Culegere de probleme de Analiz˘a numeric˘a

100 Interpolare
Pentru funct¸ia spline limitată:
⎡
⎤
2h0 h0 0 ... ... 0
⎢ h0 2(h0 ⎢ +h1) h1 ... ... ... ⎥
⎢
A = ⎢ 0 h1 2(h1 +h2) h2 ... 0 ⎥
⎢ ... ... ... ... ... 0 ⎥
⎣ ... ... ... hn−2 2(hn−2 +hn−1) hn−1 ⎦
0 ... ... 0 hn−1 2hn−1
⎡
⎢
b = ⎢
⎣
3
h0 (a1 −a0)−3f ′ (a)
3
h1 (a2 −a1)− 3
h0 (a1 −a0)
.
3
hn−1 (an −an−1)− 3
hn−2 (an−1 −an−2)
3f ′ (b)− 3
hn−1 (an −an−1)
bj = 1
hj
hj = xj+1 −xj
(aj+1 −aj)− hj
3 (2cj +cj+1)
dj = cj+1−cj
, n = 3
3hj
a0 = 0, a1 = 1
2 , a2 = 1, a3 = 0
h0 = 1
6 , h1 = 1 1 1
− =
2 6 3 , h2 = 1
2
⎡ ⎤
⎡
⎢
b = ⎢
⎣
1 0 0 0
⎢
A = ⎢ 1 1
⎣
1
3 0
0 1
⎥
5 1 ⎦
3 3 2
0 0 0 1
0
3
1 ·
3
1
2 − 3 1 ·
6
1
2
(−1)− 3 1 ·
3
1
⎤
⎥
2 ⎦
0
=
⎡
⎢
⎣
⎡ 1 1
0 0 3 6
⎢ 1 1
A = ⎢ 1 6 3
⎣
0
0 1
⎤
⎥
3 1 ⎦
3 5 2
0 0 0 1
3 1
2
0
−9 2
−15 2
0
⎤
⎥
⎦
f ′ (x) = πcosπx f ′ (0) = π f ′ (1) = −π
⎤
⎥
⎦