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Computer-Aided Analysis of Electronic Networks: Lecture 4 Overview

This lecture by Dr. J. A. Starzyk, a professor at the School of EECS at Ohio University, focuses on advanced concepts in computer-aided analysis of electronic networks. Key topics include network equations, Gaussian elimination, LU decomposition, pivoting, and detecting ill conditioning. The lecture also delves into rounding, pivoting, and scaling in sparse matrices, their data structures, and graph approaches. Concrete examples illustrate concepts like fill-ins and node reordering, essential for efficient computation in network analysis.

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Computer-Aided Analysis of Electronic Networks: Lecture 4 Overview

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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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