-
7/31/2019 LU DECOPOSITION MATRIX
1/9
STR 613: Advanced
Numerical Analysis
Instructor
Dr. Ahmed Amir Khalil
Website
www.egypteducation.org
Make your own username and password
Structural Engineering --- Numerical analysis
Enrolment key: nacairo
Has notes, Matlab notes, and assignment
Assignment has been delayed for one week
Due date: Nov 26, 2006
-
7/31/2019 LU DECOPOSITION MATRIX
2/9
=
0001.6
3
0001.42
21
2
1
x
x
=
0001.6
3
42
21
2
1
x
x
=
6
3
42
21
2
1
x
x
=
6
3
0001.42
21
2
1
x
x
= 11
2
1
x
x
Infinite no. of solutions
Unique
= 03
2
1
x
xUnique
No solution
=
0
0
42
21
2
1
x
xTrivial solution or eigen value
For a system of linear equations,
there are four possibilities:
A unique solution (consistent set of
equations). Two intersecting lines
No solution (inconsistent set of equations).
Two parallel lines
Infinite number of solutions (consistent set
of equations). Two identical lines The trivial solution (set of
homogeneous
equations, [A]{x}=0) Two lines intersecting
at the origin
-
7/31/2019 LU DECOPOSITION MATRIX
3/9
Direct Methods (continued)
Lower
?
0?
00?
000?
xxx
xx
x
Upper
?000
?00
?0
?
x
xx
xxx
Solution is easy!
2100
9310
3101
-
7/31/2019 LU DECOPOSITION MATRIX
4/9
LU Decomposition
Ax=bA=LU
LUx=b
Let Ux= y
Ly=b get yUx=y get x
L-U Decomposition
A=LU
=
?00
??0
???
???
0??
00?
333231
232221
131211
c
b
a
c
b
a
aaa
aaa
aaa
Cholesky Method
Number of equations, Number of unknowns
What if A is symmetric?
-
7/31/2019 LU DECOPOSITION MATRIX
5/9
L-U Decomposition
A=LU
=
?00
??0
???
1??
01?
001
333231
232221
131211
aaa
aaa
aaa
=
1
?1
??1
???
??
?
333231
232221
131211
aaa
aaa
aaa
=
?
??
???
???
??
?
333231
232221
131211
c
b
a
c
b
a
aaa
aaa
aaa
Doolittle
Crout
Cholesky
Example using Doolittle
13
12
)5/2(
)5/4(
330
110
125
RR
RR
5332
34
325
321
321
321
+=+
=+
+=++
xxx
xxx
xx
-
7/31/2019 LU DECOPOSITION MATRIX
6/9
Example using Doolittle
23 )3/19(5/135/190
5/95/30125
RR
1400
5/95/30
125
=U
Example using Doolittle
1400
5/95/30
125
=U
13/195/2
015/4
001
=L
-
7/31/2019 LU DECOPOSITION MATRIX
7/9
Symmetric Matrix: A=AT
Banded Symmetric Matrix
xx
x
xx
xxxxx
xxxx
xxx
xxxx
xxx
xx
xx
00000000
000000000
00000000
00000
000000
0000000
000000
0000000
00000000
00000000
M
MS
Banded Symmetric Matrix
x
x
x
xxx
xx
xxx
xx
xx
xx
xx
000
000
000
0
00
0
00
00
00
00
MS
xx
x
xx
xxxxx
xxxx
xxx
xxxx
xxx
xx
xx
00000000
000000000
00000000
00000
000000
0000000
000000
0000000
00000000
00000000
MS
(I, J-I+1)(I, J)
-
7/31/2019 LU DECOPOSITION MATRIX
8/9
Banded NON-Symmetric Matrix
xx
x
xx
xxxxx
xxxx
xxx
xxxx
xxx
xxx
xx
00000000
000000000
00000000
00000
000000
0000000
000000
0000000
0000000
00000000
M
xx
x
xx
xxxxx
xxxx
xxx
xxxx
xxx
xxx
xx
00000
000000
00000
00
000
0000
000
0000
0000
00000
M
(I, J) (I, MS+J-I)
Flow ChartA diagram
that shows
a step-by-
step
progressio
n through a
procedure
or system
-
7/31/2019 LU DECOPOSITION MATRIX
9/9
Algorithm
A fully-specified step-by-step procedure forsolving a
mathematical problem in a finite
number of steps, often involving repetition of an
operation or more than one operation.
Clear, Detailed, Pseudo-code
Use
Read or Input
Write, Print, or Output
Decisions: If, Then, Else
Loops: Do, While; Repeat, Until, For i=1 to 10
Assignment Rules
Complete Cover-Sheet, Use A4 paper,
ELSE lose 20%
Deliver on due date ELSE lose 40%
No Plagiarism ELSE lose 100%
No Cheating ELSE lose 40%
Best solution(s) may be delivered in
electronic format for a bonus 100%
