BA $`5q3%"X 9%5BWr)5B121d5qDG@r1iU9(wx$ C0DG 1i5f 0 #7 # % 50 4 450 ' (

8f (x0 + h/4) 8f (x0 h/4) + f (x0 h/2) f (x0 + h/2) 3h

h4 #7

'

34 #! (

f (x0 )

F 8 % F # t 5HR'6q$p1dR`5rD# CEA &(A%H 5f GQUWrGQCE}A(Ij6q1Y"vAC0U&%1i)5 m i1 = VSERe&( 5050g 6iR f 57621 V 6 CER'6 4 WBf C0eAU(B1iR'Akt%58kvDG( 5BAUXe1e8 3Q@7e )iR @7y DG va%C0$'AU501iV%)d0$5B`)G CE)d$6h sQRT16qd1dR`5f B 5 ' $ ' $ f 5 V C h h 0

"8h$e9Ft @B58cUfv$7 'R1d@ Q mj f 0) H G '}rRQIsVQA%CE&(6 y CEA(62 h

f (4) (x0 )

4 h f (x0 ) Ch2 . h4

m&%R'6dSU&%5$t C0"@B)Y1i57&%) 57621$'RÀ%@HvR')i55B15BAw&%1iR'$`RT6i0AU1$'C0)dDj&%$'5 H G%`}AVSIsVBA%C0&(6pC0aQ1d5BA%CEA(64 2 2 2 h f (x0 ) = h f (x0 + h) 2h f (x0 ) + h f (x0 h) = f (x0 + 2h) 4f (x0 + h) + 6f (x0 )

2 4 3 f (x0 ) h f (x0 ) = h h f (x0 ) = h h h

2 2 = h h f (x0 ) .

4f (x0 h) + f (x0 2h).

H G% I(}I

tG V Q!# % !# !# nt 0&Q1i)d5u"Xv)i17V$`5v3Q@ y 5B$'C0"%"5BDG57AE1G$'R`DGR`1i@io$t CE)d3Q)d5Sy3Q5v$'oC0A(Wr1iR'C0A 0&Q1iCE&%)G3Q& " C0R'AU1 A%CE&(606d62&()i5gSU&%5

f

x0

CSdeR "%)iFH662&(a(6q1dR1d&Q1iR'C0Ab3%0AV (6 % GH I(}IrVQA%C0&(6CEaQ1iV 57A%C0AX64 h f (x0 ) = f (4) (x0 )h4

(2h)1 (2h)2 (2h)3 + f (x0 ) + f (x0 ) 1! 2! 3! (2h)5 (2h)6 (2h)4 (5) (6) (4) + f (x0 ) + f (1 ) , + f (x0 ) 4! 5! 6! (2h)1 (2h)2 (2h)3 f (x0 2h) = f (x0 ) f (x0 ) + f (x0 ) f (x0 ) 1! 2! 3! (2h)4 (2h)5 (2h)6 (4) (5) (6) + f (x0 ) f (x0 ) + f (2 ) , 4! 5! 6! h2 h3 h1 f (x0 + h) = f (x0 ) + f (x0 ) + f (x0 ) + f (x0 ) 1! 2! 3! 5 4 h h6 h + f (4) (x0 ) + f (5) (x0 ) + f (6) (3 ) , 4! 5! 6! h1 h2 h3 f (x0 h) = f (x0 ) f (x0 ) + f (x0 ) f (x0 ) 1! 2! 3! 4 5 h h h6 + f (4) (x0 ) f (5) (x0 ) + f (6) (4 ) , 4! 5! 6! f (x0 + 2h) = f (x0 ) + f (x0 )

1 ]x0 , x0 + 2h[ 2 ]x0 2h, x0 [ 3 ]x0 , x0 + h[

5B1

4 ]x0 h, x0 [

PCER1

h0 > 0

&%AiA%CEDja%)d5wv)da%R1d)d0R`)d5f5r1862CER1 V%A(C0&(6 y C0A(63QCEA(WC=

+

4 64 (6) f (1 ) + f (6) (2 ) f (6) (3 ) + f (6) (4 ) 6! 6!

h6 .

