1

EE 616 Computer Aided Analysis of Electronic Networks

Lecture 4

Instructor: Dr. J. A. Starzyk, ProfessorSchool of EECSOhio UniversityAthens, 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

1 2 3 4 1m m

X X

X X X

X X X

X X X

X X X

X X X

X X X

X X

m

Sparse Matrices – Resistor Line

Tridiagonal Case

9

1R

5R

3R

4R

2R1V 2V

3V

1Si

SymmetricDiagonally Dominant

Nodal Matrix0

Sparse Matrices – Fill-in – Example 1

10

X X X

X X 0

X 0 X

X X X

X X 0

X 0 X

X

X

X X

X= Non zero

Matrix Non zero structure Matrix after one LU step

X X

Sparse Matrices – Fill-in – Example 1

11

X X X X

X X 0 0

0 X X 0

X 0 00

Fill-ins Propagate

XX

X

X

X

X X

X

X X

Fill-ins from Step 1 result in Fill-ins in step 2

Sparse Matrices – Fill-in – Example 2

12

3V

Node Reordering Can Reduce Fill-in - Preserves Properties (Symmetry, Diagonal Dominance) - Equivalent to swapping rows and columns

1V 2V

0

x x x

x x x

x x x

Fill-ins

2V 1V

3V

0

x x 0

x x x

0 x x

No Fill-ins

Sparse Matrices – Fill-in & Reordering

13

Where can fill-in occur ?

x x x

x x x

x x x

Multipliers

Already Factored

Possible Fill-inLocations

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

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 Approach

Markowitz 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)