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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Capitolul 4<br />
Rezolvarea numerică a sistemelor<br />
algebrice liniare<br />
4.1 Descompunere LU<br />
A =<br />
a11 w T<br />
v A ′<br />
A = LU<br />
<br />
1 0<br />
=<br />
v/a11 In−1<br />
<br />
a11 wT 0 A ′ −vwT /a11<br />
MatriceaA ′ −vw T /a11 se numes¸te complement Schur al lui a11.<br />
<br />
1 0<br />
=<br />
<br />
1 0<br />
A =<br />
v/a11 In−1<br />
v/a11 In−1<br />
a11 w T<br />
0 L ′ U ′<br />
a11 wT<br />
<br />
=<br />
0 a ′ −vwT /a11<br />
<br />
a11 wT <br />
=<br />
1 0<br />
v/a11 L ′<br />
Problema 4.1.1 Calculat¸i <strong>de</strong>scompunerea LU a matricei<br />
⎡ ⎤<br />
2 3 1 5<br />
⎢<br />
A = ⎢ 6 13 5 19 ⎥<br />
⎣ 2 19 10 23 ⎦<br />
4 10 11 31<br />
Solut¸ie.<br />
2 3 1 5<br />
3 4 2 4<br />
1 16 9 18<br />
2 4 9 21<br />
54<br />
0 U ′
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