calvi@picard.ups − tlse.fr

9;: <>= ? @ A B <C9D? :

E A F GH? : B <>9)? : I IKJCLM9;: NCI

OQP/RSUT#VWYX0Z\[][_^)`aDbca]d3P.efhgiZjVk

Sous la direction de : Jean − Paul CALVI Laboratoire Emile Picard Bureau 241 Bˆatiment 1R2 Universit´e Paul Sabatier 31062 TOULOUSE Cedex 04 Tel. : 05.61.55.76.66

calvi@picard.ups − tlse.fr

7%1 7*0%! 7 ;*7 !*, + %1

R)f WYW [! #"$"&%Yg' \[(" W7V *)#k+, [!"

--.-/.-.--.-.---.-. 0

1 V *) k2+, [43Y[ `f65g fhk5[87 O)[$g&, Z^ [ -/.-.--.-.---.-. 0

:9 ; k&^ [$g W V< f ^iZ\Vk=30[.>,[@? [$gA7 O)[_gB, Zj^ [ .-/.-.--.-.---.-. C

ED

PGF0[!, W \[("/H

.---.-.--.-/.-.--.-.---.-.

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4 OQ%!QPSR 7h%17TPJP_ QP

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C ^k

-.--.-.---.-. X0

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U:9 Y

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.-/.-.--.-.---.-. X`

UED a f<" [-30[ T

m (x 1 , . . . , x k )

-.--.-/.-.--.-.---.-. Nb

c

3 QP &dfe3 QP hg %1 ! *0%!7 4 c

9 Y L!ZYkYZj^iZ\Vk[32% WYg Vi2 #j!, [ -.--.-/.-.--.-.---.-. <9

9 k

^M%23Y[/3Y[!"lW_Vk23YZ^ ZjVk2"

.-.--.-/.-.--.-.---.-. <9

9:9

R\L!"iV m%0^ ZjVk[3Y[!"\"A)Q" ^Mj!, [!"D^igiZm3YZ f<5hVk,f<%UF

-.--.-.---.-. <C

9ED PGF0[!, W \[("/H .---.-.--.-/.-.--.-.---.-. 9n

9I0 bcV,J, [_k ^ fhZ\g [("K"&%Yg' \[!"Dg&L("&%2 j^ f ^M" .-/.-.--.-.---.-. 99

Pr´ eface

Y fhk2"GW$[ ,JL(, VZjgi[ k0V%2" f< m \Vk" ^ig fhZ^ [_g 3Y[ fhWYWYg V F0Zm, f ^ Z\Vhk-3:%Yk0[ V Vk2W ^ ZjVk

f

^{3YVk&^}

Vk kY[ W$VkYkYf\^G \[("hf6 \[(%0gB"[ ^ W$[! # \[!"G3Y[$" [(" 32L_giZL$[!" "B%2W!W$[!"B"iZ[!" ]^% [$kSW$[_g ^ fhZ\k2"W VZjk&^M"

1 #%2"]W0gBL(W_Z#"&L(, [$k&^ L ^ fhk&^/3YVk0k2L("

k

W$V<%YW2 \[!"

(x i , f (x i )), ∀ i = 1, ..., k

^\[ W0giVi2 mj(, [ W$Vhk2"iZm" ^ [ ^ g V%h[$g\%YkY[4V Vk2W ^ Z\Vhk

P

^{^} [! # \[4]^%Y[

P ^(j) (x i ) = f ^(j) (x i ) ∀ i = 1, ..., k, ∀ j = 0, ..., n i − 1

V[ j[("

f(x i )

^" Vk&^'3Y[(" f< j[(%Yg&"K3YVkYkL$[(" [ ^$ \[!"

n i ∈ N

^{^} [! #"K]^%Y[

f ^{(n i −1)} (x i )

[8F#Z#" ^ [$k&^

)V<%2"SW$Vk2" Z#32j_gi[_giVk2"J3Y[!%UF fhWYW0giVQW XY[("J3YZ L$g [$k&^ [!" H fhWYWYg VNF#Zm, f ^iZ\Vk W,fhgS3Y[!" W7V *)^7

k2+, [(" 3Zjk ^i[$g W V \f ^ ZjVkQ[ ^ fhWYWYg V F0Zm, f ^ Z\VhkQW,fhg 30[("V Vk2W ^ ZjVk2" "iW \Z\k0[("3 Z\k&^i[$giW7V \f ^ ZjVk

\[!" VVkW_^ ZjVk2" "iW2 jZ\kY[!"k EL_^ f k ^W,f<"G30[(" V Vk2W ^ ZjVk2"W7V *)#k+, Z\f< \[!" , f Z#"T3Y[(" VVkW_^ ZjVk2"W V67

m) k2+<, Z\f< \[!"cW,fhg$, Vg&W$[ f6%UF

`Zjk&^ [$g W V< f ^iZ\Vk f W7V%Yg$i2%#^'3Y[ gi[(, W2 \f<W$[_g$%YkY[4VVk2W ^ Z\Vhk

f

3YVkYk2L_[WYfhg$%YkY[4V Vk2W ^ ZjVk W2 m%2"$"iZm, W2 j[ [$kQ%Y[/3 E%YkW$f< #W!%2 3k^%2,JL$g Z#]^%Y[ WYfhg [8F#[(, W2 j[ %Yk W f6 #W(% 3 Z\k&^ML!5g f< j[ 3Y[

32L_giZ<L$[ V% " Z#, W2 \[!, [_k ^ W V%0g Vi0^i[$kYZjg %YkY[ gi[_WYgBL!"i[_k ^ f ^ ZjVk 5g f WYXYZ#]^%Y[ f WYWYgiVQW X2L_[ 3Y[

f V Vhk2W_^iZ\Vk

f

1 V<%Yg)Zm # m%2" ^igi[$g Z\k&^ [_giW7V f ^iZ\Vk W V< m) k2+, Zf< j[ kYV%2"\L_^B%23YZj[$giVhk2"l Zjk&^ [$g W V< f ^iZ\Vk 3Y[ ` f 7

5gifhk25[l7 O)[_gB, Zj^ [ [ ^ [$k W,f g ^ ZmW(%2 jZ\[_g \[lW f<"$3Y[/>,[@? [$gA7 O)[_gB, Zj^ [V["i[(% \[("$ j[("$32L_giZL$[!"

WYg [(, Z#j$g [(")gi[$k&^igi[$k&^ [_kJ? [(% W2%0Z#"QW7V%Ygl Z\k&^ [_giW7V f ^ Z\Vhk W7V *)#k+, Z\f< \[]WYfhg , VgBW_[ f<%QF

\[4W$f<"'3Y[("'VVk2W ^ ZjVk2"$"iW \Z\k0[("$W(%iYZ#]^%Y[!"

`[("l, f ^iX2L(, f ^iZ#]^%Y[("'%0^iZ# jZ#"BL!"$V Vk&^ fhW0W [! S f< #5<j(iYg [- jZ\k2L$fhZ\g [4L( mL(, [$k&^ fhZjgi[ f ^ X2L_VgiZj[

f< m5L(iYg Z#]^%Y[-L( mL(, [$k&^ fhZjgi[/3Y[!"QW7V m) k2+, [(" \[-W f< mW(%2 3YZ L_gi[$k&^iZ\[( 3 E%YkY[.VVk2W_^iZ\Vk %0kY[

fhgiZ\f<i2 j[

"!$#%& '%(*),+.- &

/)kY[/V Vk2W ^ ZjVk W V< m) k2+, Z f< j[Q[!" ^\%YkY[-W$V, iYZ\k,f Z#"iVhk[ \Z\kL fhZjgi[ 3Y[/, Vk2+, [(" T#Z

f

^{[!" ^}

%YkY[ V VkW_^ ZjVk W V< m) k2+, Zf< j[kYVhkk^%2 m \[ Z# [ F0Zm" ^i[h%0k.%Yk0Z#]^%Y[

n ∈ N

[_^T%Yk-%YkYZm]Q%0[

(n+1)

⁷

%YW2 j[_^

(a ₀ , a ₁ , . . . , a n ) ∈ R ⁿ × R ^∗

^{^i[( ]^%Y[}

∀t ∈ R

^{Vk f Zj^}

f (t) = a ₀ +a ₁ t +. . .+a n t ⁿ

Y fhk" W$[-W f6"A7@

n

^{[(" ^} fhWYW7[( mL- j[.30[(5g&L.3Y[

f

^{Y f k2"} j[-W$f<"DV

f ≡ 0

Vk3YZj^\]^%Y[.3Y[(5

f = −∞

0 kf j[(" WYg VWYgiZmL_^BL("'"B%YZ fhk&^ [("/H

T#Z

P, Q

"iVk&^\3Y[(" W7V m) k2+<, [!" Vkf[H 1

deg(P Q) = deg(P ) + deg(Q)

1

deg(P + Q) ≤ sup {deg(P), deg(Q)}

1

T#Z

deg(P) 6= deg(Q)

f< jVgB"

deg(P + Q) = sup{deg(P), deg(Q)}

P k W,fhg ^ ZmW(%2 jZ\[$g [$k"i[(, i2 \['3Y[!"cW7V m) k2+, [(" 3Y[\3Y[!5gBL

≤ n

^{[(" ^ %Yk} [("iWYf<W$[ 7 [!W_^iVgiZj[(

3Y[/3YZm, [_k2"iZjVk

n + 1

^{0 k j[]kYVh^ [}

R n [X]

³²

54 67'%(*),+.- 98,;:<=#>?),=A@CBDE#>- F%GH

D´ efinition :

T#VZj[$k&^

X = {x 1 , . . . , x k }

^%Yk [$k"i[(, i2 \[[3Y[,I gBL_[( #" 30Z#" ^iZ\k2W ^M" [ ^ W7V%Yg W X,f<]^%Y[

1 ≤ i ≤ k

^{%Yk [$k&^ Zj[$g}

n i ≥ 1

^{Xk ^ fhk&^} 30VkYk2L_[h%0kY[hVVhk2W_^iZ\Vk V#W V<%Yg f<]^%Y[! # \[

f 3L$giZL_[

f ^{(n i −1)} (x i )

[8F#Z#" ^ [ [$k W,fhg ^ Z#W!%2 \Zj[$g j[(" 32L_giZ<L$[!" W0gBL(W!L(3Y[_k&^ [(" [8F#Z#" ^ [_k ^ [ ^

"iVhk ^ 32L!Z,k0Z\[(" 3Yfhk2" %Yk hVZ#" Z\k,f<5h[ 3Y[

x i

Vhk W XY[$g&W XY[ %Yk W V< m) k2+, [ 1 L_giZmZ,fhk&^ \[!"

