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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4.3. Sisteme <strong>de</strong> ecuat¸ii 65<br />
TJm = 2<br />
= TJl<br />
5<br />
TJk 1−TJ x(1) −x (0) < ε<br />
T 7 7 7<br />
, , , x1 =<br />
5 5 5<br />
7<br />
5<br />
k 2<br />
x (0) = 0, x (1) =<br />
5<br />
3<br />
5<br />
· 7<br />
5<br />
2k 7<br />
= ·3·<br />
5k−1 5<br />
< ε<br />
k 2<br />
·21 < ε, k(ln2−ln5)+ln21 > lnε<br />
5<br />
Pentru Gauss-Sei<strong>de</strong>l x (0) = 0<br />
x (1)<br />
2<br />
x (1)<br />
3 = 1<br />
5<br />
1 (1)<br />
= (7−a21x 1<br />
5<br />
x (1)<br />
1<br />
1 7<br />
= (7) =<br />
5 5<br />
(0) 7 1 7<br />
−a23x 2 ) = − ·<br />
5 5 5<br />
(7−a31x (1)<br />
1 −a32x (1)<br />
2 ) = 7<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
U = ⎝<br />
x (1) −x (0) ∞ =<br />
<br />
1 1<br />
= 7 − =<br />
5 25<br />
28<br />
25<br />
1 7 28 35−7−28<br />
− · − =<br />
5 5 5 25 25<br />
<br />
<br />
<br />
7 28<br />
,<br />
5 25 ,0<br />
<br />
<br />
<br />
7<br />
=<br />
5<br />
x (k) = (D −L) −1 Ux (k−1) +(D −L) −1 b<br />
a11x (k)<br />
1 = −a12x (k−1)<br />
2 −···−a1nx (k−1)<br />
n +b1<br />
a21x (k)<br />
1 +a22x (k)<br />
2 = −a23x (k)<br />
3 −···+b2<br />
...<br />
an1x (k)<br />
1 +an2x (k)<br />
2 +···+annx (k)<br />
n = bn<br />
⎛<br />
D = ⎝<br />
⎛<br />
5 0 0<br />
0 5 0<br />
0 0 5<br />
⎞<br />
⎛<br />
⎠, L = ⎝<br />
⎞<br />
0 0 0<br />
−1 0 0<br />
−1 −1 0<br />
0 −1 −1<br />
5 0 0<br />
0 0 −1 ⎠, E = D −L = ⎝ 1 5 0<br />
0 0 0<br />
1 1 5<br />
<strong>de</strong>tE = 125, E T ⎛ ⎞<br />
5 1 1<br />
= ⎝ 0 5 1 ⎠<br />
0 0 5<br />
⎛<br />
⎞<br />
⎠<br />
⎞<br />
⎠<br />
= 0
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