1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 18 LU Decomposition and Matrix Inversion

2. EXAMPLE

3. Eliminate Column 1 PIVOTS

4. Eliminate Column 1

5. Eliminate Column 2 PIVOTS

6. Eliminate Column 2 Upper Triangular Matrix [ U ] Modified RHS { b }

7. LU Decomposition PIVOTS Column 1 PIVOTS Column 2

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

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

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

11. LU Decomposition [ L ]{ y }{ b } [ A ]{ x }{ b }

12. LU Decomposition Lyb

13. Modified RHS { b }

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

15. Matrix Inversion

16. [A][A] -1 [A] [A] -1 =[I] If [A] -1 does not exist [A] is singular

17. Matrix Inversion

18. Solution

19. Matrix Inversion [A] [A] -1 =[I]

20. Matrix Inversion

23. To calculate the invert of a nxn matrix solve n times :

24. Matrix Inversion For example in order to calculate the inverse of:

25. Matrix Inversion First Column of Inverse is solution of

26. Matrix Inversion Second Column of Inverse is solution of

27. Matrix Inversion Third Column of Inverse is solution of:

28. Use LU Decomposition

29. Use LU Decomposition – 1 st column Forward SUBSTITUTION

30. Use LU Decomposition – 1 st column Back SUBSTITUTION

31. Use LU Decomposition – 2 nd Column Forward SUBSTITUTION

32. Use LU Decomposition – 2 nd Column Back SUBSTITUTION

33. Use LU Decomposition – 3 rd Column Forward SUBSTITUTION

34. Use LU Decomposition – 3 rd Column Back SUBSTITUTION

35. Result

36. Test It