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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 ].

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ECIV 301

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  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

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