GAUSSIAN ELIMINATION,row reduction,numerical analyssi

1. Eng.Ansam Hadi Rashed
GAUSSIAN
ELIMINATION
2019
Al- Mustansiriya university
college of engineering
Computer Department
“Mathematics is the door and key
to the sciences. Roger Bacon”

2. Gaussian elimination
(also called row reduction) is a method used to solve systems of linear equations. It
is named after Carl Friedrich Gauss, a famous German
mathematician who wrote about this method,
Gaussian elimination include two methods
• Without pivot
• With pivot

3. Solving the linear system without pivot
We denote this linear system by Ax = b.
The augmented matrix
for this system is

4. Solving the linear system
1- eliminate x1 from equations 2, 3, and 4
R2=R2-2R1 M2,1=2
R3=R3+R1 M3,1=-1
R4=R4-2R1 M4,1=2
This will introduce zeros into the positions
below the diagonal in column 1, yielding

5. Solving the linear system
2- eliminate x2 from equations 3 and 4
R3=R3-2R2 M3,2=2
R4=R4-3R2 M4,2=3
This reduces the augmented matrix to

6. Solving the linear system
2- eliminate x3 from equation 4
R4=R4+R3 M4,3=-1
This reduces the augmented matrix to
Ms values take revers sign

7. Solving the linear system
3- Return this to the familiar linear system

8. Solving the linear system
There is a surprising result involving matrices associated with this
elimination process
U which resulted from the
elimination process.
L the lower triangular matrix constructed from M values ..

9. A = LU
when the process of Gaussian elimination without pivoting is applied to solving
a linear system Ax = b, we obtain A = LU

10. Solving linear system using LU Decomposition method
A = LU



11. Solving linear system using LU Decomposition method
 This implies

12. Solving linear system using LU Decomposition method
Ax=b A=LU
LUx=b
Ly=b

13. Solving linear system using LU Decomposition method
Ax=b A=LU
LUx=b to find Xs Ly=b && Ux=y

14. Example : Solve the following system of linear equations, by LU decomposition
method
Sol:
A=LU

15. LU
=L33

16. Therefore, we get,
Now, let , then implies L y =b U x=y

19. In the case in which partial pivoting is used P as a permutation matrix.
For example , consider
The matrix P A is obtained from A by switching around rows of A
The result LU = P A means that the LU - factorization is valid for the matrix A with its
rows suitably permuted.
Solving linear system with pivot

21. Solving the linear system with pivot

24. OPERATIONS COUNT

25. 2019
THANK YOU
ANY QUESTION??