mC0&()1iC0&%1

17 max f (6) (x) . 90 x[x0 2h0 ,x0 +2h0 ]

h h0

4 f (x0 ) h f (4) (x0 ) Ch2 . h4

(BA%r@7kv5058VXkv6i( C0R`Xi1 X QC0R`1 5r1 3$Tf n Q CEeA% C0A(Wr1iR'C0Ab3Q@rlXA%R`5g"(0) B}E e@B)dR`l(5B)bSU&%5 5r1

f : R R x0 R h>0

3QC0A(&%A%A%@75v 67 CEPA( CERWs1 1dR`CEA1d)iCER'6nCER'6# V WBC0AU1iR'A%DG5BAU1q3Q5r@B1f)dR 6iy C00aR`% 1 $`5x 1 = x 0 + h x 2 = x 0 + 2h

g

g (x) = f (x0 ) +

2 f (x0 ) h f (x0 ) (x x0 ) + h 2 (x x0 )(x x1 ). h 2h

g (xj ) = f (xj ) 0 [x0 , x1 ] 1 [x1 , x2 ]

C0&%SE)&(5 i1 57"$'6e

j = 0, 1, 2

5r1i5BA 3Q@73Q&(R`)d5SU&pt R'$f5BsQR'621i5

f (0 ) = g (0 ) ,

f (1 ) = g (1 ).

!

GQ 5rPsQCERTR6q1 1d5 $TnC0A(Wr1iR'C0Ar1d3%5B@r$l(SUA%&%R'5g 5 "(v)r [0 , 1 ]

r(x) = f (x) g (x) r ( ) = 0x x

5r1e3%C0A(W

uG T e@H3Q&%R')i5g3Q&v"XCERÀU1w}wSU&pt R'$6iR

I% 4@73Q&%R')d5f3Q&i"XCERÀU1 GSE&(5

r (x) =

r (t)dt =

f (t)dt.

C0DG"(0)i57) y 57W4$'5f3Q@ y 5B$'C0"%"5BDG5BAU183Q5wkIU$'C0)H j}E7f{"8hA 9F58f 7 1d&j3Q5$T83Q@Bl(A%R`1iR'C0Aj3%&f"XCE$ÌPA%tEDG5 VBR'$E5H6q1Wr$TvR')!SU&%5 7 5 2 ) y 0)3Q@rl(A(R1dR`CEAi3%5w$et C0"@B)Y1d5B&%) V%A%CE&(6 y CEA(6 "Xv)i17V%"(t[x0 ,x2 ]

|f (x) g (x)| 2h3 max |f (t)|

x [x0 , x2 ].

g

g (x0 ) = f (x0 )

nt 0&Q1i)d5

h

5B1

g (x1 ) = f (x0 ) + h f (x0 ) = f (x0 + h) = f (x1 ),

g (x2 ) = f (x0 ) + 2h f (x0 ) + 2 h f (x0 )

m0)83Q@rl(A(R1dR`CEAi3%5w$et C0"@B)Y1d5B&%) V(A%C0&(6 y C0A(62 h

= 2f (x0 + h) f (x0 ) + 2 h f (x0 ).

2 h f (x0 ) = h f (x0 + h) f (x0 )

5B"X1j A%CE&(6fCEaQ1i57A%C0A(6fa%R`57A kmg&%RT6iSU&%5 PCER1 $Tv5r1j C0A("%Wr&(1iR'R'6dC0SUA &%5 3Q@Bl(A%5762R'51 v) WBC0AU1iR' A %DG5BAU183Q@B)dR y 0a%$'50VUA%C0&(6h"C0& y C0A(6h&Q1dR`$'RT6257)h$'541i9%@7C0)dFBDG5f3%5CE$`$'5f"XCE&%)C0aQ1d5BA%R') 1d5B$SU&%5 R 5E 1d5B$SU&%5 R 5E G% 5 A(C0& 57v&v&Q1dR`$'R'6i5B)$'5g1i9(@B5rC01G)dFB"%DG &%RT5g 6iSU3Q&%5 5 CE$`$'575g 621G AU1iC0R' Aa%%1i5BDGA(5BR`) AU13Q@7)iR y va%$'50VkA%CE&(6 " C0& mgy &%C0RTAX6iSU6&%n "XCEWB&%C0)8 y 1i57$SU&%5 m0)8WrC0AX62@HSE&(5BAU17VQ"%&(R'6dSU&%5 57621WBC0AU1iR'AP&%50VQA(C0&(6 y C0A(6= f (x0 + 2h) 2f (x0 + h) + f (x0 ), g (x2 ) = f (x0 + 2h) = f (x2 ) r r(x) = f (x) g (x) r(x0 ) = r(x1 ) = r(x2 ) = 0 0 [x0 , x1 ] r (0 ) = 0 r (1 ) = 0 r r f (0 ) = g (0 ), f (1 ) = g (1 ). 1 [x1 , x2 ] r (0 ) = r (1 ) = 0 [0 , 1 ] r r ( ) = 0.x