J

R n [ X ]

KMLONPRQSKTLOUWV%XYK[Z\KMX]N_^%`_abKMcedRfhgjiE^\Q\N_`k^\QRLlPmK

λP ∈ R

n [ X ]

^KjN

P + Q ∈ R

n [ X ]

n

deg λP = deg P si λ 6= 0 deg λP = −∞ si λ = 0

KYN"XY^\iEiAK[^\QoVpN_^%Prqs^\Pm`_L

−∞ ≤ deg P

Vtcu^%`vLLOawfhKMx

P ≤ n

^%QyVEfhKMx

λP ≤ n ∀ λ ∈ R

n

fhKjx

( P + Q ) ≤ max{deg P , deg Q}

fh^%QRXzLkaWfmKjx

P ≤ n

^KYNfhKjx

Q ≤ n

Vtcu^%`vL{fhKMx

( P + Q ) ≤ n

|

VA}RV\LOK~XjV%Qm^%QRaulPmK[KMLON

{1, t, t ² , . . . , t ⁿ }

^

_

W$Vhk23YZj^iZ\Vk"$"B%YZhf k ^i[("/H

P ^(j) (x i ) = f ^(j) (x i ) ∀ 0 ≤ j ≤ n i − 1, ∀ 1 ≤ i ≤ k.

0 k Z#, W V" [

n ₁ +n ₂ +. . .+n k

W$Vk3YZj^iZ\Vk2" 3YVk2WK" Z0Vk [(%#^ E%YkYZ#W_Zj^BL;[_k 5L$kL$g f6 Vk

3YVZ^GW X0[$gBW X0[$g %Yk W7V m) k2+, [h3Y[K3Y[!5gBL

d = n 1 +n 2 +. . .+n k −1

^{0 k fQ[ [!W_^iZ[!,} [_k&^'H

)$*,+ .- +-

0/132546879:2$;=<>29?;=<@7A2;B1DCFE.1HG.<JI.KL2NMPO=2QO=2&RTSVU$OXWTUS6Y[ZT<F9\182]7^]E.<_OT6a`

9b6&E.<c7edfhgafji

bc[W7V *)#k+, [Q[(" ^)fhW0W [! #L]W7V m) k2+, [43 Z\k&^ [_giW7V f ^ Z\Vhk=3Y[ ` f<5g f k25[87 O)[_gB, Zj^ [

Exemples :

T#Z

n i = 1 ∀i

^{Vk gi[_^igiV%} [- \[]W7V m) k2+, [ 3 Z\k&^i[$giW7V \f ^ ZjVk=3Y[ ` f<5gifhk25[

T#Z

k = 1 n 1 = n

\[("'W_Vk23YZ^ Z\Vhk2"K"iVhk ^-H

P (x 1 ) = f (x 1 ), P ⁰ (x 1 ) = f ⁰ (x 1 ),

P ⁽ⁿ⁻¹⁾ (x 1 ) = f ⁽ⁿ⁻¹⁾ (x 1 ).

0 kg [_^ g V%h[. j[]W V *) k2+, [ 3Y[lkf )Q \Vhg S VhgB3Yg [

n − 1

^H

P (x) =

n−1

X

j=0

(f ^(j) (x 1 ))

j ! × (x − x 1 ) ^j .

`[(" W7V *)#k+, [!"30[mkfN)Q jVg[ ^ 3Y[ ` f<5hg fhk25h[ " Vk&^ 3YVk2W 3Y[(%QFW f6"W,fhg ^ ZmW(%2 jZ\[$g&" 3Y[

` f<5gifhk25[ 7 O)[_gB, Zj^i[

Illustration g´ eom´ etrique des conditions lorsque n i = 2 ∀ i

^H

b [(" ^- j[ W f<" W,f g ^ ZmW(%2 jZ\[_g W$Vhk2"iZm32L$g&L [$k 32L_^ fhZ# W2 #%2" \VZjk "iV%"- j[ kYV, 3 Z\k&^ [_giW7V f ^iZ\Vk

3Y[->,[ ? [_gA7 OD[$g&, Z^ [

T#[[3YVhkYkY[$g

P (x i )

^{[_^}

P ⁰ (x i )

^{g [ Z\[_k ^ "} [[3YVkYkY[_g f^ fhk25[_k ^i[[32% 5g f WYXY[[3Y[

P

^f<%

W7VZ\k&^

x i

R)f WYW [! \Vk"$ EL(]^%,f ^iZ\Vk[3Y[/ \f ^ fhk25[_k&^ [

y = P ⁰ (x i ) × (x − x i ) + P (x i )

Y fhk"lW_[-W$f<" j[ WYgiVi #j(, [ W7V"BL gi[ Z\[$k&^ W XY[$g&W XY[$gl%Yk W V *) k2+, [43YVk&^DVkf Zm, W7V"BL

\[!" ^ fhk5[$k&^ [!"

`

0 0.5 1 1.5 2 2.5 3

0.5 1 1.5 2 2.5 3

QUKLE.<c79bS5ZT9 6&E.<_gDbcVk2" Z#32L_giVk2"\ fhWYW \Z#W$f ^ ZjVk

ϕ : R d [X] −→ R ^{d + 1}

P 7−→ (P (x 1 ), P ⁰ (x 1 ), ..., P ^{(n 1 −1)} (x 1 ), ..., P (x k ), ..., P ^{(n k −1)} (x k ))

RD[!, f gB]^%YVk2" 3 f<i7VgB3S]^%Y[

ϕ

^{[(" ^ \Zjk2L f Z\gi[}

ϕ(λP + µQ) = λϕ(P ) + µϕ(Q)

^{bc[! f}

WYg V Zj[$k&^\3Y[/ f \Zjk2L f giZj^BL 3Y[/ f 3L$giZ f ^ ZjVk H

(f + g) ^(k) = f ^(k) + g ^(k) et (λf ) ^(k) = λf ^(k)

1 V<%Yg , Vk&^ g [$gJ [8F#Z#" ^ [$kW$[[ ^ :%0kYZ#W_Zj^ML 32% W V *) k2+, [

P

^{32% ^ X2L_VgBj!, [} Zm $"B% ^ 3Y[

, Vk&^ gi[_g ]^%Y[

ϕ

^{[(" ^-iYZ? [(W ^ Zh[ ;} W_ZW7V%Yg4, Vk&^ gi[_g

ϕ

^{iYZ? [(W_^iZh[} Zm G"B% ^/30[J3L(, Vk&^ gi[_g ]^%Y[

Ker ϕ = {0}

^a ^i^i[$k&^ ZjVkHP k 5<L$k2L_g f< "iZ

Ker ϕ = {0}

^{Vk W [!%0^ " [(%2 j[(, [$k&^}

3YZjgi[ ]^%Y[

ϕ

^{[!" ^Z\k ? [!W_^ Z[ , fhZm" W$V<, , [ Z#W_Z f[3YZm,} [_k2"iZjVk 3Y[S [("iWYf<W$[J3Y[S32L$WYfhg ^ [(" ^ L(5f< \[ / f43YZm, [_k2"iZjVk 3Y[' [(" W,f<W$[\3 fhgig Z<L$[ Vhk f

ϕ

^{i0Z? [(W ^ Z[}

⇐⇒ ϕ

^{Z\k ? [!W_^ Z[}

⇐⇒ ϕ

"B%0g? [(W ^ Z[

Y L ^ [$g&, ZjkYVk2"l3YVk2W

Ker ϕ

^{H3T VZj^}

P ∈ Ker ϕ

f< jVgB"

ϕ(P ) = (0)

^{30Vk2W [_k W,fhg ^} ZmW(%2 jZ\[$g

P (x 1 ) = 0, P ⁰ (x 1 ) = 0, ..., P ^{(n 1 −1)} (x 1 ) = 0

^3YVk2W

x 1

[(" ^ %Yk0[Qg f6W$Z\k0[\, %2 j^iZ\W2 j[$3 VhgB3Yg [

≥ n 1

0 kW [!%0^ 30Vk2W.Vf6W_^ VhgiZ#" [$g' \[ W7V m) k2+, [

P

^W,fhg

(x − x 1 ) ^{n 1}

^Vkf f< jVgB"

P (x) =

(x − x 1 ) ^{n 1} Q(x)

^{f [!W.3Y[!5}

Q ≤ d − n 1

P k %0^ Zm \Zm" fhk&^ \[ Vf Zj^ ]^%Y[

P (x 2 ) = 0, P ⁰ (x 2 ) = 0, ..., P ^{(n 2 −1)} (x 2 ) = 0

^{Vhk W [!%0^} 32L!32%YZ\g [

P (x) = (x − x 2 ) ^{n 2} R(x)

⁰ g' \[(" W7V *)#k+, [!"

(x − x 1 ) ^{n 1}

^{[_^}

(x − x 2 ) ^{n 2}

"iVk&^DWYgi[!, Zj[$g&"

[$k&^igi[ [!%UFW fhg \[ "i[(% )W7V m) k2+<, [ Z\g gBL!32%2W_^iZ#i2 j[]^%YZ\3YZ#Zm"i[

(x − x 1 ) ^{n 1}

^{[(" ^}

(x − x 1 )

[_^ 3Y[ \f[, !, [S, fhkYZ#j_gi[ \[S"i[!%2 ;W V *) k2+, [ Z\gig&L(32%W_^ Zmi2 \[ ]^%YZ 3YZ Z#"i[

(x − x 2 ) ^{n 2}

^{[(" ^}

(x − x 2 )

^{0 g}

x 1

[ ^

x 2

" Vk&^K3YZm" ^ Zjk2W_^B" 30Vk2W

(x − x 1 ) 6= (x − x 2 )

^{0 k W7[(%0^} f< \Vg&" L(W$g Z\g [

P (x) = (x − x 1 ) ^{n 1} (x − x 2 ) ^{n 2} T (x)

P kW_Vk&^ ZjkQ%Yfhk&^)Vhkf gigiZ[[H

P (x) = (x − x 1 ) ^{n 1} (x − x 2 ) ^{n 2} ...(x − x k ) ^{n k} S(x)

Y L ^ [$g&, ZjkYVk2"$ j[43Y[(5hgBL-3Y[

S

^{HYVkf 3Y[(5}

P ≤ d.

