Graphics Programs Reference
In-Depth Information
EXAMPLE 2.5
Use Doolittle's decompositionmethod to solve the equations
Ax
=
b
, where
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1
4
1
7
13
5
A
=
b
=
16
−
1
2
−
12
Solution
We first decompose
A
by Gauss elimination. The first pass consists of the
elementary operations
row2
←
row2
−
1
×
row1(eliminates
A
21
)
row3
←
row3
−
2
×
row1(eliminates
A
31
)
Storing themultipliers
L
21
=
1 and
L
31
=
2inplace of the eliminated terms, weobtain
⎡
⎣
⎤
⎦
1 4 1
12
A
=
−
2
2
−
9
0
The second pass of Gauss eliminationuses the operation
←
−
−
.
×
row3
row3
(
4
5)
row2(eliminates
A
32
)
Storing the multiplier
L
32
=−
4
.
5inplace of
A
32
, we get
⎡
⎣
⎤
⎦
1
4
1
A
=
[
L
\
U
]
=
12
−
2
2
−
4
.
5
−
9
The decompositionis now complete, with
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1 00
110
2
1 4
1
L
=
U
=
0 2
−
2
−
4
.
51
00
−
9
=
b
by forward substitution comes next. The augmented coeffi-
cientform of the equations is
Solution of
Ly
⎡
⎣
⎤
⎦
1 007
110
L
b
=
13
2
−
4
.
51 5
The solutionis
y
1
=
7
y
2
=
−
y
1
=
−
=
13
13
7
6
y
3
=
5
−
2
y
1
+
4
.
5
y
2
=
5
−
2(7)
+
4
.
5(6)
=
18
Search WWH ::
Custom Search