The document is a presentation on LU decomposition. It contains the following key points: 1. LU decomposition involves factorizing a matrix A into lower triangular matrix L and upper triangular matrix U, such that A = LU. 2. The LU decomposition method involves decomposing the matrix into L and U, then using forward and backward substitution to solve systems of equations. 3. An example is provided to demonstrate solving a system of equations using 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
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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]
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5. LU Decomposition Method
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    ULA 
1. Decompose
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Cont…

6. Decompose
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[U] =
LU Decomposition Method4
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7. Decompose
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[U] =
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[L] =
LU Decomposition Method5
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8. LU Decomposition Method
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b
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2. Forward substitution:
Given [L] and [B] find [Y]
    BYL 
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3. Backward substitution
Given [U] and [Y] find [X]     YXU 
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9. Solution:
Example
Problem:
Solve the following system of equations by LU decomposition.
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][A
Row3-Row112=Row3
Row2-Row14=Row2
8312
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5147
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321
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xxx
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10. 3
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23 RowRowRow 
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Example8
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11. ][][]][[ YforBYL 
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Now use forward substitution:
496.86
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Example9
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12. Now use backward substitution:
][][]][[ XforYXU 
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44.2244.2751
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Example10

13. Why LU Decomposition?
Total computational time for LU Decomposition is proportional to
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100
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onDecompostiLU
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 n
n
Total computation time for Gauss Elimination is proportional to
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Now, if n=100
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So LU Decomposition > Gauss Elimination
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14. )
2
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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?
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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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