Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

3.3. LU FACTORIZATION 31
appropriately:
This continues:
⎡
⎢
⎣
−3
8
−2
−1
⎡
⎢
⎣
−1
8
2
1
⎤ ⎡
⎥
⎦ ⇒ ⎢
⎣
⎤ ⎡
⎥
⎦ ⇒ ⎢
⎣
− 12
5
8
−2
−1
−3
8
−2
−1
⎤
⎥
⎦
⎤ ⎡
⎥
⎦ ⇒ ⎢
⎣
We then perform a permuted backwards substitution on the augmented system
⎛
27 1
0 0
10 5
− 12 ⎞
5
⎜ 4 2 1 2 8
⎟
⎝
17
0 0 0
9
− 2 ⎠
3
5 7 1
0
2 4 2
−1
This proceeds as
Fill in your own values here.
3.3 LU Factorization
We examined G.E. to solve the system
where A is a matrix:
− 12
5
8
− 2 3
−1
⎤
⎥
⎦
x 4 = −2 9
3 17 = −6
17
x 3 = 10 (
− 12
27 5 − 1 )
−6
= . . .
5 17
x 2 = 2 (
−1 − 1 −6
5 2 17 − 7 )
4 x 3 = . . .
x 1 = 1 (
8 − 2 −6
)
4 17 − x 3 − 2x 2 = . . .
⎡
A =
⎢
⎣
Ax = b,
⎤
a 11 a 12 a 13 · · · a 1n
a 21 a 22 a 23 · · · a 2n
a 31 a 32 a 33 · · · a 3n
.
. . . ..
⎥
. ⎦
a n1 a n2 a n3 · · · a nn
We want to show that G.E. actually factors A into lower and upper triangular parts, that is A = LU,
where ⎡
⎤ ⎡
⎤
1 0 0 · · · 0
u 11 u 12 u 13 · · · u 1n
l 21 1 0 · · · 0
0 u 22 u 23 · · · u 2n
L =
l 31 l 32 1 · · · 0
, U =
0 0 u 33 · · · u 3n
.
⎢
⎣
.
. . . ..
⎥ ⎢
. ⎦ ⎣
.
. . . ..
⎥
. ⎦
l n1 l n2 l n3 · · · 1
0 0 0 · · · u nn
We call this a LU Factorization of A.
.