1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 15 Solution of Systems of Equations

2. Objectives Introduction to Matrix Algebra Express System of Equations in Matrix Form Introduce Methods for Solving Systems of Equations Advantages and Disadvantages of each Method

3. Last Time Linear Equations in Matrix Form

4. # Equations = # Unknowns = n Square Matrix n x n

5. Last Time Solution of Linear Equations Express In Matrix Form Upper Triangular What is the characteristic? Solution by Back Substitution

6. Last Time Solution of Linear Equations Objective Can we express any system of equations in a form 0

7. Last Time Background Consider (Eq 1) (Eq 2) Solution 2*(Eq 1) (Eq 2) Solution !!!!!! Scaling Does Not Change the Solution

8. Last Time Background Consider (Eq 1) (Eq 2)-(Eq 1) Solution !!!!!! (Eq 1) (Eq 2) Solution Operations Do Not Change the Solution

9. Last Time Gauss Elimination Example Forward Elimination

10. Last Time-Correction Gauss Elimination -

11. Substitute 2 nd eq with new

12. Last Time-Correction Gauss Elimination -

13. Substitute 3rd eq with new

14. Last Time-Correction Gauss Elimination -

15. Substitute 3rd eq with new

16. Last Time-Correction Gauss Elimination

17. Last Time Gauss Elimination Forward Elimination

18. Last Time-Correction Gauss Elimination Back Substitution

19. Gauss Elimination – Potential Problem Forward Elimination

20. Gauss Elimination – Potential Problem Division By Zero!! Operation Failed

21. Gauss Elimination – Potential Problem OK!!

22. Gauss Elimination – Potential Problem Pivoting

23. Partial Pivoting a 32 >a 22 a l2 >a 22 NO YES

24. Partial Pivoting

25. Full Pivoting In addition to row swaping Search columns for max elements Swap Columns Change the order of x i Most cases not necessary

26. EXAMPLE We will work directly on the coefficient matrix