⇒ d ≥ n 1 + n 2 + ... + n k + deg S

C

⇒ d ≥ (d + 1) + deg S

⇒ deg S = −∞

⇒ S = 0

⇒ P = 0

1 V%0g[W f< mW(%2 j[$g j[("W7V *)#k+, [!" 3 Z\k&^i[$giW7V \f ^ ZjVk3Y[ `f65g fhk5[87 O)[$g&, Z^ [ Zm )[ F#Z#" ^i[

%YkY[[,JL_^iXYVQ3Y[ 5L_k2L$gif< \[ W_[( # j[ 3Y[!" 3YZ L$g [$k2W_[(" 3YZ Z#"BL_[(" ]^%Y[kYV%" k :L ^M%230Z\[$g Vk2" W,f6"

W f g [( # j[]kYV%2" [_k&^ g f kY[_g fhZ^ ^ g VW jVZ\k

\L fhk2, VZjk2" k0V%2"'W XY[$g&W XY[$g Vk2"\3Y[("DW7V m) k2+, [("

3 Z\k&^ [_giW7V f ^ Z\Vhk W,fhg[ \f ,JL_^ X0VU3Y[ 30[ > [@? [$g 7 O)[_gB, Zj^i[ , VZjk2" 5 jVi,f< j[W2%0Z#"B]^% [( m \[ " [

gi[!" ^igi[$Zjk&^Qf6% W$f<" V

n i = 2

)GHE#>'yGHF%'.) 8, E#H@CBDE#>- F%GH

D´ efinition

H,T#VZ\[_k&^

k

g&L$[! #"

x i

W V%0g

i = 1, . . . , k

^{1 V" Vk2" W7V%Yg}

x ∈ R

ω(x) =

k

Y

i=1

(x − x i ) et l i (x) =

k

Y

j=1 j6=i

(x − x j ) x i − x j

.

l i

[(" ^\ E%YkYZ#]^%Y[]W7V *)#k+, [ 3Y[/3Y[(5g&L

k

^{^i[( ]^%Y[}

l i (x j ) =

1 si i = j 0 sinon

)$*,+

.- +- 430< ZL1 U&RhZT1H6a9:U#7;=6aW.ZT<F92

l i (x) =

k

Y

j=1 j6=i

(x − x j ) x i − x j

= ω(x) ω ⁰ (x i )

1 x − x i

, ∀x ∈ R

QUKLE.<c79bS5ZT9 6&E.<_g 1 V%0g

x ∈ R

l i (x) =

k

Y

j=1 j6=i

(x − x j ) x i − x j

=

k

Y

j=1 j6=i

(x − x j )

k

Y

j=1 j6=i

1 x i − x j

= ω(x) x − x i

k

Y

j=1 j6=i

1

x i − x j

b

Y [ W2 m%2"

ω ⁰ (x) = (x − x ₂ )(x − x ₃ ) . . . (x − x k ) + (x − x ₁ )(x − x ₃ ) . . . (x − x k ) + . . . + (x − x ₁ )(x − x ₂ ) . . . (x − x _k−1 )

=

k

X

j=1 k

Y

l=1 l6=j

(x − x l )

3YVkW

∀i = 1, . . . , k, ω ⁰ (x i ) = Q k j=1 j6=i

(x i − x j )

0

kf iYZj[$k

∀i = 1, . . . , k, l i (x) = _ω ^ω(x) 0 (x i ) 1 x−x i

D´ efinition

^H T#VhZ\[$k&^

X = {x 1 , ..., x k }

^{%Yk [$k2" [(, i \[-3Y[?I} gBL$[! #"\3YZ#" ^ Zjk2W_^B"D[ ^\%YkY[

VVk2W ^ ZjVk

f

32L8Z,kYZj[[ ^ 3L$giZ f<i2 j[ [$kW_[(" W VhZ\k&^M" 0 kW XY[$g&W XY[ %0k%YkYZm]^%Y[ W V *) k2+, [

P ∈ R 2k − 1 [X]

^{^ [! ]Q%0[}

∀i = 1, . . . , k ,

P (x i ) = f(x i ) P ⁰ (x i ) = f ⁰ (x i )

bc[ W V< m) k2+, [][(" ^Qf WYW [! #L4 j[]W V *) k2+, [ 3 Z\k&^ [_giW7V f ^iZ\Vk 3Y[-> [@? [$g 7 O)[_gB, Zj^i[

)$*,+ .- +- c ,2CFE.1HG <JI.KL2

P ∈ R 2k − 1 [X]

^WTUSj6Y[ZT<F9182]7 ^]E.<_OT6a9 6&E.<c7

∀i = 1, . . . , k ,

P (x i ) = f (x i ) P ⁰ (x i ) = f ⁰ (x i )

2]79?O=E.<F<JU0CcZTS

P (x) =

k

X

i=1

f(x i )A i (x) +

k

X

i=1

f ⁰ (x i )B i (x)

ZTWT2]^

A i (x) = [1 − ^ω _ω ⁰⁰ 0 (x ^(x i ⁱ ) ⁾ (x − x i )]l ² _i (x)

²⁹

B i (x) = (x − x i )l ² _i (x)

O=2]7CFE.1HG.<JI.KL2]7g

QUKLE.<c79bS5ZT9 6&E.<_g 1 V%0gJ3L(, Vk&^ gi[_g W$[^iX2L$Vg&j(, [ Zm ckYV<%2"S"B% ^S3Y[ L_giZ*Z,[$g j[(" ^ g VZ#"

W$Vhk23YZj^iZ\Vk"$]Q%0Z "iVk&^-H

P ∈ R 2k − 1 [X]

^Z[430[(5

P ≤ 2k − 1

U

∀ 1 ≤ i ≤ k, P (x i ) = f (x i )

9

∀ 1 ≤ i ≤ k, P ⁰ (x i ) = f ⁰ (x i )

7 T#Z

P (x) = P k

i=1 f (x i )A i (x) + P k

i=1 f ⁰ (x i )B i (x)

f< \VhgB"$3 fhWYg&j(" j[("DWYgiVhWYgiZmL_^ML!"

"B%0g\ j[(" W7V m) k2+<, [!" Vkf 3Y[(5

P ≤ max 1≤i≤k {deg A i , deg B i }

Calculons donc le degr´e de A i et le degr´e de B i :

3Y[!5

(l i ) = k − 1

^{30Vk2W 3Y[!5}

(l _i ² ) = 2(k − 1) = 2k − 2

^{[ ^$3Y[!5}

(1 − ^ω _ω ⁰⁰ 0 (x ^(x i ⁱ ) ⁾ (x − x i )) = 1

Nn

bcV,J, [430[(5

(P Q) = deg P + deg Q

^{Vk fJ30[(5}

(A i ) = 1 + 2k − 2 = 2k − 1

Y [/, (, [ 3Y[!5

(B i (x)) = deg (x − x i ) + deg (l ² _i (x)) = 1 + 2k − 2 = 2k − 1

0 kf iYZj[$k 3Y[!5

P ≤ 2k − 1

7 L$giZ*Z,Vk2"$, fhZjk ^i[$k,f k ^']^%Y[

∀ 1 ≤ i ≤ k, P (x i ) = f(x i )

T#VZ^

i ₀ ∈ {1, . . . , k}

]^%Y[( mW$Vk2]^%Y[

eVk&^ g Vk2"\]Q%0[

P (x i 0 ) = f(x i 0 )

P (x i ₀ ) =

k

X

i=1

f(x i )A i (x i ₀ ) +

k

X

i=1

f ⁰ (x i )B i (x i ₀ )

Y Zm" ^ Zjk25%YVhk2" W f<"/H

1

i = i 0 :

A i (x i 0 ) = A i 0 (x i 0 ) =

1 − ω ⁰⁰ (x i 0 )

ω ⁰ (x i ₀ ) (x i 0 − x i 0 )

l _i ² ₀ (x i 0 ) = l _i ² ₀ (x i 0 ) = 1 et B i (x i 0 ) = B i 0 (x i 0 ) = (x i 0 − x i 0 )l ² _{i 0} (x i 0 ) = 0

1

i 6= i 0

H Y

fhk2"\W$[/W f<"

l i (x i 0 ) = 0

^3YVhk2W

A i (x i 0 ) = 0

^{[_^}

B i (x i 0 ) = 0

aQZjk2"iZ kYV%"DV<i0^ [_kYVk2"

P (x i ₀ ) = f (x i ₀ ), ∀ 1 ≤ i 0 ≤ k

9 7 ; kYV<%2" gi[(" ^ [ L_giZmZY[$g\]Q%0[

P ⁰ (x i 0 ) = f ⁰ (x i 0 )

^{1 V%Yg'W$[! f} 3L$giZVk"

P (x i 0 )

P ⁰ (x i ₀ ) =

k

X

i=1

f(x i )A ⁰ _i (x i ₀ ) +

k

X

i=1

f ⁰ (x i )B _i ⁰ (x i ₀ )

0 g

B _i ⁰ (x i 0 ) = l _i ² (x i 0 )+2(x i 0 −x i )l i (x i 0 )l ⁰ _i (x i 0 )

^{3YVk2W "iZ}

i = i 0 , B _i ⁰ ₀ (x i 0 ) = l ² _i (x i 0 ) = 1

[_^\" Z

i 6= i 0 , B _i ⁰ (x i 0 ) = 0

^W$fhg

l i (x i 0 ) = 0

bc[(W_Z, Vk&^igi[4]^%Y[

P k

i=1 f ⁰ (x i )B _i ⁰ (x)

|x=x i 0 = f ⁰ (x i ₀ )

T#Z k0V%2" , Vk&^ giVhk2" ]^%Y[

P k

i=1 f(x i )A ⁰ _i (x)

|x=x i 0 = 0

^kYV%" f<%YgiVhk2" 32L!, Vk&^igBL[]^%Y[

P ⁰ (x i 0 ) = f ⁰ (x i 0 )

