Beispiel - SAM - ETH Zürich

⎛ ⎞
f 1 (y)
f 2 (y)
f 3 (y)
f(y) =
f 4 (y)
⎜ .
⎟
⎝f d−1 (y) ⎠
f d (y)
=
⎛ ⎞
p 1 (y)
q 1 (y)
0
0
⎜ ⎟
⎝
0. ⎠
0
+
⎛ ⎞
0
p 2 (y)
q 2 (y)
0
⎜ ⎟
⎝
0. ⎠
0
+
⎛
0
0
p 3 (y)
q 3 (y)
⎜ ⎟
⎝ ⎠
0.
0
⎞
+···+
⎛ ⎞
0.
0
0
⎜p d−2 (y)
⎟
⎝q d−2 (y) ⎠
0
+
⎛ ⎞
0.
0
0
⎜ 0
⎟
⎝p d−1 (y) ⎠
q d−1 (y)
.
Numerische
Mathemtik
Beweis. (vgl. [16, Theorem 9.3])
Konstruktiv fürf = (f 1 ,...,f d ) T mit beliebigena i ∈ R
p i (y) = f i (y)+r i (y) , q i (y) = −r i+1 (y) ,
∫ y i ( ∂f1
r i (y) = +···+ ∂f )
i−1
(y
∂y 1 ∂y 1 ,...,y i−1 ,τ,y i+1 ,...,y d ) dτ , 2 ≤ i ≤ d−1 ,
i−1
a i
⇒
r 1 (y) ≡ 0 ,
∂p i
= ∂f i
+ ∂f 1
+···+ ∂f i−1
,
∂y i ∂y i ∂y 1 ∂y i−1
∂q i
= − ∂f 1
−···− ∂f i
= − ∂p i
.
∂y i+1 ∂y 1 ∂y i ∂y i
1 ≤ i ≤ d−1 .
R. Hiptmair
rev 35327,
24. Juni
2011
4.2
p. 454