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PETE 603

PETE 603. Lecture Session #27 Tuesday, 7/27/10. 13. 28. 14. 29. 15. 30. 25. 10. 26. 11. 27. 12. 7. 22. 8. 23. 9. 24. 19. 4. 20. 5. 21. 6. 1. 16. 2. 17. 3. 18. A3 Ordering. This ordering is also known as red/black or checkerboard ordering. A3 Incidence Matrix.

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PETE 603

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  1. PETE 603 Lecture Session #27 Tuesday, 7/27/10

  2. 13 28 14 29 15 30 25 10 26 11 27 12 7 22 8 23 9 24 19 4 20 5 21 6 1 16 2 17 3 18 A3 Ordering This ordering is also known as red/black or checkerboard ordering.

  3. A3 Incidence Matrix Where D are diagonal matrices and P are 4-diagonal matrices (for 2D).

  4. 5 22 10 27 14 30 18 6 23 11 28 15 2 19 7 24 12 29 16 3 20 8 25 13 1 17 4 21 9 26 D4 Ordering D4 ordering follows alternate diagonals.

  5. D4 Incidence Matrix Where D are diagonal matrices, and P are 4-diagonal matrices.

  6. A3 and D4 Orderings The A3 and D4 orderings of the grid blocks “partitions” the matrix equation, Ax = y, as follows: For A3, elimination of the submatrix P2, forms the upper triangular submatrix D2’. For D4, during elimination of submatrix P2, coefficients are added to submatrix D2 (shown as circles on Dalton and Mattax, Fig 6.7). Elimination on the modified D2 is done to form the upper triangular submatrix D2’.

  7. A3 and D4 Orderings Afterwards, we have: The second equation gives: and back substitution gives:

  8. A3 and D4 Comparison • D4 is usually more efficient than A3. • D4 also more complicated to program and has larger computer memory requirement, so A3 sometimes is preferred to D4.

  9. Direct/Iterative Methods • For direct methods • Computer time increases rapidly with the number of unknowns [O(n) to O(n)3] • Roundoff error increases with the number of unknowns CPU Time Iterative Direct Numberof Unknowns

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