Numerical Method LU Decomposition

2. Green University of Bangladesh
Presented by:
Md. Al-Amin
ID: 172015031
Dept. of CSE
Presented to:
Dr. Md. Monirul Islam
Distinguished Professor
Dept. of CSE
Course Name: Numerical Methods
Course Code: CSE-211
Topic: LU Decomposition

3. Outline
 What is LU decomposition?
 LU decomposition method
 Example
 Why LU decomposition?
 Conclusion
1

4. What is LU Decomposition?
LU decomposition was introduced by mathematician Tadeusz Banachiewicz in 1938.
In numerical analysis and linear algebra, LU decomposition of a matrix is the
factorization of a given square matrix into two triangular matrices, one upper
triangular matrix and another lower triangular matrix, such that the product of these
two matrices gives the original matrix.
Assume
[A] = Original matrix
[L] = Lower triangular matrix
[U] = Upper triangular matrix
[A] = [L][U]
2

5. LU Decomposition Method
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
    ULA 
1. Decompose
3
Cont…

6. Decompose
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[U] =
LU Decomposition Method4
Cont…

7. Decompose
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[U] =
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[L] =
LU Decomposition Method5
Cont…

8. LU Decomposition Method
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1
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001
b
b
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y
y
y


2. Forward substitution:
Given [L] and [B] find [Y]
    BYL 
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y
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3. Backward substitution
Given [U] and [Y] find [X]     YXU 
6

9. Solution:
Example
Problem:
Solve the following system of equations by LU decomposition.
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3112
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][A
Row3-Row112=Row3
Row2-Row14=Row2
8312
62944
5147
321
321
321

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xxx
xxx
xxx
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Cont…

10. 3
32
85
23 RowRowRow 
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41.1500
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][U
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1656.212
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Example8
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11. ][][]][[ YforBYL 
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8
62
51
0656.212
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3
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1
y
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Now use forward substitution:
496.86
612496.7068
266
20462
51
3
2
1
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496.86
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3
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y
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Example9
Cont…

12. Now use backward substitution:
][][]][[ XforYXU 
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496.86
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12.1
44.2244.2751
92.3
32
25.140266
61.5
41.15
496.86
1
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Example10

13. Why LU Decomposition?
Total computational time for LU Decomposition is proportional to
3333.343333
100
3
100
3
onDecompostiLU
2
3
2
3

 n
n
Total computation time for Gauss Elimination is proportional to
23
23
nn

Now, if n=100
2
3
3
n
n
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3333.338333
2
100
3
100
23
nEliminatioGauss
2323
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
nn
So LU Decomposition > Gauss Elimination
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Cont…

14. )
2
n
3
n
(m
23
)n(m
3
n 2
3

5
1033.8 
If the [B] vector changes then what will be happed…
Let m = the number of times the [B] vector changes
The computational times are proportional to
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1069.1 LU Decomposition = Gauss Elimination =
Gauss Elimination =LU decomposition =
Now LU Decomposition < Gauss Elimination
Why LU Decomposition?
12

15. Conclusion
LU Decomposition is provides an efficient means to compute the
matrix inverse. The inverse has a number of valuable application in
engineering practice. It also provides a means for evaluating
system condition.
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