1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 20 Solution of Linear System of Equations - Iterative Methods

2. Iterative Methods Recall Techniques for Root finding of Single Equations Initial Guess New Estimate Error Calculation Repeat until Convergence

3. Gauss Seidel

4. First Iteration: Better Estimate

5. Gauss Seidel Second Iteration: Better Estimate

6. Gauss Seidel Iteration Error: Convergence Criterion:

7. Jacobi Iteration

8. First Iteration: Better Estimate

9. Jacobi Iteration Second Iteration: Better Estimate

10. Jacobi Iteration Iteration Error:

11. Example

12. Determinants Are composed of same elements Completely Different Mathematical Concept

13. Determinants Defined in a recursive form 2x2 matrix

14. Determinants Defined in a recursive form 3x3 matrix

15. Determinants Minor a 11

16. Determinants Minor a 12

17. Determinants Minor a 13

18. Solution of Small Systems of Equations – Cramer’s Rule 1. Compute

19. Solution of Small Systems of Equations – Cramer’s Rule 2. Compute

20. Solution of Small Systems of Equations – Cramer’s Rule 3. Compute

21. Solution of Small Systems of Equations – Cramer’s Rule 4. Compute

22. Solution of Small Systems of Equations – Cramer’s Rule If D =0 solution does NOT exist

23. Singular Matrices If D=0 solution does NOT exist Regardless of Method

24. Singular Matrices For Example {x} does not exist [A] -1 does not exist

25. Determinants and LU Decomposition {x} is not affected

26. Determinants and LU Decomposition {x} is not affected

27. Determinants and LU Decomposition

28. Example

29. After Elimination [A] becomes