Lu Decomposition

Numerical Method
LU Decomposition

1. 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

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

3. 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

4. LU Decomposition Method 1. Decompose 3 Cont…

5. Decompose [U] = LU Decomposition Method 4 Cont…

6. Decompose [U] = [L] = LU Decomposition Method 5 Cont…

7. LU Decomposition Method 2. Forward substitution: Given [L] and [B] find [Y] 3. Backward substitution Given [U] and [Y] find [X] 6

8. Solution: Example Problem: Solve the following system of equations by LU decomposition. Row3-Row1 12 =Row3 Row2-Row1  4=Row2 7 Cont…

9. Example 8 Cont…

10. Now use forward substitution: Example 9 Cont…

11. Now use backward substitution: Example 10

12. Why LU Decomposition? T otal computational time for LU Decomposition is proportional to  T otal computation time for Gauss Elimination is proportional to  Now, if n=100 So  LU Decomposition > Gauss Elimination 11 Cont…

13. 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 LU Decomposition = Gauss Elimination = Gauss Elimination = LU decomposition = Now LU Decomposition < Gauss Elimination Why LU Decomposition? 12

14. 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. 13

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