1. 1 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

2. 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. 3 Rounding

4. 4 Scaling and Equilibration

5. 5 Example -1

6. 6 Sparse Matrix Technology

7. 7 General Goals for SMT

8. 8 m Sparse Matrices – Resistor Line Tridiagonal Case

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

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

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

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

13. 13 Where can fill-in occur ? Multipliers Already Factored Possible Fill-in Locations 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

14. 14 Determination of Pivots

15. 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. 16 Data Structures

17. 17 Data Structures (cont’d)

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

19. 19 Sparse Matrices – Graph Approach Markowitz Products

20. 20 Graph Theoretic Interpretation (cont’d)

21. 21 Sparse Matrices – Graph Approach

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

23. 23 Diagonal Pivoting

24. 24 Diagonal Pivoting (cont’d)