1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 17 Solution of Systems of Equations

2. Last Time Linear Equations in Matrix Form

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

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

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

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

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

8. Last Time Gauss Elimination Example Forward Elimination

9. Last Time Gauss Elimination Forward Elimination

10. Last Time Gauss Elimination Back Substitution

11. Last Time GE – Potential Problem Forward Elimination

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

13. Gauss Elimination – Potential Problem OK!!

14. Gauss Elimination – Potential Problem Pivoting

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

16. Partial Pivoting

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

18. EXAMPLE

19. Eliminate Column 1 PIVOTS

20. Eliminate Column 1

21. Eliminate Column 2 PIVOTS

22. Eliminate Column 2

23. LU Decomposition PIVOTS Column 1 PIVOTS Column 2

24. LU Decomposition As many as, and in the location of, zeros Upper Triangular Matrix U

25. LU Decomposition PIVOTS Column 1 PIVOTS Column 2 Lower Triangular Matrix L

26. LU Decomposition = This is the original matrix!!!!!!!!!!

27. LU Decomposition Lyb

28. Lyb

30. Ax=b A=LU -LU Decomposition Ly=b- Solve for y Ux=y- Solve for x