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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56 Rezolvarea numerică a sistemelor algebrice liniare<br />
ai,i−1 = li,i−1, i = 2,n (4.2)<br />
aii = li,i−1ui−1,i +lii, i = 2,n (4.3)<br />
ai,i+1 = liiui,i+1<br />
Ordinea <strong>de</strong> obt¸inere este (4.2), (4.4), (4.3) alternativ<br />
Algoritmul:<br />
P1 l11 := a11<br />
u12 := a12/l11<br />
P2 fori = 2 ton−1<br />
li,i−1 := ai,i−1<br />
P3 ln,n−1 = an,n−1<br />
lii = aii −li,i−1ui−1,i<br />
ui,i+1 = ai,i+1/lii<br />
ln,n = ann −ln,n−1un−1,n<br />
4.2 Descompunere LUP<br />
Aici rolul luia11 va fi jucat <strong>de</strong>ak1.<br />
Efectul QA, Q matrice <strong>de</strong> permutare<br />
<br />
ak1 w<br />
QA =<br />
T<br />
v A ′<br />
<br />
1 0<br />
=<br />
v/ak1 In−1<br />
<br />
ak1 wT 0 A ′ −vwT /ak1<br />
<br />
(4.4)<br />
Matricea A ′ − vw T /ak1 se numes¸te complementul Schur al lui ak1 s¸i este<br />
nesingulară.<br />
Determinăm mai <strong>de</strong>parte <strong>de</strong>scompunerea LUP a complementului Schur<br />
Definim<br />
P ′ (A ′ −vw T /ak1) = L ′ U ′ .<br />
P =<br />
1 0<br />
0 P ′<br />
<br />
Q<br />
care este tot o matrice <strong>de</strong> permutare.<br />
Avem acum<br />
<br />
1 0<br />
PA =<br />
0 P ′<br />
<br />
1 0<br />
QA =<br />
0 P ′<br />
<br />
1 0<br />
<br />
=<br />
1 0<br />
P ′ v/ak1 P ′<br />
ak1w T<br />
v/ak1 In−1<br />
0 A ′ −vw T /ak1<br />
<br />
ak1 wT 0 A ′ −vwT /ak1<br />
<br />
=<br />
<br />
=