Culegere de probleme de Analiz˘a numeric˘a
Culegere de probleme de Analiz˘a numeric˘a
Culegere de probleme de Analiz˘a numeric˘a
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4.1. Descompunere LU 55<br />
⎛<br />
⎝<br />
13 5 15<br />
15 10 23<br />
10 11 31<br />
A ′ −vw T /a11 =<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
⎛<br />
⎠−⎝<br />
3<br />
1<br />
2<br />
9 18<br />
9 21<br />
2 3 1 5<br />
6 13 5 19<br />
2 19 10 23<br />
4 10 11 31<br />
⎞<br />
⎠ <br />
⎛<br />
<br />
3 1 5 = ⎝<br />
⎛<br />
= ⎝<br />
<br />
4<br />
−<br />
1<br />
13 5 15<br />
15 20 23<br />
⎞<br />
⎛<br />
⎠−⎝<br />
10 11 31<br />
⎞<br />
4 2 4<br />
16 9 18 ⎠<br />
4 9 21<br />
2 3 1 5<br />
3 4 1 4<br />
1 4 1 2<br />
2 1 7 17<br />
<br />
(2,4) =<br />
2 3 1 5<br />
3 4 2 4<br />
1 4 1 2<br />
2 1 7 3<br />
9 18<br />
9 21<br />
A ′ −vw T /a11 = 17−7·2 = 3<br />
⎞<br />
⎟<br />
⎠ =<br />
⎛<br />
⎜<br />
⎝<br />
1 0 0 0<br />
3 1 0 0<br />
1 4 1 0<br />
2 1 7 1<br />
⎞⎛<br />
⎟⎜<br />
⎟⎜<br />
⎠⎝<br />
<br />
8 16<br />
−<br />
2 4<br />
2 3 1 5<br />
0 4 2 4<br />
0 0 1 2<br />
0 0 0 3<br />
9 3 15<br />
3 1 5<br />
6 2 10<br />
<br />
=<br />
⎞<br />
⎟<br />
⎠<br />
⎞<br />
⎠ =<br />
1 2<br />
7 17<br />
Problema 4.1.2 (Sisteme tridiagonale) Dat¸i algoritmul <strong>de</strong> <strong>de</strong>scompunere LU pentru<br />
o matrice tridiagonală.<br />
Timp liniar<br />
El. Gaussiană<br />
Factorizare Crout vii = 1<br />
Factorizare Doolittlelii = 1<br />
Exemplu. Crout<br />
⎛<br />
⎜<br />
L = ⎜<br />
⎝<br />
l11 0 ... 0<br />
l21 l22 ... 0<br />
.<br />
. . .. .<br />
0 0 ... lnn<br />
a11 = l11<br />
⎞<br />
⎟<br />
⎠<br />
⎛<br />
1 u2 ...<br />
⎜ 0 1<br />
U = ⎜<br />
⎝ .<br />
. . ..<br />
⎞<br />
0<br />
⎟<br />
. ⎟<br />
un−1,n<br />
⎠<br />
0 0 ... 1<br />
(4.1)
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