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Transcript of 1 EE 616 Computer Aided Analysis of Electronic Networks Lecture 4 Instructor: Dr. J. A. Starzyk,...
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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
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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
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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
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1R
5R
3R
4R
2R1V 2V
3V
1Si
SymmetricDiagonally Dominant
Nodal Matrix0
Sparse Matrices – Fill-in – Example 1
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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
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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
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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
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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
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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