Solving linear equation systems using direct methods: Gauss Jordan and Aitken methods
- 1. Solving linear equation systems using direct methods: Gauss Jordan and Aitken methods Presented by: Sulaiman M. KARIM Student No: 4013031617 E-mail: Sulaiman_musaria@yahoo.Com
- 2. Outlines INTRODUCTION METHODS OF SOLUTION GAUSS - JORDAN METHOD LINEAR SYSTEM OF EQUATIONS GAUSS - JORDAN METHOD THEORY EXAMPLES AITKEN METHOD EXAMPLES
- 3. A system of equations is a collection of two or more equations with a same set of unknowns (x1,x2,… xn). In solving a system of equations, we try to find values for each of the unknowns ,that will satisfy every equation in the system. The equations in the system can be linear or non-linear. Here we are interested in solving linear equations systems using two direct methods: 1- Gauss Jordan method 2- Aitkem method Introduction
- 4. There are two classes of methods for solving linear systems equations:- Class 1: Direct methods : The most known methods are:- * Gauss - Elimination method, * Gauss - Jordan method, * Matrix inversion method, * Aitken method. 2- Indirect methods (or iterative methods): * Jacobi’s method . * Gauss - Seidel method . Methods of solution
- 5. Gauss elimination method (with back substitution) is named after Carl Gauss how formulated this method in 1810. Gauss – Jordan method is an extension of this methods proposed by W. Jordan in 1888. Using a sequence of row reduction techniques, a system of linear equations can be resolved without back substitutions. Gauss – Jordan method provides a direct method for obtaining the solution of linear system of equations. The methods of Gauss-Jordan and Gauss elimination can look almost identical, the former requires approximately 50% fewer operations. Gauss - Jordan method
- 6. Linear system of equations
- 7. Gauss - Jordan method theory
- 8. Example (1): system of two equations
- 9. Example (2): system of three equations
- 10. Example (2): system of three equations
- 11. Example (3): system of three equations
- 12. Example (3): system of three equations
- 13. 2- Aitken Method
- 14. 2- Aitken Method
- 15. 2- Aitken Method
- 16. Aitken Method- Example (1)
- 17. Aitken Method- Example (1)
- 18. Aitken Method- Example (1)
- 19. Aitken Method- Example (2)
- 20. Aitken Method- Example (2)
- 21. Aitken Method- Example (2)