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EE 616 Computer Aided Analysis of Electronic Networks

Lecture 3

Instructor: Dr. J. A. Starzyk, ProfessorSchool of EECSOhio UniversityAthens, OH, 45701

09/12/2007

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Review and Outline

Review of the previous lecture* Network scaling

* Thevenin/Norton Analysis

* KCL, KVL, branch equations

* Sparse Tableau Analysis (STA)

* Nodal analysis

* Modified nodal analysis

Outline of this lecture* Network Equations and Their Solution -- Gaussian elimination -- LU decomposition(Doolittle and Crout algorithm) -- Pivoting

-- Detecting ILL Conditioning

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A is n x n real non-singular X is nx1; B is nx1;

bxM

SeY

Problems:

Direct methods: find the exact solution in a finite number of steps

-- Gaussian elimination, LU decomposition, Crout, Doolittle)

Iterative methods: produce a sequence of approximate solutions hopefully converging to the exact solution

-- Gauss-Jacobi, Gauss-Seidel, Successive Over Relaxation (SOR)

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Gaussian Elimination Basics

Reminder by 3x3 example

11 12 13 1 1

21 22 23 2 2

3 331 32 33

M M M x b

M M M x b

x bM M M

11 1 12 2 13 3 1M x M x M x b

21 1 22 2 23 3 2M x M x M x b

31 1 32 2 33 3 3M x M x M x b

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Gaussian Elimination Basics – Key idea

Use Eqn 1 to Eliminate x1 from Eqn 2 and 3

11

13

11

132

11

121 M

bx

M

Mx

M

Mx

11 1 12 2 13 3 1M x M x M x b

21 21 2122 12 2 23 13 3 2 1

11 11 11

M M MM M x M M x b b

M M M

31 31 3132 12 2 33 13 3 3 1

11 11 11

M M MM M x M M x b b

M M M

Multiply equation (*) by –M21 and add to eq (2)

Eq.1 divided by M11

(*)

Multiply equation (*) by –M31 and add to eq (3)

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GE Basics – Key idea in the matrix

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11 12 13

21 21 2122 12 23 12 2 2 1

11 11 11

31 3132 12 33 12

311 113 3

0

0

xb

M M M

M M MM M M M x b b

M M M

M MM M M M

MM M x b

1

111

bM

Pivot

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GE Basics – Key idea in the matrix

11 12 13

21 2122 12 23 12

11 11

3132 12

1131 2133 12 23 12

11 112122 12

11

0

0 0

M M M

M MM M M M

M M

MM M

MM MM M M M

M MMM M

M

11

212 1

11

2

3132 12

1131 213 1 2 1

11 112122 12

113

x b

Mb b

Mx

MM M

MM Mb b b b

M MMM M

Mx

Continue this step to remove x2 from eqn 3

8

11

11 12 13

21 21 2122 12 23 12 2 2 1

11 11 11

31 3132 12 33 12

311 113 3

0

0

xb

M M M

M M MM M M M x b b

M M M

M MM M M M

MM M x b

1

111

bM

22M 23M

32M33M

3b2b

GE Basics – Simplify the notation

Remove x1 from eqn 2 and eqn 3

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22 23 2

32 3233 23 3 2

111 12 13

2 2

1

2

3

2 2

0

0 0

x

M M b

M MM M b b

M M

bM

x

M

x

M

GE Basics – Simplify the notation

Remove x2 from eqn 3

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GE Basics – GE yields triangular system

AlteredDuring

GE

11 12 13 1 1

22 23 2 2

33 3 3

0

0 0

U U U x y

U U x y

U x y

11

2

3

22

11 12 1

2

3

2

3 3

3

30

0 0

x

M M b

b

M

M M M

x

x b

~ ~

11

33

33

yx

U

2 23 32

22

y U xx

U

1 12 2 13 31

11

y U x U xx

U

11 12 13 1 1

22 23 2 2

33 3 3

0

0 0

U U U x y

U U x y

U x y

GE Basics – Backward substitution

12

1

1

212 12

11

3 322

2

13 3 1

11 2

by

Mb by

M

M Mby b

M Mb

32

1 1

212 2

113 3

31

11 22

1 0 0

1 0

1

y bM

y bM

y bMM

M M

GE Basics – RHS updates

3

2

1

3

2

1

3231

21

1

01

001

b

b

b

y

y

y

LL

L

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GE basics: summary

(1) M x = b

U x = y Equivalent systemU: upper triangle

(2) Noticed that:Ly = b L: unit lower triangle

(3) U x = yLU x = b M x = b

GE

Efficient way of implementing GE: LU factorization

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Solve M x = bStep 1

Step 2 Forward Elimination

Solve L y = bStep 3 Backward Substitution Solve U x = y

=M = L U

Gaussian Elimination Basics

Note: Changing RHS does not imply to recompute LU factorization

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LU decomposition

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LU decomposition

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LU decomposition – Doolittle example

5052

1183

1241

6442

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LU factorization (Crout algorithm)

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LU factorization (Crout algorithm)

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Properties of LU factorization

Now, let’s see an example:

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LU decomposition - example

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Relation between STA and NA

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Pivoting for Accuracy:

Example 1: After two steps of G.E. MNA matrix becomes:

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Pivoting for Accuracy:

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Pivoting for Accuracy:

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Pivoting for Accuracy:

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Pivoting Strategies

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Error Mechanism

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Detecting ILL Conditioning

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Detecting ILL Conditioning