1 / 36

ECIV 301

ECIV 301. Programming & Graphics Numerical Methods for Engineers Lecture 18 LU Decomposition and Matrix Inversion. EXAMPLE. Eliminate Column 1. PIVOTS. Eliminate Column 1. Eliminate Column 2. PIVOTS. { b }. Eliminate Column 2. Upper Triangular Matrix. Modified RHS. [ U ].

darius
Download Presentation

ECIV 301

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  7. LU Decomposition PIVOTS Column 1 PIVOTS Column 2

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

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

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

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

  12. LU Decomposition L y b

  13. { b } LU Decomposition Modified RHS

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

  15. Matrix Inversion

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

  17. Matrix Inversion

  18. Matrix Inversion Solution

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

  20. Matrix Inversion

  21. Matrix Inversion

  22. Matrix Inversion

  23. Matrix Inversion • 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 – 1st column • Forward SUBSTITUTION

  30. Use LU Decomposition – 1st column • Back SUBSTITUTION

  31. Use LU Decomposition – 2nd Column • Forward SUBSTITUTION

  32. Use LU Decomposition – 2nd Column • Back SUBSTITUTION

  33. Use LU Decomposition – 3rd Column • Forward SUBSTITUTION

  34. Use LU Decomposition – 3rd Column • Back SUBSTITUTION

  35. Result

  36. Test It

More Related