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This document summarizes the Gaussian elimination method for solving systems of linear equations. It discusses: 1) Gaussian elimination involves eliminating variables from equations to put the system in triangular form without or with pivoting. 2) When solving without pivoting, the process introduces zeros below the diagonal and results in A = LU decomposition, where L is lower triangular and U is upper triangular. 3) To solve for x, the system is decomposed into Ly = b and Ux = y and then solved step-by-step. 4) Pivoting may be used by permuting rows to choose optimal elements for elimination and ensures the method remains numerically stable.
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Gaussian
- 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