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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58 Rezolvarea numerică a sistemelor algebrice liniare<br />
Verificare.<br />
⎛<br />
0 0 1 0<br />
⎜ 1 0 0 0<br />
⎝ 0 0 0 1<br />
0 1 0 0<br />
⎞⎛<br />
⎟⎜<br />
⎟⎜<br />
⎠⎝<br />
3 5 5 4 2<br />
1 0.4 −2 0.4 −0.2<br />
4 −0.2 0.5 4 −0.5<br />
2 0.6 0 0.4 −3<br />
2 0 2 0.6<br />
3 3 4 −2<br />
5 5 4 2<br />
−1 02 3.4 −1<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
⎟<br />
⎠ =<br />
5 5 4 2<br />
−2 0.4 −0.2<br />
0 4 −0.5<br />
−3<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
⎟<br />
⎠<br />
1<br />
0.4 1 0<br />
−0.2 0.5 1<br />
0.6 0 0.4 1<br />
Definit¸ia 4.2.2 Spunem că matricea A n × n este diagonal dominantă pe linii<br />
dacă<br />
n<br />
|aii| > |aij|, i = 1,n<br />
j=1<br />
j=i<br />
Problema 4.2.3 Să se rezolve sistemul<br />
folosind <strong>de</strong>scompunerea Cholesky.<br />
x1 +2x2 +x3 = 4<br />
2x1 +5x2 +3x3 = 10<br />
x1 +3x2 +3x3 = 7<br />
Solut¸ie. Calculând radicalii pivot¸ilor s¸i complementele Schur se obt¸ine:<br />
⎡ ⎤<br />
1 2 1<br />
⎡ ⎤<br />
1 2 1<br />
⎡ ⎤<br />
1 2 1<br />
B = ⎣ 5 3 ⎦ ∼ ⎣ 1 1 ⎦ ∼ ⎣ 1 1 ⎦.<br />
3 2 1<br />
Sistemele echivalente sunt<br />
⎧<br />
⎨<br />
⎩<br />
y1<br />
2y1+y2<br />
y1 +y2 +y3<br />
=<br />
=<br />
=<br />
4<br />
10<br />
7<br />
⎞<br />
⎟<br />
⎠
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