LU
B=
9 6 3
6 3 1
1 0 1
.
LU B
L=
1 0 0
2
31 0
1
9
2
31
, U =
9 6 3
0−1−1
0 0 4
3
.
A
Bx =c c =
1
2
3
LU
B
LU
A=
0 3 1
4 1 1
2 2 4
.
LU P
L U P A =LU
P=
0 1 0
1 0 0
0 0 1
, L =
1 0 0
0 1 0
1
2
1
21
, U =
4 1 1
0 3 1
0 0 3
.
Ax =b b =
1
5
6
LU
A=
b1c1
a2b2c2
an−1bn−1cn−1
anbn
δk= det Akk= 1, . . . , n
δ0= 1 δk
δk=bkδk−1−akck−1δk−22≤k≤n.
LU A
1
a2δ0
δ11
a3δ1
δ21
an
δn−2
δn−11
δ1
δ0c1
δ2
δ1c2
δn−1
δn−2cn−1
δn
δn−1
Ax =b
A= (ai,j )1≤i,j≤nn
∀i∈ {1,2, . . . , n},|ai,i|>X
i6=j
|ai,j |.
A
(x1, . . . , xn)A
x1
x2
xn
=
0
0
0
A[ai,j ]1≤i,j≤kk=
1, . . . , n
10 7 8 7
7 5 6 5
8 6 10 9
7 5 9 10
u1
u2
u3
u4
=
32
23
33
31
1
1
1
1
10 7 8 7
7 5 6 5
8 6 10 9
7 5 9 10
u1+δu1
u2+δu2
u3+δu3
u4+δu4
=
32,1
23,9
33,1
30,9
9,2
−12,6
4,5
−1,1
(10−4u1+u2= 1
u1+u2= 2 (t= 3)
u1= 1,00010 . . . '1, u2= 0,99990 . . . '1.
10−4u1= 0, u2= 1.1
u1=u2= 1
1
/
3
100%