1 V<%Yg'W$[4VfhZ\g [ , Vk&^ g Vk2"$]^%Y[

A ⁰ _i (x i ₀ ) = 0

A ⁰ _i (x i 0 ) = − ^ω _ω ⁰⁰ 0 (x ^(x i ⁱ ) ⁾ l _i ² (x i 0 ) + 2

1 − ^ω _ω ⁰⁰ 0 (x ^(x i ⁱ ) ⁾ (x i 0 − x i )

l i (x i 0 )l ⁰ _i (x i 0 )

T#Z

i 6= i 0

f6 \Vg&"

l i (x i 0 ) = 0

^3YVkW

A ⁰ _i (x i 0 ) = 0

¹ fhg'W$Vk&^igi[/"iZ

i = i 0

Vk f

A ⁰ _i (x i 0 ) = A ⁰ _{i 0} (x i 0 ) = − ^ω _ω ⁰⁰ 0 (x ^(x i ⁱ 0 ⁰ ) ⁾ l _i ² ₀ (x i 0 ) + 2

1 − ^ω _ω ⁰⁰ 0 (x ^(x i ⁱ 0 ⁰ ) ⁾ (x i 0 − x i 0 )

l i 0 (x i 0 )l ⁰ _{i 0} (x i 0 )

0 g

l i ₀ (x i ₀ ) = 1

^{3 V}

A ⁰ _{i 0} (x i ₀ ) = − ^ω _ω ⁰⁰ 0 (x ^(x i ⁱ 0 ⁰ ) ⁾ + 2l ⁰ _{i 0} (x i ₀ )

)V<%2"'W XY[$g&W XYVk2" S, Vk&^ g [$g$]^%Y[

A ⁰ _{i 0} (x i 0 ) = 0

^H

A ⁰ _{i 0} (x i 0 ) = 0 ⇐⇒ − ω ⁰⁰ (x i ₀ )

ω ⁰ (x i 0 ) + 2l ⁰ _{i 0} (x i 0 ) = 0

⇐⇒ l ⁰ _{i 0} (x i 0 ) = ω ⁰⁰ (x i 0 ) 2ω ⁰ (x i ₀ ) Calculons l ⁰ _{i 0} (x)

^H

l i 0 (x) = _ω 0 (x i 0 ^ω(x) )(x−x i 0 )

3YVk2W

l _i ⁰ ₀ (x) = _ω 0 (x ¹ i 0 )

ω ⁰ (x)(x−x i 0 )−ω(x) (x−x i 0 ) ²

1 V<%Yg ]^%Y[

l ⁰ _{i 0} (x i ₀ )

^{"iVZ^ iYZj[$k 32L!Z,k0Z Z# Vf<%0^ ]^%Y[}

x i ₀

"iVhZj^g f<W_Z\kY[\3YV%2i \[\32%k^%2,JL$g f ^ [(%0g

N(x) = ω ⁰ (x)(x − x i 0 ) − ω(x)

^{Z [}

N (x i 0 ) = 0 et N ⁰ (x i 0 ) = 0

ω(x i 0 ) = 0

^3YVk2W

N(x i 0 ) = 0

^{[ ^\W$V,J, [}

N ⁰ (x) = ω ⁰⁰ (x)(x − x i 0 ) + ω ⁰ (x) − ω ⁰ (x) = ω ⁰⁰ (x)(x − x i ₀ ) N ⁰ (x i ₀ ) = 0

)V<%2" [_kYVk2" 3Y[ , Vk&^igi[$g ]Q%0[

x i 0

[(" ^[iYZ\[_k5g f<W_Z\kY[ 3YV%2i2 j[ 3Y[

N

^{KY [ W$[ VfhZ^ Vk}

" f Zj^ ]^% Z# [8F#Zm" ^ [ %Yk W V< m) k2+, [

Q ∈ R k − 2 [X]

^{^i[( ]Q%0[}

N (x) = (x − x i 0 ) ² Q(x)

^30[(5

Q = deg N − 2 et deg N ≤ k

P k3L$giZ fhk&^

N

^{Vk Vi0^iZ\[_k ^}

N ⁰ (x) = 2(x − x i ₀ )Q(x) + (x − x i ₀ ) ² Q ⁰ (x)

= (x − x i ₀ )[2Q(x) + (x − x i ₀ )Q ⁰ (x)].

0 kV<i0^ Zj[$k&^Df< \VhgB"K :L!5&f< \Z^MLl"B%YZhf k ^i[SH

ω ⁰⁰ (x)(x − x i 0 ) = (x − x i 0 )[2Q(x) + (x − x i 0 )Q ⁰ (x)]

= ⇒ ω ⁰⁰ (x) = 2Q(x) + (x − x i 0 )Q ⁰ (x).

P kWYfhg ^iZ#W(% \Z\[_g W V%0g

x = x i 0

Vkf

ω ⁰⁰ (x i 0 ) = 2Q(x i 0 )

^{Z [}

Q(x i 0 ) = ^{ω 00 (x} ₂ ^{i 0 )}

bcV,J, [

l ⁰ _{i 0} (x i 0 ) = 1 ω ⁰ (x i ₀ )

N (x) (x − x i ₀ ) ²

= Q(x) ω ⁰ (x i ₀ ) ,

kYV%")f hVk2"

l _i ⁰ ₀ (x i ₀ ) = _2ω ^{ω 00} 0 ^(x (x ⁱ i ⁰ 0 ⁾ )

bc[/]^%YZ WYgiV% [/]^%Y[

A ⁰ _{i 0} (x i 0 ) = 0

fhZ\k"iZ]^%Y[- \[ ^iX2L$Vg&j(, [

*&- %&

f 1 (x) = e ^x

^{W V<%Yg}

x ∈ [−3; 3]

f1 P1

Point d’interpolation Interpolation avec 1 point

–30 –20 –10 10 20

–3 –2 –1 1 x 2 3

f1 P1

Points d’interpolation Interpolation avec 2 points

0 5 10 15 20

–3 –2 –1 1 2 3

x

f1 P1

Points d’interpolation Interpolation avec 3 points

0 5 10 15 20

–3 –2 –1 1 2 3

x

f1 P1

Points d’interpolation Interpolation avec 4 points

0 5 10 15 20

–3 –2 –1 1 2 3

x

U

f 2 (x) = _(3+x) ¹ 2

W V<%Yg

x ∈] − 3; 3]

f2 P2

Point d’interpolation Interpolation avec 1 point

–1 –0.5 0.5 1 1.5 2

y

–3 –2 –1 1 2 3

x

f2 P2

Points d’interpolation Interpolation avec 2 points

–1 –0.5 0.5 1 1.5 2

y

–3 –2 –1 1 2 3

x

N9

f2 P2

Points d’interpolation Interpolation avec 3 points

–1 –0.5

0.5 1 1.5 2

y

–3 –2 –1 1 2 3

x

f2 P2

Points d’interpolation Interpolation avec 4 points

–1 –0.5

0.5 1 1.5 2

y

–3 –2 –1 1 2 3

x

9

f ₃ (x) = |x|cos(x)

^W7V%Yg

x ∈ [−3; 3]

f3 P3

Point d’interpolation Interpolation avec 1 point

–3 –2 –1 0 1

y

–3 –2 –1 1 x 2 3

f3 P3

Points d’interpolation Interpolation avec 2 points

–2 –1 0 1 2

y

–3 –2 –1 1 2 3

x

f3 P3

Points d’interpolation Interpolation avec 3 points

–3 –2 –1 0 1

y

–3 –2 –1 1 x 2 3

f3 P3

Points d’interpolation Interpolation avec 4 points

–3 –2 –1 0 1

y

–3 –2 –1 1 x 2 3

'?- - &) G"F%# & "!$# %& # &H!tG"yG"

;

k&^ [$g WYgBL ^ Vk"l j[("'5g f WYXY[("/H

D 1

`Vg&"B]^% Z# k E)f ]^% E%Yk W7VZ\k&^l3 Z\k&^ [_giW7V f ^iZ\Vk \[.3Y[(5g&L 32%W V *) k2+, [Zjk ^i[$g W V67

\f ^ [!%Ygc[!" ^'L(5f< W[(" ^ W7V%Yg&]^%YVZ" Vk[5g f WYXY[][(" ^' f^ fhk25[_k&^ [43Y[4 \f.V Vk2W ^ Z\Vhk

[_kW_[ W7VZjk ^

1 1 V%Yg \fVVk2W ^ Z\Vhk [8F#W VhkY[$k&^ Zj[( m \[ Vk g [(, fhgB]^%Y[=]^%Y[ fhW0WYgiV F#Z#, f ^ ZjVk [(" ^ ^ gBj!"

"if ^ Zm"&VfhZm" fhk&^ [ f [!W ]^%,f ^igi[ W7VZjk ^B" k2L$fhk2, VZ\k2" [! # \[ [!" ^ W_Vgig [(W ^ [ f h[(W ^igiVZm"

0 k

f- Z#, WYgi[!"B"iZjVkJ]Q%0[ W[(" ^$"B%Yg ^ V%#^K f W V<"iZj^iZ\VkJ3Y[!" W VhZ\k&^M"h]^%YZ Zm, W7Vg ^i[ [_^ k0Vk= \[

kYV<, iYgi[

1 1

V%Yg= \f V Vk2W ^ Z\Vhk.g f ^iZ\Vk0kY[( m \[ fhW0WYgiV F#Z#, f ^ ZjVk [(" ^ f<W(W_[$W0^ f<i2 \[ W7V%Yg ]^%,f ^ g [

W7VZ\k&^B" [8FUW$[_W0^ML-"B%Yg\ Z\k&^i[$g f< m \[ 9f

1 P kZ,k W7V%YgG \f'V Vhk2W_^iZ\Vk

f (x) = |x|× cos(x)

^Z# kYV<%2"flVf6 # #%-L_^B%23YZj[$gG j[ 30V, f Z\kY[

3Y[-32L8Z,kYZj^iZ\Vk 3Y[- f 32L_giZL$[

f ⁰ (x)

