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EE 616 Computer Aided Analysis of Electronic Networks Lecture 4

EE 616 Computer Aided Analysis of Electronic Networks Lecture 4. Instructor: Dr. J. A. Starzyk, Professor School of EECS Ohio University Athens, OH, 45701. 09/16/2005. Note: some materials in this lecture are from the notes of UC-berkeley. Review and Outline.

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EE 616 Computer Aided Analysis of Electronic Networks Lecture 4

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  1. EE 616 Computer Aided Analysis of Electronic NetworksLecture 4 Instructor: Dr. J. A. Starzyk, Professor School of EECS Ohio University Athens, OH, 45701 09/16/2005 Note: some materials in this lecture are from the notes of UC-berkeley

  2. Review and Outline • Review of the previous lecture • * Network Equations and Their Solution • -- Gaussian elimination • -- LU decomposition (Doolittle and Crout algorithm) • -- Pivoting • -- Detecting ILL Conditioning • Outline of this lecture • * Rounding, Pivoting and Network scaling • * Sparse matrix • -- Data Structure • -- Markowitz product • -- Graph Approach

  3. Rounding

  4. Scaling and Equilibration

  5. Example -1

  6. Sparse Matrix Technology

  7. General Goals for SMT

  8. Sparse Matrices – Resistor Line Tridiagonal Case m

  9. Sparse Matrices – Fill-in – Example 1 Nodal Matrix 0 Symmetric Diagonally Dominant

  10. X X Sparse Matrices – Fill-in – Example 1 Matrix Non zero structure Matrix after one LU step X X X X X= Non zero

  11. X X X X X Sparse Matrices – Fill-in – Example 2 Fill-ins Propagate X X X X X Fill-ins from Step 1 result in Fill-ins in step 2

  12. Fill-ins 0 No Fill-ins 0 Sparse Matrices – Fill-in & Reordering Node Reordering Can Reduce Fill-in - Preserves Properties (Symmetry, Diagonal Dominance) - Equivalent to swapping rows and columns

  13. Fill-in Estimate = (Non zeros in unfactored part of Row -1) (Non zeros in unfactored part of Col -1) Markowitz product Sparse Matrices – Fill-in & Reordering Where can fill-in occur ? Already Factored Possible Fill-in Locations Multipliers

  14. Determination of Pivots

  15. Sparse Matrices – Data Structure • Several ways of storing a sparse matrix in a compact form • Trade-off • Storage amount • Cost of data accessing and update procedures • Efficient data structure: linked list

  16. Data Structures

  17. Data Structures (cont’d)

  18. Sparse Matrices – Graph Approach Structurally Symmetric Matrices and Graphs

  19. Sparse Matrices – Graph ApproachMarkowitz Products

  20. Graph Theoretic Interpretation (cont’d)

  21. Sparse Matrices – Graph Approach

  22. Sparse Matrices – Graph Approach Discuss example 2.8.1 (Page 73 ~ 74)

  23. Diagonal Pivoting

  24. Diagonal Pivoting (cont’d)

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