Culegere de probleme de Analiz˘a numeric˘a

4.2. Descompunere LUP 57
=
1 0
=
P ′
ak1 w
v/ak1 In−1
T
0 P ′ (A ′ −vwT /ak1)
=
1 0
P ′
ak−1
v/ak1 In−1
wT 0 L ′ U ′
1 0
=
P ′ v/ak1 L ′
ak1 wT 0 U ′
= LU
De notat că în acest rat¸ionament atât vectorul coloană cât s¸i complementul
Schur se înmult¸esc cu matricea de permutareP ′ .
Problema 4.2.1 Să se calculeze descompunerea LUP a matricei
⎡ ⎤
2 0 2 0.6
⎢ 3 3 4 −2 ⎥
⎣ 5 5 4 2 ⎦
−1 −2 3.4 −1
Solut¸ie.
1 2 0 2 0.6
2 3 3 4 −2
3 5 5 4 2
4 −1 −2 3.4 −1
3 5 5 4 2
2 0.6 0 1.6 −3.2
1 0.4 −2 0.4 −0.2
4 −0.2 −1 4.2 −0.6
3 5 5 4 2
1 0.4 −2 0.4 −0.2
2 0.6 0 1.6 −3.2
4 −0.2 −1 4.2 −6
3 5 5 4 2
1 0.4 −2 0.4 −0.2
2 0.6 0 1.6 −3.2
4 −0.2 0.5 4 −0.5
3 5 5 4 2
1 0.4 −2 0.4 −0.2
4 −0.2 0.5 4 −0.5
2 0.6 0 1.6 −3.2
3 5 5 4 2
2 3 3 4 −2
1 2 0 2 0.6
4 −1 −2 3.4 −1
3 5 5 4 2
2 0.6 0 1.6 −3.2
1 0.4 −2 0.4 −0.2
4 −0.2 −1 4.2 −0.6
3 5 5 4 2
1 0.4 −2 0.4 −0.2
2 0.6 0 1.6 −3.2
4 −0.2 −0.5 4 −0.5