^{W V<%Yg'W XYVZ#"} Z\g$ j[(" W VhZ\k&^M"'3 Z\k&^ [_giW7V f ^ Z\Vhk 0 k kY[ W7[(%0^DW,f6")WYg [$k230gi[(L_giV W$fhg

f ⁰

^{k [("} ^QW,f6"\3L!Z,kYZj[ [$k W_[ W VZjk&^ a [!W \[4W X0VZ*F 3Y[ kYV<"DW7VZjk ^B" f WYWYgiV F#Z#, f ^iZ\Vk Vi0^ [_k^%Y[k[!" ^)W,f<"Df<"B" [ WYg&L(W$Zm"i[

aQZjk2"iZ Vk W$Vk" ^ f ^ [/]^%Y[. Zjk&^ [$g W V< f ^iZ\VkW7V *)#k+, Z\f< \[Dk [(" ^QW,f6"liYZj[$k f<3,f W0^ML_[ fhWU7

WYg VNF#Zm, f ^iZ\Vk 3Y[/W$[$g ^ fhZjkY[("$VVkW_^ ZjVk2"

1

V%0gQfhW0WYgiVQW XY[$g'%Yk0[/V Vhk2W_^iZ\Vk W,fhg'3Y[!")W7V *)#k+, [!" Zm 3[8F#Zm" ^ [/3 f<%0^igi[!"l,JL_^iXYVU30[("

kYVh^ f<,J, [_k ^-W_[( # j[("/3Y[!" "iW \Z\k0[(" ]^%YZ "iVk&^ 3Y[("W V *) k2+, [(" W,fhg-, VhgBW$[$f<%UF 4]Q%0[ kYV<%2"

f< m \Vk2" L_^M%3YZ\[_g$3,fhk2"' \[4W XYfhWYZj^igi[/"B%0Z fhk&^

$

4 "!$#%&R'.) pGHF%'?) 8 A\?Hz

C ^k

T#VZ^

f : [a, b] −→ R

^{0 k} 30Zj^']^%Y[

f

^[(" ^\32L_giZhf6i2 \[ "&%Yg f i " Z H

f

^{[(" ^\32L$g Z} f<i2 j[4"B%Yg f i f<% "i[_k2"l%"B%Y[!

U

f

^{[(" ^\32L$g Z f<i2 j[ S30giVZ^ [][$kf Z [}

x→a lim ⁺

f(x) − f(a)

x − a existe

[ ^)Vhk k0Vh^ [/W$[ ^i^i[- jZ#, Zj^i[

f ⁰ (a)

9

f

^{[(" ^\32L$g Z f<i2 j[ S5f<%2W XY[ [$ki Z [}

x→b lim ⁻

f (x) − f (b)

x − b existe

[ ^)Vhk k0Vh^ [/W$[ ^i^i[- jZ#, Zj^i[

f ⁰ (b)

Y [/ f , (, [4, f kYZ#j_gi[ Vk 32L!Z,k0Zj^' \[!"'32L$g ZL_[("\3 VgB3Yg [/"B%YW L$g Z\[(%0g\3:%Yk0[/V Vhk2W_^iZ\Vk 32L 7

Z,kYZj[4"B%Yg f i

D´ efinition :

^{0 k3YZj^ ]^% :%0kY[ V Vk2W ^ Z\Vhk}

f : E → F

^{[(" ^[3Y[} W( \f<"B" [

C ⁿ

"&%Yg ;

n ∈ N , I ⊂ E

^J"iZ

f

^{[(" ^ kV VZm" 32L_giZ f<i \[ "B%Yg ; [} ^="iZl" f

n ^i`eme

^{32L$g Z<L$[}

f ⁽ⁿ⁾

^{[(" ^}

W$Vhk ^iZ\k^%Y[

C ⁰

^{32L(" Z#5kY[ [$k2" [(, i2 j[ 3Y[!" V Vk2W ^ ZjVk2" W_Vk&^} ZjkQ%0[(" [_^

C ^∞

[_k2"i[!, i2 \[3Y[(" VVk2W_^iZ\Vk"

3YVk&^ ^ V<%0^ [!"\ j[("$32L_giZL$[!"l"&%2W(W_[("&"iZh[(" " Vk&^\W$Vk&^ Zjk^%Y[("

0 kf f< jVgB" \[(" Zjk2W( m%2"iZjVk2"K"&%YZ fhk&^ [!"/H

C ^∞ ⊂ . . . ⊂ C ⁿ⁺¹ ⊂ C ⁿ ⊂ . . . ⊂ C ¹ ⊂ C ⁰ .

N_

0 kf j[(" WYg VWYgiZmL_^BL("'"B%YZ fhk&^ [("/H

T#Z

f, g ∈ C ⁿ

^{"B%Yg ;} f< \VhgB" Vk f H

1

f + g ∈ C ⁿ

^{"B%Yg ; [_^DVk f}

(f + g ) ⁽ⁿ⁾ = f ⁽ⁿ⁾ + g ⁽ⁿ⁾ ,

1

∀ α ∈ K, αf ∈ C ⁿ

^{"B%0g ; [_^}

(αf ) ⁽ⁿ⁾ = αf ⁽ⁿ⁾ ,

1

T#Z

∀ t ∈ I, f(t) 6= 0

f< \VhgB"

1

f ∈ C ⁿ

^{"B%Yg ;}

4 54 ),FtGHF\'?)98& R'.) pGHF%'?) - '?)+?- F\%& Gz# '?) ! &

D´ efinition :

^{0 k} 32L8Z,kYZj^ W7V%Yg

m ≥ 1

^{\f V} Vk2W_^iZ\Vk

M ⁺ m : R −→ R x 7−→

x ^m si x > 0 0 sinon.

Graphe des fonctions M ⁺ 1 , M ⁺ 2 , M ⁺ 3

H

M1+

M2+

M3+

–0.5 0.5 1 1.5 2 2.5 3

y

–2 –1 1 2

x

)$*,+

.- +-

∀m ≥ 2, M _m ⁺

2]79?O=USj6aW.Z182Q29"E.< Z

(M _m ⁺ ) ⁰ = m(M _m−1 ⁺ ).

QUKLE.<c79bS5ZT9 6&E.<_g

M _m ⁺ |]0,+∞[

[(" ^ %YkY['V Vk2W ^ Z\Vhk W7V m) k2+<, [$3YVhk2WQ[! # \[ f WYW,fhg ^ Z\[_k&^

C ^∞

M _m ⁺ |]−∞,0[

[(" ^\ f V Vk2W ^ ZjVk kQ% # \[ 3YVk2W [( m \[ fhW0W,fhg ^iZ\[_k ^

C ^∞

0 kW7[(%#^Df< \Vg&"$W f< mW(%2 j[$g$ j[("$32L_giZ<L$[!".H

TU%Yg

]0, +∞[, (M _m ⁺ ) ⁰ = m(M _m−1 ⁺ )

^[ ^'"B%Yg

] − ∞, 0[, (M _m ⁺ ) ⁰ = 0 = m(M _m−1 ⁺ )

;

T"B% ^43YVhk2W 3 EL_^ f<i2 jZ\g\ f V VgB, % \[ [$k

n

a %#^ gi[!, [_k&^ 3YZ^

(M _m ⁺ ) ⁰ (0) = m(M _m−1 ⁺ )(0)

Z [

(M _m ⁺ ) ⁰ (0) = 0

0

k3L(, Vk&^ gi[4W_[_^ ^ [/VVgB, % \[ W,f g gBL(W!%Ygig [$k2W_[ "&%Yg

m ≥ 2

^H

X`

Initialisation : m = 2.

0 kf

M ₂ ⁺ (x) =

x ² si x > 0 0 sinon.

x→0 lim ⁺

M ₂ ⁺ (x) − M ₂ ⁺ (0)

x − 0 = lim

x→0 ⁺

x ² x = 0

x→0 lim ⁻

M ₂ ⁺ (x) − M ₂ ⁺ (0)

x − 0 = lim

x→0 ⁻

0 x = 0

3YVkW

M ₂ ⁺

^{[!" ^l32L$g Z f<i2 j[ "B%Yg}

R

^{[_^DVk fSiYZ\[_k}

(M ₂ ⁺ ) ⁰ (0) = 0.

H´ er´ edit´ e :

^Vhk "B%0WYW V<"i[= X ) W V ^ X2j!"i[ g f Z\[/?&%"B]^% f<% g f k25

m ≥ 2

^{[_^ Vk \f , Vk&^ g [}

W7V%Yg

m + 1

0 k f

M _m+1 ⁺ = M _m ⁺ × M ₁ ⁺

^30Vk2W

M _m+1 ⁺

^{[(" ^} 32L_giZhf6i2 \[ "B%0g

R

^{W$V,J, [} WYgiVQ32%YZ^S3Y[

VVk2W ^ ZjVk2"$32L_giZ f<i \[("$"&%Yg

R

(M _m+1 ⁺ ) ⁰ (0) = (M _m ⁺ ) ⁰ (0) × M ₁ ⁺ (0) + M _m ⁺ (0) × (M ₁ ⁺ ) ⁰ (0) = 0.

)$*,+ .- +- 3

M _m ⁺ ∈ C ^m−1 ( R ) , ∀ m ≥ 1.

QUKLE.<c79bS5ZT9 6&E.<_g

Y

L(, Vk&^ g Vk2"'W$[ ^ XL$Vg&j(, []W,fhg gBL!W(%Yg gi[$kW$[-"B%Yg

m ≥ 1

b [(" ^ g fhZW7V%Yg

m = 1

^{W f g$ f V Vk2W ^ Z\Vhk}

M ₁ ⁺

^[!" ^lW$Vk&^ Zjk^%Y[ VhZ\g$5gifhWYXY[

TU%YW0W V" Vk2" , fhZjk ^i[$k,f k ^ ]^%Y[

M _m ⁺ ∈ C ^m−1

f< \Vg&")Zm TVf6%0^ , Vk&^ g [$g ]^%Y[

M _m+1 ⁺ ∈ C ^m

0 g

M _m+1 ⁺ ∈ C ^m ⇐⇒ (M _m+1 ⁺ ) ⁰ ∈ C ^m−1

Y fhWYg&j(" j[ ^ XL$Vg&j(, [ WYg&L(W(L!3Y[$k&^ Vk f

(M _m+1 ⁺ ) ⁰ = (m + 1)M _m ⁺

^{[_^ W$V,J, [}

M _m ⁺ ∈

C ^m−1

Vhk[_k32L!32%YZ^']Q%0[

(M _m+1 ⁺ ) ⁰ ∈ C ^m−1

4 ),FtGHF\'?) 8,& R'?) GHF\'?)H%F),&

`[D^i[$gB, [)fhk25 \fhZ#" "iW2 jZ\kY[ 32L!"iZ#5hkY[l%0kY[l \f ^i^i['3Y[li7VZm"Y[8F#Z#i2 j['%0^ Zm \Z#"&L$[ W,f g j[(" 3Y[(" 7

"iZjk,f ^i[(%Yg&"DW7V%Yg ^ig f<W_[$g'%YkY[/W_V%YgBi7[ W,f<"&" fhk&^)W,fhg$3Y[!")W7VZ\k&^B"$3YVkYk2L!"

