Download Lecture 4 - Ohio University

EE 616
Computer Aided Analysis of Electronic Networks
Lecture 4
Instructor: Dr. J. A. Starzyk, Professor
School of EECS
Ohio University
Athens, OH, 45701
09/16/2007
Note: some materials in this lecture
are from the notes of UC-Berkeley
1
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
2
Rounding
3
Consider a linear system
Ax  b
Scaling and Equilibration
Consider a linear
system
4
Example -1
5
Sparse Matrix Technology
6
General Goals for SMT
7
Sparse Matrices – Resistor Line
1
m
8
2
X X
X X X


X X X

X X


X







3
X
X
m 1
4
X
X
X
X










X

X X
X X 
m
Tridiagonal Case
Sparse Matrices – Fill-in – Example 1
R4
V3
V2
V1
R1
R2
R5
iS 1
R3
Nodal Matrix
0
0
0
9
Symmetric
Diagonally
Dominant
Sparse Matrices – Fill-in – Example 1
Matrix Non zero structure
X X X
X X 0 


 X 0 X 
X= Non zero
10
Matrix after one LU step
X
X

 X
X X

X X
0

X
0 X 
Sparse Matrices – Fill-in – Example 2
Fill-ins Propagate
X
X
X

0

0
X
X
X
X
X0
X
X
X
X
X
X
X0
X

X0

X0 

X0 
Fill-ins from Step 1 result in Fill-ins in step 2
11
Sparse Matrices – Fill-in & Reordering
V3
V1
V2
0
V3
V2
V1
0
x
x

 x
x
x
x

 0
x
x
x
x
x
x

x Fill-ins

x 
0

x No Fill-ins

x 
Node Reordering Can Reduce Fill-in
- Preserves Properties (Symmetry, Diagonal Dominance)
- Equivalent to swapping rows and columns
12
Sparse Matrices – Fill-in & Reordering
Where can fill-in occur ?
13
Already Factored
Multipliers








x
x
x
x
x
x

 Possible Fill-in

x  Locations

x
x

Fill-in Estimate = (Non zeros in unfactored part of Row -1)
 (Non zeros in unfactored part of Col -1)
Markowitz product

Determination of Pivots
14
Sparse Matrices – Data Structure

Several ways of storing a sparse matrix in a
compact form

Trade-off
–
–

15
Storage amount
Cost of data accessing and update procedures
Efficient data structure: linked list
Data Structures
16
Data Structures (cont’d)
17
Sparse Matrices – Graph Approach
Structurally Symmetric Matrices and Graphs
18
Sparse Matrices – Graph Approach
Markowitz Products
19
Graph Theoretic Interpretation (cont’d)
20
Sparse Matrices – Graph Approach
21
Sparse Matrices – Graph Approach
Exercise
Construct the elimination graph corresponding to the following matrix
22
Sparse Matrices – Graph Approach
Exercise
23
Diagonal Pivoting
24
Diagonal Pivoting (cont’d)
25