Gauss Elimination & Gauss Jordan Methods in Numerical & Statistical Methods

Contents:
1.Gauss Elimination Method with example
2.Gauss Jordan Method with example

1. Subject :- Numerical and Statistical Methods Topic :- Gauss Elimination & Gauss Jordan Method

2. Contents System of Equations System of Linear Equation Solving Linear System of Equations Gauss Elimination Method Gauss Jordan Method Applications of Gaussian Method References

3. System of Equations A set of equations is called a system of equations . The solutions must satisfy each equation in the system. If all equations in a system are linear, the system is a system of linear equations , or a linear system .

4. System of Linear Equations Representation of system of linear equation : a 11 x 1 + a 12 x 2 + ··· + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + ··· + a 2n x n = b 2 . . . . a m1 x 1 + a m2 x 2 + ··· + a mn x n = b m

5. System of Linear Equations In matrix form : A X = B

6. Solving Linear System of Equations Linear System of Equations Direct Methods Gauss Elimination Method Gauss Jordan Method Iterative Methods Gauss Seidal Method Gauss Jacobi Method

7. Carl Friedrich Gauss 1777-1855

8. Gauss Elimination Method Write the system of equation in matrix form. Form the augmented matrix [a | b] Use row operations to transform the augmented matrix into the form Row Echelon Form (REF) Row Echelon Matrix

9. An elementary row operation is one of the following: Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row. Inspect the resulting matrix and re-interpret it as a system of equations No Solution Infinite no. of solutions Exactly one solution

10. Example : Q : Solve the following set of equations using Gauss Elimination Method x + y + z = 6 2x – y + z = 3 x + z = 4 Solution:

11. Now re-interpret the augmented matrix as a system of equations, starting at the bottom and working backwards ( back substitution ). 0x + 0y + z = 3 so z = 3 0x + y + 0z = 2 so y = 2 x+ y +z = 6. . Substitute the values z = 3 and y = 2 into the equation and get x = 1

12. Gauss Jordan Method Write the augmented matrix of the system Use row operations to transform the augmented matrix into the form Reduced Row Echelon Form (RREF) Reduced Row Echelon Matrix

13. An elementary row operation is one of the following: Interchange two rows. Multiply a row by a nonzero constant. Add a multiple of a row to another row. Inspect the resulting matrix and re-interpret it as a system of equations No Solution Infinite no. of solutions Exactly one solution

14. Example : Q : Solve the following set of equations using Gauss Jordan Method x + y + z = 5 2x + 3y + 5z = 8 4x + 5z = 2 Solution:

15. Applications of Gaussian Method To solve a system of linear equations with many unknowns Channel Decoding Algorithm Robust Fingerprint Image Enhancement Solving linear equations on mesh-connected processors Scheduling Algorithm

16. REFERENCES www.epcc.edu/Gauss-Jordan_Method www.Pages.pacificcoast.net/cazelais/Gauss-Jordan_elimination_Method.pdf www.personal.soton.ac.uk/workbook_8_3_gauss_elim.pdf

17. Thank You