D´ efinition :

T#VZj[$k&^

I = [a, b]

^{[ ^}

a = x 0 < x 1 < . . . < x k < x k+1 = b

^%YkY[

"B%i 3YZ Z#"iZjVk

σ

^3Y[

[a, b]

0 k3L!Z,kYZ^

S m (x ¹ , . . . , x k ) = {s ∈ C ^{m − 1} ([a, b])/s |[ x _i , x _i+1 ] ∈ R m [X], i = 0, . . . , k}.

`[("/L( mL(, [$k&^M"/3Y[

S m (x ₁ , . . . , x k )

^" fhWYW7[( m \[_k ^/ j[(" VVkW_^ ZjVk2"4" W2 \ZjkY[("43 Vg&3Ygi[

m

f6"B"iV 7 W$ZmL$[!" \fS"&%2i 3YZ Z#" Z\Vk

σ

P %! fhg

S m (x 1 , . . . , x k ) contient R _m [X].

NC

g

(x 7→ M ⁺ _m (x − x i )) ∈ S m (x 1 , . . . , x k )

QUKLE.<c79bS5ZT9 6&E.<_g L Zm3Y[$k&^ [

U\1 V%Yg, Vhk ^igi[_g ]Q%0[h \f VVk2W ^ Z\Vhk

M _m ⁺ (.−x i ) : t 7→ M _m ⁺ (t−x i )) ∈ S m (x 1 , . . . , x k )

Vk 3YVhZj^'32L!, Vk&^igi[$g$]^%Y[JH

f

M _m ⁺ (. − x i ) ∈ C ^m−1

i

M _m ⁺ (. − x i )

W$VZjk2W$Zm3Y[Df [(W %Yk W7V m) k2+<, ['3Y[l3Y[(5g&L

≤ m

^{"B%Yg W} X,f<]^%Y[ Zjk ^i[$g 7 f< m \[

f

M _m ⁺ (x − x i ) =

(x − x i ) ^m si x > x i

0 sinon

M _m ⁺ (x − x i ) = (M _m ⁺ ◦ f )(x)

^{f [!W}

f :

R −→ R x 7−→ x − x i

f ∈ C ^∞

^{[ ^}

M _m ⁺ ∈ C ^m−1

^3YVhk2W

M _m ⁺ ◦ f ∈ C ^m−1

i

M _m ⁺ (. − x i ) _{|[x l} ,x _l+1 ] =

(. − x i ) ^m si x l ≥ x i

0 si x l < x i

3YVhk2W/3,fhk2" ^ V<%2"' \[("$W$f<" Vhk ^igiV%h[/%YkW7V *)#k+, [ 3Y[/3Y[(5g&L

≤ m

Exemple : Graphe de la fonction M ⁺ 2 (. − x 1 ) avec x 1 = −1

^{Vi0^ [_k^% WYfhg}

^ gifhk2"& f ^iZ\Vk[3Y[ h[(W_^i[(%Yg

(x 1 , 0)

^{W,fhg ^ Z\g 32%5hg} fhWYX0[-30[

M ₁ ⁺

0 1 2 3 4

–3 –2 –1 1

x

P %! K43

S m (x 1 , . . . , x k )

2]79\;=< 7AE.; 7` 2]7:CcZ=^]2#WT2]^9E.Sj6&21,O=2

C ^m−1 ([a, b])

^g

QUKLE.<c79bS5ZT9 6&E.<_g 1 fhg'32L8Z,kYZ^ Z\Vhk Vk f

S m (x 1 , . . . , x k ) ⊂ C ^m−1 ([a, b])

T#VZj[$k&^

s ₁ , s ₂ ∈ S m (x ₁ , . . . , x k )

f6 \Vg&" Z# Vf<%#^l, Vk&^ g [$g$]^%Y[

s ₁ + s ₂ ∈ S m (x ₁ , . . . , x k )

[_^

∀λ ∈ R, λs 1 ∈ S m (x 1 , . . . , x k ).

bcV,J, [

s 1 , s 2 ∈ S m (x 1 , . . . , x k ), s 1 , s 2 ∈ C ^m−1 ([a, b])

^{[ ^}

s 1 _{|[ xi,xi+1]} , s 2 _{|[ xi,xi+1]} ∈

R m [X] ∀i = 0, . . . , k

M (X 1 , . . . , X K )

^Nb

Y fhWYg&j("$ \[!"cWYg VWYg Z#L_^BL(" "B%YgK j[(" W( f6"B"i[!"

s 1 + s 2 ∈ C ^m−1 ([a, b])

^{[_^}

λs 1 ∈ C ^m−1 ([a, b])

Y [ W2 m%2" Vhk f Q% ]Q%0[

R m [X]

^{[(" ^ %Yk} [("iWYf<W$[ [!W_^ VhgiZ\[! 3YVkW

s 1 + s 2 ∈ R m [X]

[_^

λs ₁ ∈ R m [X]

^{P k5W,fhg ^} ZmW(%2 jZ\[$g

(s ₁ + s ₂ ) _{|[x i} ,x _i+1 ] ∈ R m [X]

^{[_^}

(λs ₁ ) _{|[x i} ,x _i+1 ] ∈ R m [X], ∀i = 0, . . . , k

0 kf , Vk&^ g&L4]Q%0[

s 1 + s 2 ∈ S m (x 1 , . . . , x k )

^{[ ^}

λs 1 ∈ S m (x 1 , . . . , x k )

4 z;8,

m (x ₁ , . . . , x _k )

)$*,+ .- +- 83 2<c7A2K 182!O=2]7 E.<J^9b6&E.<c7

{(t 7→ t ^l ) ∀ 0 ≤ l ≤ m, (t 7→ M ⁺ m (t − x i )) ∀ 1 ≤ i ≤ k}

E.SjKL2l;=<J2 ]Z.7A2 O=2

S m (x 1 , . . . , x k )

^g

[<LCcZTSj96&^;=1H6&2S

1Z OT6aKL2<c76&E.< O=2

S m (x 1 , . . . , x k )

2]79"U&RhZT182

1 + m + k

^g

QUKLE.<c79bS5ZT9 6&E.<_g Y fhWYgBj!" f WYgiVhW V" Zj^iZ\Vk WYg&L(W(L!3Y[$k&^i[ Vk f ]^%Y[

{(t 7→ t ^l ) ∀ 0 ≤ l ≤ m, (t 7→ M _m ⁺ (t − x i )) ∀ 1 ≤ i ≤ k}

^{[!" ^l%Yk" V%2" 7} [$k"i[(, i2 \[/3Y[

S m (x 1 , . . . , x k )

;

gi[!" ^i[ S, Vk&^ gi[_g$]^%Y[]^ V%0^

s ∈ S m (x 1 , . . . , x k )

^{" :L!W$g Zj^'3Y[/,} fhkYZ#j_gi[4%Yk0Z#]^%Y[SH

(∗) s(t) =

m

X

i=0

a i t ⁱ +

k

X

i=1

b i M _m ⁺ (t − x i ) t ∈ [a, b].

T#VZ^$%YkY[/V Vk2W ^ ZjVk

s ∈ S m (x ₁ , . . . , x k )

^{3YVk0k2L$[ 0 k W} V<"i[

p i (t) = s(t), t ∈ [x i , x i+1 ], i = 0, . . . , k

f< jVgB" Vk f

p 0 , . . . , p k ∈ R _m [X]

P kWYfhg ^iZ#W(% \Z\[_g

p 0 ∈ R _m [X]

^3YVk2W

p 0

" EL(W_giZj^\3Y[/, fhkYZmj$gi[ %YkYZ#]^%Y[SH

p 0 (t) =

m

X

i=0

a i t ⁱ , t ∈ [x 0 , x 1 ].

Y [ W2 m%2"

p 0 (x 1 ) = p 1 (x 1 ) = s(x 1 )

^{[ ^lW_V,J, [}

s ∈ C ^m−1 [a, b]

^{Vk f}

p ⁽ⁱ⁾ ₁ (x ₁ ) = p ⁽ⁱ⁾ ₀ (x ₁ ) ∀i = 0, . . . , m − 1.

bcV,J, [

p 1 ∈ R _m [X] p 1 − p 0 ∈ R _m [X]

W$[=]^%YZZ#, W2 \Zm]Q%0[J]^%Y[

p 1 − p 0

" EL(W$g Zj^ 3Y[

, fhkYZ#j_gi[ %0kYZ#]^%Y[SH

p 1 (t) − p 0 (t) = b 1 (t − x 1 ) ^m

W$fhg

x 1

gif<W$ZjkY[

m ^{i` eme}

^3Y[

p 1 −p 0

[_^

deg (p 1 −p 0 ) ≤ m = ⇒

^W

1 (t)−p 0 (t) = (t−x 1 ) ^m Q(t)

f [!W

deg Q = deg (p 1 − p 0 ) − m = 0

1

V<%Yg

t ∈ [x 0 , x 2 ], s(t) =

p 0 (t) si x 0 ≤ t ≤ x 1

p 1 (t) si x 1 ≤ t ≤ x 2

<n

1 f g\W_Vk2"&L(]^%Y[$k&^ Vk f

s(t) =

P m

i=0 a i t ⁱ si x 0 ≤ t ≤ x 1

P m

i=0 a i t ⁱ + b ₁ (t − x ₁ ) ^m si x ₁ ≤ t ≤ x ₂

Vk f f< \Vg&"

s(t) =

m

X

i=0

a i t ⁱ + b 1 M _m ⁺ (t − x 1 ) si x 0 ≤ t ≤ x 2 .

