Beispiel - SAM - ETH Zürich

Abschätzungen für Legendre-Polynome
Numerische
Mathemtik
Erinnerung: Für{τ j } s j=1
ˆ= Gauss-Knoten in[−1,1] ➤ P(t) ˆ=s. Legendre-Polynom (Grads)
✎ Notation: P n ˆ= Legendre-Polynom vom Gradn ∈ N 0
Rekursionsformel: (n+1)P n+1 (t)−(2n+1)tP n (t)+nP n−1 (t) = 0 , (2.2.68)
Rodrigues-Formel: P n (t) = 1 d n
2 n n! dt n(t2 −1) n . (2.2.69)
(Start der Rekursion mit P 0 ≡ 1,P 1 (t) = t)
Legendre-Polynome auf[−1,1]
✄
1
Legendre polynomials on [−1,1]
R. Hiptmair
P 0 (x) = 1 ,
P 1 (x) = x ,
0.8
0.6
0.4
rev 35327,
25. April
2011
P 2 (x) = 1/2(3x 2 −1) ,
0.2
P 3 (x) = 1/2(5x 3 −3x) ,
P 4 (x) = 1/8(35x 4 −30x 2 +3) ,
P 5 (x) = 1/8(63x 5 −70x 3 +15x) ,
P 6 (x) = 1/(16)(231x 6 −315x 4 +105x 2 −5) .
P n
(t)
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
t
n = 0
n = 1
n = 2
n = 3
n = 4
n = 5
n = 6
n = 7
Fig. 66
2.2
p. 197