Polynôme de Lagrange

Problème

∈ ! !

" # ! !

! $ % &

' ( )

$ )

* +

&

&

, - - , ,

. / )

&

&

, - - , ,

.+

&

&

, - - , , _.+ )

&

&

, - - , ,

.+

Polynôme de Lagrange

* ! 0! 0 % % ! )

* + 0 , 0 * + * + )

0 , 0 +

0 , 0 + 0 +. 0 +

0 + 0 +.

0 $ 0 0 $ % )

0 +

&

& 0 +

&

&

!" " )

/ $ 0 % % )

0 +∏

≠= −

−

. 1

1

0 + 0 1 +. ≠1

! 2 $ *

* + 0

=. 3 * +

4 4 4

5 5 5 4

6 6 6

0 $ ! )

* +

& & ₅ & ₆

& &

5 &

6

,

& & ₅ & ₆

& &

5 &

6

,

& & & ₆

5&

5&

5&

6 5,

& & &

5 6&

6&

6&

5 6

#$ %%

/ ) → 7 $ ! ! 3

&π

. . π

.

7 *π

5

π 5 *π

6

π 6

* +

&π

&π

&π

&π ×. , ,π

&π .,π

.&π × ,

,π π,π π

&.

×.

* +& 6

π & π

6 +&6

π ,

*π

6 +. 89 π

6 +. 8.8 :*π

5 +. 99 π

5 +. 9

& % " "!

= + = − +

$ %

;

)

∂

∂∂

∂

. .

.

.

.

.

.

. .

. .

=

= ⇔

− − − =

− − − =

⇔

+ =

+ =

=

=

= =

= = =

3 / 2 !

< )

=

= =

=

= . =

. =

−

−

=

−

−

= = = =

= =

= =

=

= =

5

' % "

% = 3 /

! &2& % 2

3 > 3 ! &2&

;

2 3 )

? + − +3 +

=

; ?

2 3 )

∂

∂ +& − +3 +

=

: ∂

∂ +& − +3 +

=

∂

∂ +& − +3 +

=

=

=

=

=

−

−

− ⁶ 3 ⁵ +. : − −3 −

=

=

=

+.

=

=

=

=

−

−

− ⁵ 3 +.

7 / )

=

=

=

=

+ +

= ⁶ 3 ⁵

=

=

=

=

+ +

= ⁵ 3

3 +

+

=

=

=

=

@ / ' 3 0 /

=A +4 A +=B4

@ = +

= =

= = =

= = =

5

5 6

4 +

=

=

=

= . : := :5 :=₅ & :=

6 6 :=

9 8

0 =

9 C

C 8

8 CC

4

9 C5 9

: ) +. 6C6 :3+. 8DD6 : + DDE

-2 -1 0 1 2 3 4

0 2 4 6 8 10 12 14

#$ %%

* ! % F

G ?! /

*Gγ+7 Hγ 7 !

I % 3 *Gγ+7

γ 7

I ! * G * .. ⁵

J 3

G ⁵ 96 5 C E 8 6 EE8 EC D6

* ,γ G + 7 * +&γ G , 7

@ γ 7 * G

J 3

G 96 5 C E 8 6 EE8 EC D6

G 5 DD69 6 5D 6 E 6 6E95 6 889E 9 C8D

* K%L C 6D9 58 C E6 D .

* 6 6 5 D. . 5 C 8. 5 56C6 D96D 5 9

γ+& 6.6 7 ++DC8E

!H7 + 9DCC 0! * G *Gγ+ 9DCC : G + .. *+ 9 5.66 ⁵