Y

[-, !, [ Vkf

p ⁽ⁱ⁾ ₂ (x 2 ) = p ⁽ⁱ⁾ ₁ (x 2 ) ∀i = 0, . . . , m − 1.

bcV,J, [

p 2 ∈ R _m [X] p 2 − p 1 ∈ R _m [X]

W$[=]^%YZZ#, W2 \Zm]Q%0[J]^%Y[

p 2 − p 1

" EL(W$g Zj^ 3Y[

, fhkYZ#j_gi[ %0kYZ#]^%Y[SH

p 2 (t) − p 1 (t) = b 2 (t − x 2 ) ^m

W$fhg

x ₂

gif<W$ZjkY[

m ^{i` eme}

^3Y[

p ₂ −p ₁

^{[_^}

deg (p ₂ −p ₁ ) ≤ m = ⇒

^W

₂ (t)−p ₁ (t) = (t−x ₂ ) ^m R(t)

f [!W

deg R = deg (p 2 − p 1 ) − m = 0

1

V<%Yg

t ∈ [x 0 , x 3 ], s(t) =

( p ₀ (t) si x ₀ ≤ t ≤ x ₁ p 1 (t) si x 1 ≤ t ≤ x 2

p 2 (t) si x 2 ≤ t ≤ x 3

1 f g W$Vk2"&L(]^%Y[_k ^ Vk f

s(t) =

( P m

i=0 a i t ⁱ si x ₀ ≤ t ≤ x ₁

P m

i=0 a i t ⁱ + b 1 (t − x 1 ) ^m si x 1 ≤ t ≤ x 2

P m

i=0 a i t ⁱ + b 1 (t − x 1 ) ^m + b 2 (t − x 2 ) ^m si x 2 ≤ t ≤ x 3

Vk f f< \Vg&"

s(t) =

m

X

i=0

a i t ⁱ +

2

X

i=1

b i M _m ⁺ (t − x i ) si x ₀ ≤ t ≤ x ₃ .

P kW_Vk&^ ZjkQ%Yfhk&^lW_[_^ ^ [/, L ^ XYVQ3Y[ Vk Vi0^ Zj[$k&^$Z,k,f6 \[(, [$k&^

∗

eVk&^igiVk2"' :%Yk0Z#W$Z^ML 30[

s

^H

TU%YW0W V" Vk2"\]^%Y[

P m

i=0 a i t ⁱ + P k

i=1 b i M _m ⁺ (t−x i ) = P m

i=0 a ⁰ _i t ⁱ + P k

i=1 b ⁰ _i M _m ⁺ (t−x i ) si x 0 ≤ t ≤ x k+1

[_^\, Vk&^ g Vk2"$]^%Y[

a i = a ⁰ _i

^{[ ^}

b i = b ⁰ _i ∀i

Y fhWYg&j("\ X^) W7Vh^ Xj("i[ Vkf H

h(t) =

m

X

i=0

a i t ⁱ +

k

X

i=1

b i M _m ⁺ (t − x i ) −

m

X

i=0

a ⁰ _i t ⁱ +

k

X

i=1

b ⁰ _i M _m ⁺ (t − x i ) si x 0 ≤ t ≤ x k+1

= 0

⇐⇒ h(t) = P m

i=0 (a i −a ⁰ _i )t ⁱ + P k

i=1 (b i −b ⁰ _i )M _m ⁺ (t −x i ) = 0 si x 0 ≤ t ≤ x k+1

efhZjk&^ [$kYfhk&^ L_^B%23YZjVk2"' EL(5&f6 \Zj^BL "&%Yg'W X,f<]^%Y[ Z\k&^ [_g f< # j[

[x i , x i+1 ]

^{W V%0g}

i = 0, . . . , k

^H

M (X 1 , . . . , X K )

^#

1

TU%0g

[x 0 , x 1 ] h(t) = P m

i=0 (a i − a ⁰ _i )t ⁱ + P k

i=1 (b i − b ⁰ _i )M _m ⁺ (t − x i ) = 0

0 g

M _m ⁺ (t − x l ) = 0, ∀ l = 1, . . . , k

^{3 V}

h(t) = P m

i=0 (a i − a ⁰ _i )t ⁱ

^{W$[ ]Q%0Z}

Zm, W2 jZ#]^%Y[

a i − a ⁰ _i = 0

^{3YVhk2W Vkf , Vk&^} g&L4]Q%0[

a i = a ⁰ _i ∀ i.

ef Z\k&^ [_k,fhk&^ , Vk&^ g Vk2"$]^%Y[

b i = b ⁰ _i ∀ i :

1

TU%0g

[x 1 , x 2 ] h(t) = P k

i=1 (b i − b ⁰ _i )M _m ⁺ (t − x i ) = 0

0 g

M _m ⁺ (t − x l ) = 0, ∀l = 2, . . . , k

^{[_^}

M _m ⁺ (t − x 1 ) 6= 0

^{3 V}

h(t) = (b ₁ − b ⁰ ₁ )M _m ⁺ (t − x ₁ )

W_[.]^%YZZm, W2 jZ#]^%Y[

b ₁ − b ⁰ ₁ = 0

1

TU%0g

[x 2 , x 3 ] h(t) = P k

i=2 (b i − b ⁰ _i )M _m ⁺ (t − x i ) = 0

0 g

M _m ⁺ (t − x l ) = 0, ∀ l = 3, . . . , k

^{[ ^}

M _m ⁺ (t − x ₂ ) 6= 0

^3V

h(t) = (b 2 − b ⁰ ₂ )M _m ⁺ (t − x i )

W_[/]Q%0ZZm, W \Z#]^%Y[

b 2 − b ⁰ ₂ = 0

P k W$Vk&^ Zjk^%,fhk&^\W$[_^ ^ [/,JL_^ X0VU3Y[4"&%Yg\W X,f<]^%Y[ Z\k&^ [_g f< # j[ Vkf H

1

TU%0g

[x k , x k+1 ] h(t) = (b k − b ⁰ _k )M _m ⁺ (t − x k ) = 0

0 g

M _m ⁺ (t − x k ) 6= 0

^{3 V}

h(t) = (b k − b ⁰ _k )M _m ⁺ (t − x k )

^{W$[ ]^%YZ Zm,} W2 jZ#]^%Y[

b k − b ⁰ _k = 0

0 kf , Vk&^ gBL4]^%Y[ W7V%Yg ^ V<%0^

i a i = a ⁰ _i

^{[_^}

b i = b ⁰ _i

),FtGHF\'?) 8! ,# '&-

Y

fhk2" W$[ W X,fhWYZ^ g [ Vk W$Vk2" Z#32j_gi[ [("iWYf<W$[

S ₃ (x ₁ , . . . , x k )

3Y[("JVVk2W ^ Z\Vhk2"S"iW2 jZ\kY[!"

3 VgB30gi[

9

f<"&"iVQW$Z#L f "B%2i 3YZ Zm"iZ\Vhk

σ

^H

a = x 0 < x 1 < . . . < x k < x k+1 = b

^bc[

"iVhk ^'30Vk2W 3Y[!"KV Vk2W ^ Z\Vhk2" 3Y[4W! f<"&"i[

C ² ([a, b])

3YVk&^$ \f gi[!" ^ g Z#W ^ Z\Vhk W X,f<]^%Y[ Zjk ^i[$g hf6 # \[

[x i , x _i+1 ]

^[!" ^l%YkW7V m) k2+<, [/3Y[-3Y[(5hgBL

≤ 3

^{Y f} WYgBj!" \[]^iX2L$Vg&j(, [ _

kYV%2"l" f Vk" ]^%Y[

S 3

[(" ^ %Yk [(" W,f<W_[$3Y[$3YZ#, [$k"iZ\Vhk

4+ k

^{0 k} h[(%0^/g&L(" V%23Yg [$ \[ WYg Vi2 mj(, [ 3 Z\k&^i[$giW7V \f ^ ZjVk

"B%0Z fhk&^/H

Probl` eme

^{H k ^} fhk&^3YVhkYk2L$[!"3Y[(" f< \[!%YgB"

f 0 , f 1 , . . . , f k , f k+1

^ giV<%[_g

s ∈ S 3 (x 1 , . . . , x k )

^ [! # j[ ]^%Y[

s(x i ) = f i pour i = 0, . . . , k + 1.

RD[!, f gB]^%YVk2" ]Q%Vk Zm, W7V" [

k + 2

W$Vhk23YZj^iZ\Vk" 3YVkW "iZVk [!%0^ [$k 5L_k2L$gif< %YkY[

"iV< #%0^iZ\Vk[%Yk0Z#]^%Y[ Zm Vf<%230g f g fX? V<%0^ [_g W_Vk23YZ^ Z\Vhk2"

54 G"!8, 8,& A'.)8,FtGHF\'?)

)Vh^iVk2"

s i

f g [(" ^ giZmW_^iZ\Vk3Y[ fSVVk2W ^ Z\Vhk W XY[_gBW X2L_[

s

^{J Z\k&^ [_g f< # j[}

[x i , x i+1 ]

^{W V%0g}

i = 0, . . . , k

DV%2"'" f hVk2" ]^%Y[

s i

[!" ^\%Yk W V< m) k2+, [ 3Y[/3Y[!5gBL

≤ 3

`[ W0giVi2 mj(, []Z#, W V<"i[ j[("$W$Vhk23YZj^iZ\Vk"$3Y[ g f<W!W$Vg&3Y[(, [$k&^\"B%YZ fhk&^ [("/H

W7V%Yg

i = 1, . . . , k

s i−1 (x i ) = s i (x i ) (continuit´e de s),

s ⁰ _i−1 (x i ) = s ⁰ _i (x i ) (existence et continuit´e de la d´eriv´ee premi´ere),

s ⁰⁰ _i−1 (x i ) = s ⁰⁰ _i (x i ) (existence et continuit´e de la d´eriv´ee seconde).

6D

)Vh^ f ^ ZjVk2" H 0 k W V<"i[ W7V%Yg

i = 0, . . . , k + 1

( s(x i ) = f i impos´ee par le probl`eme `a r´esoudre s ⁰ (x i ) = m i pure notation

s ⁰⁰ (x i ) = M i pure notation et pour i = 1, . . . , k + 1, h i = x i − x i−1 .