= h f (x0 + h) h f (x0 )

f (t)

m&%R'6dSU&%5r5 1e3%57C062A(1!W &(Aj"C0$'IUA(t0DG53Q5h3Q5BcE)i@3%5B&QswR'$E5H6q1Wr$TvR')!SU&%5

r (x) = r (x) r ( ) =

r (t)dt. r (t) = f (t) g (t) =

g

x

r (x) =

f (t)dt.

IE Q!# % !# !# p58r5 18 W7063QC0CSAXR W I(5w UD#C00AA(%6iR`RTF73Q)i@B5g )dC0vAA(X6v$"X DGt v""%($`5 06 3Q 'C0v)c0g&(5B50sP 57j )iFH$'5e 6W7$`5w "CSC0RR'AU1g}EV E

x [x0 , x1 ] r(x0 ) = 0

x [x1 , x2 ]

6251d)d0R1d5H GQ IvGSI

x

mv)8WBC0A(6i@7SU&%57AU1 PCER1

f (x) g (x) = r(x) = r(x) r(x0 ) =x x1

r (s)ds.x0

|f (x) g (x)| s [x0 , x1 ]

nt 0"%)iFH6$`5w"C0R'AU1w}0Vx0 s

|r (s)|ds

x0

|r (s)|ds h max |r (s)|.

r (0 ) = 0

5B13%C0A(Ws 0

x0 sx1

mv)8WBC0A(6i@7SU&%57AU1 PCER1|r (s)| | t [x0 , x1 ]0

r (s) = r (s) r (0 ) =x1

r (t)dt.

nt 0"%)dF76$'5w"XCERÀU1 Gt

|r (t)|dt|

x0

|r (t)|dt h max |r (t)|.x0 tx1

H GQ ItIUI

t

5B18"(v)WrCEA(6i@7SU&%5BAU1|r (t)| |

r (t) =

f (u)du,

x2

f

576R'A%@BcUv$'R1d@76 H % G IBGQIrV H GQ ItIUIg5B1 H G% I0UIR`DG"%$'RTSE&(5BAU1e3QC0AXW CE57D y 57W$TC0)dDj&%$'5g3Q5h!IP$'C0)H PC0R`1 1i5B$!SE&(5 3QCEA%A%@EVB"%&%RT6dSE&(5 621"X1iv))ddC0C0RTA( 66uCEWBR'5g6e)iWr@HC062&( AU1d$1YRÀ1! %DG5BAU183Q@7)iR y va%$'50VQR'$p5rsQRT6q1d5 H GQ IvPSI CSR 576218$`5g"C0$'IPA%t0DG5w3Q5f3Q57c0)d@g3%5B&Qsr3%@rl(A%R"(v)x0 x0 ux2

|f (u)|du|

|f (u)|du 2h max |f (u)|.

H GQ IvcI

|f (x) g (x)| 2h3 max |f (s)|.x0 sx2

x [x0 , x2 ] [x0 , x]

f (x) = G(x) +

f ( ) (x x0 )3 , 6

G

mv)8WBC0A(6i@7SU&%57AU17VP$`CE)d6dSU&%5

G(x) = f (x0 ) + f (x0 )(x x0 ) + x [x0 , x2 ]

V%A%CE&(6 y CEA(6

f (x0 ) (x x0 )2 . 2

|f (x) G(x)| = |

f ( ) (x x0 )3 | 6 (x2 x0 )3 max |f (t)| x0 tx2 6 4 = h3 max |f (t)|. 3 x0 tx2

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Introduction à l’analyse numérique

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