/#+-+-

E.6&2<F9

α < β

O=2;T4LSVU]217g 6

P ∈ R 3 [X]

^{ZT18E.SV7 E.< Z}

P ⁰⁰ (x) = P ⁰⁰ (α) (β − x)

β − α + P ⁰⁰ (β) (x − α) β − α

9Uj

QUKLE.<c79bS5ZT9 6&E.<_g `[(" , [(, i0gi[("S3Y[ :L!5&f< \Z^ML=" Vk&^ 30[(" W V *) k2+, [(" 3Y[ 3Y[!5gBL

≤ 1

1 V<%Yg , Vk&^ g [$g ]^% Z# m" " Vk&^KL!5&f<%UF Zm "B% ^$30[ , Vk&^ gi[_g ]^% Z# m" " Vk&^KL(5f<%UF [$k W VZjk&^M"

0 g W V<%Yg

x = β

^{[_^}

x = α

VhkfS EL(5f< \Z^ML

Remarque :

<sup>`f VVg&, %2 \[ 9 W$Vg gi[(" W Vhk23 W$[! # \[ 3Y[</sup> Z\k&^i[$giW7V \f ^ ZjVk 3Y[` f 7 5gifhk25[ fhW0W2 \Zm]Q%L$[ W7VZjk ^B"$3YZ#" ^ Zjk2W_^B"

P k[ [ ^ \f V VhgB, %2 j[ 5L_k2L$gif< \[]W7V%Yg

k

W7VZ\k&^M"

x i

[(" ^-H

P (x) =

k

X

i=1

f (x i )l i (x), ∀x ∈ R

[_^ "iZ'V][(" ^[%YkY[ V Vk2W ^ Z\Vhk W7V m) k2+, Zf6 \[=3Y[3Y[(5hgBL

≤ k − 1

f6 \Vg&"JV][!" ^ L(5&f< j[ "iVk W7V m) k2+, [ 3 Z\k&^i[$giW7V \f ^ ZjVk=3Y[ ` f<5gifhk25[

1 V<%Yg

k = 2

P (x) = f (x 1 )l 1 (x) + f (x 2 )l 2 (x)

= f (x 1 ) x − x ₂ x 1 − x 2

+ f(x 2 ) x − x ₁ x 2 − x 1

.

)V<%2" "if Vk2" ]^%Y[

s i−1

[(" ^K%Yk W V< m) k2+, ['3Y[l30[(5g&L

≤ 3

3YVkW)[_kfhWYW \Z#]^%,f k ^h j[\ j[(,J, [

s i−1

"&%Yg' Z\k&^ [_g f< m \[

[x i−1 , x i ]

^{Vhk ^igiV% [JH}

s ⁰⁰ _i−1 (x) = s ⁰⁰ _i−1 (x i−1 ) (x i − x) h i

+ s ⁰⁰ _i−1 (x i ) (x − x i−1 ) h i

.

0 g

s ⁰⁰ _i−1 (x i−1 ) = s ⁰⁰ (x i−1 )

^{[_^ 3Y[ \f., (, [ ,} fhkYZ#j_gi[

s ⁰⁰ _i−1 (x i ) = s ⁰⁰ (x i )

^{W,f gBW$[ ]Q%0[ \[!"}

V Vk2W ^ ZjVk2"$"i[ gif<W(W_VgB3Y[_k&^ iYZj[$k W X,f<]^%Y[.i V%#^'3 Zjk&^ [$g f< # j[

>Zjk,f< \[!, [_k&^ Vhkf \f g [( \f ^ ZjVk H

s ⁰⁰ _i−1 (x) = M i−1

(x i − x) h i

+ M i

(x − x i−1 ) h i

, pour x ∈ [x i−1 , x i ].

0

1

V<%Yg ^ giV<%[_g

s i−1

VkZjk&^Mj(5hgi[43Y[!%UF V VZ#" W$[_^ ^ [/VVgB, %2 \[ H

s i−1 (x) = ^M _h ⁱ⁻¹

i

(x i −x) ³ h i + ^M _h ⁱ

i

(x−x ³ _i−1

6 + C i−1 (x − x i−1 ) + ˜ C i−1 .

)V<%2" 3Y[ Vk2" 30Vk2W=32L ^ [_gB, Z\kY[_g \[!" W$Vhk2" ^ fhk&^ [("

C i−1

[_^

C ˜ i−1

1

V<%Yg \[!" 32L ^ [_gB, Z\kY[_g

kYV%")f6 # \Vhk2"K%0^iZ# \Zm"i[_gK \[!"$3Y[(%UF W_Vk23YZ^ ZjVk2"K"&%YZ fhk&^ [!"/H

f i−1 = s(x i−1 ) = s i−1 (x i−1 ) et f i = s(x i ) = s i−1 (x i ).

`[(" WYg [(, Z#j_gi[("$L!5&f< jZj^ML!" " Vk&^l3 [(" f<% WYg Vi2 mj(, [ ]^% Vhk 30VZj^ g&L(" V%23Yg [][_^' \[!"$"i[!W$Vk230[("

f<%[VfhZ^']Q%0[-"&%Yg' Z\k&^ [_g f< m \[

[x i−1 , x i ]

^{f V VkW_^ ZjVk}

s i−1

W$VZjk2W$Zm3Y[ f [(W- f VVkW_^ ZjVk

s i

Y L ^ [$g&, ZjkYVk2"$ j[("'W$Vhk2" ^ fhk&^ [("-H,[$k VfhZm" fhk&^

x = x i−1

W2%YZ#"

x = x i

Vk Vi#^ Z\[_k&^/H

f i−1 = s i−1 (x i−1 ) = M i−1

h i

(x i − x i−1 ) ³

6 + ˜ C i−1

= M _i−1

6 h ² _i + ˜ C i−1

f i = s _i−1 (x i ) = M i

h i

(x i − x _i−1 ) ³

6 + C _i−1 (x i − x _i−1 ) + ˜ C _i−1

= M i−1

6 h ² _i + C i−1 × h i + ˜ C i−1

bc[(W_ZT30VkYkY[

C ˜ i−1 = f i−1 − M i−1

h ² _i

6 et C i−1 = f i − f _i−1 h i

− h i

6 (M i − M i−1 ).

^9U

P k32L_giZhf k ^

s i−1

Vhk V<i0^ Zj[$k&^/H

s ⁰ _i−1 (x) = −3 M i−1

6 × h i

(x i − x) ² + 3 M i

6 × h i

(x − x _i−1 ) ² + C _i−1

3YVkW

s ⁰ _i−1 (x i ) = M i

2 (x i − x i−1 ) + C i−1

= M i

2 (x i − x i−1 ) + f i − f i−1

h i

− h i

6 (M i − M i−1 )

= h i

3 M i + h i

6 M i−1 + f i − f i−1

h i

.

<_

0 kf

s i (x) = M i (x i+1 −x) ³

6h i+1 + M i+1 (x−x i ) ³

6h i+1 + C i (x − x i ) + ˜ C i .

P k3L$giZ fhk&^

s i

Vk Vi#^ Z\[_k&^.H

s ⁰ _i (x) = M i (−3) (x i+1 − x) ² 6h i+1

+ 3M _i+1 (x − x i ) ² 6h i+1

+ C i

3YVkW

s ⁰ _i (x i ) = −M i

2 h _i+1 + C i

= −M i

2 h _i+1 + f i+1 − f i

h i+1

− h i+1

6 (M _i+1 − M i )

= −M i

3 h i+1 − M i+1

6 h i+1 + f i+1 − f i

h _i+1 .

; , W7V" Vk2" WYg&L(" [$k&^[ f W_Vk&^ Z\k^%YZ^ML 3Y[ f 32L$g ZL_[W0gi[(, Z#j_gi[ [$k

x i

W[(" ^ 3YZ\g [

s ⁰ _i−1 (x i ) = s ⁰ _i (x i )

^{0 k} Vi0^iZ\[$k&^Df< jVgB" %Yk"A)Q" ^Mj!, [- \Zjk2L f Z\gi[ HYW V<%Yg

i = 1, . . . , k, h i

6 M i−1 + h i

3 M i + f i − f i−1

h i

= −h i+1

3 M i − h i+1

6 M i+1 + f i+1 − f i

h i+1

⇐⇒ h i

6 M i−1 + (h i + h i+1 )

3 M i + h i+1

6 M i+1 = f i+1 − f i

h i+1

− f i − f i−1

h i

.

^9UI9

e%2 ^ ZjW2 \ZjVk2"

3.3

^cW,fhg

_h ⁶

i +h i+1

H

⇐⇒ _h ^{h i}

i +h i+1 M i−1 + 2M i + _h ^{h i+1}

i +h i+1 M i+1 = _h ⁶

i +h i+1 ( ^{f i+1} _{h i+1} ^{−f i} − ^{f i −f} _h ⁱ⁻¹

i )

0

kW7V" [JH

µ i = _h ^{h i}

i +h i+1 , λ i = _h ^{h i+1}

i +h i+1 , d i = _h ⁶

i +h i+1 ( ^{f i+1} _h ^{−f i}

i+1 − ^{f i −f} _h ⁱ⁻¹

i )

= ⇒ (∗) µ i M i−1 + 2M i + λ i M i+1 = d i , ∀i = 1, . . . , k.

`['"&)Q" ^Bj(, [

(∗)

^f

k + 2

Z\k2W_VkYk^%Y[!";[ ^

k

^{L!]Q%Yf ^ ZjVk2" W_Vk23YZ^} Z\Vhk2" gi[!" ^ [_k&^ 3YVhk2W 4ZUF0[_g P k5<L$k2L_g f< W$[!"lW_Vk23YZ^ Z\Vhk2"K"iVhk ^\3Y[/ \f V VhgB, [

2M 0 + λ 0 M 1 = d 0 et µ k+1 M k + 2M k+1 = d k+1

V

0 ≤ λ 0 , µ k+1 ≤ 1

^{[_^}

d 0 , d k+1

"iVk&^l3Y[(" f< \[!%YgB"'3YVhkYk2L$[!"

1

V%Yg$ZF#[$glW$[("\W$Vhk23YZ*7

^ ZjVk2" 3Y[!%UF W7V"B" Z#iYZm \Z^ML("h"iZm, W \[(" " V gi[$k&^ k0V%2"4H" VZj^cVk W7V"i[ ^iV%2"K j[("KW_V#[ W_Z\[_k ^B"

L(5f<%UF

0

^{" VZj^ VkW7V" [}

λ 0 = µ k+1 = 1 d 0 = d 1

[_^

d k+1 = d k