lecture #12 eee 574 dr. dan tylavsky optimal ordering

Lecture #12

EEE 574

Dr. Dan Tylavsky

Optimal Ordering

© Copyright 1999 Daniel Tylavsky

Optimal Ordering

XX

XX

XX

XX

XX

XX

XXXXXXX

DzzzzzL

zDzzzzL

zzDzzzL

zzzDzzL

zzzzDzL

zzzzzDL

UUUUUUD

*L,D,U Indicate positions of native nonzeros. z Indicates positions where fill occurs.

– When we discussed fill we observed that depending on the ordering we got different amounts of fill.

Table of Factors*

kik

ij

ij jiy

jiyY

1

23

45

6

7


Optimal Ordering– Re-label nodes: native node 1 <=> native node 7.

XXXXXXX

XX

XX

XX

XX

XX

XX

DLLLLLL

UD

UD

UD

UD

UD

UD

Table of Factors

– There is a graphical relationship between the elimination of a node and the fill that occurs after a node is eliminated in Gauss elimination/LU factorization.

12

3

45

67


Optimal Ordering– When column/row k of a matrix is eliminated, the fill produced corrresponds to new branches in the network graph created by:

• removing node k and all of its connected branches from the network graph.• mutually interconnecting in the (resulting network graph) all nodes upon which the removed branches were incident.


Optimal Ordering

xxx

xxx

xxx

xxx

xxx– Ex:

xxzx

xxx

xxx

zxxx

xxx

Eliminate node 1.

Eliminate node 2.

2

34

5

1

2

34

5


Optimal Ordering

xxzzx

xxx

zxxx

zxxx

xxx

Eliminate node 3.

xxzzx

xxx

zxxx

zxxx

xxx

Eliminate node 4.

xxzx

xxx

xxx

zxxx

xxx

34

5

4

5


Optimal Ordering– Defn: Node Valency - number of branches incident on the node or, equivalently, the number of non-zero

off-diagonal terms in the row of the matrix corresponding to the node of interest.

1 2 3

4 56

78 Node Valency1 12 13 14 45 36 37 28 3


Optimal Ordering– Bill Tinney’s sub-optimal ordering methods:

• Tinney #1: Number nodes in ascending order of valency.

1 2 3 4

5

6

78Node Valency1 12 13 14 25 36 37 38 4

xxxxx

xxxx

xxxx

xxxx

xxx

xx

xx

xx


Optimal Ordering

xxxxx

xxx

xxxx

xxxx

xxx

xx

xx

xx

– Look at the fill graphically.

4

5

6

78

1 2 3 4

5

6

78


Optimal Ordering

xxxxx

xxx

xxxx

xxxx

xxx

xx

xx

xx

4

5

6

78

5

6

78

8

6

7

xzzxxxx

zxxxx

zxxxx

xxxx

xxx

xx

xx

xx


Optimal Ordering– Teams (Think Pair Share) - Using Tinney #1 ordering scheme, find the fill that will occur during

factorization for the matrix associated with the network graph shown below.

The ordering you get will depend on the order in which you process the nodes.


Optimal Ordering– Bill Tinney’s sub-optimal ordering methods:

• Tinney #2: Number nodes in ascending order of valency accounting for valency changes due to fill.

4

1 2 3

5

6

7 8


Optimal Ordering

1 2 3

4 5 6

7 8

xxx

xxxx

xxxx

xxxx

xxxxx

xx

xx

xx

– No fill with Tinney #2 for this simple problem!


Optimal Ordering– Teams (Think Pair Share) - Using Tinney #2 ordering scheme, find the fill that will occur during

factorization for the matrix associated with the network graph shown below.

Ordering you get will depend on the order in which you process the nodes.


Optimal Ordering– Teams (Think Pair Share) - Find the fill due another Tinney #2 ordering. (This yields a form known as

border block diagonal form.)

1 2

3

4

5

6

7

8

9

10

1112

Is there an ordering scheme consistent with Tinney #2 that has fill?


Optimal Ordering Node Ordering for Generating Network Equivalents.

• Order External Buses First.• Order Boundary Buses Second.• Order Internal Buses Third.• Order according to Tinney #1 or 2.

I

B

E

I

B

E

IITBI

BIBBTEB

EBEE

I

I

I

E

E

E

YY

YYY

YY

IYE

0

0

Internal

External

Boundary Buses


Optimal Ordering After reduction of rows associated with the external system:

I

B

E

I

B

EEBEE

IITBI

BIBBTEB

EE

I

I

I

E

E

E

I

UU

YY

YY

I

IL

L 0ˆ

0

Only submatrix where fill occurs.

Premultiplying by the inverse of L. (In practice perform forward substitution.)

I

B

E

I

B

ETEB

EE

I

I

I

I

I

I

IL

L1

0


Optimal Ordering

Writing:

I

B

E

I

B

EEBEE

E

E

E

E

E

E

I

UU 0

Yields:

I

B

E

I

B

E

IITBI

BIBB

I

I

I

E

E

E

YY

YY

Iˆ

I

B

I

B

IITBI

BIBB

I

I

E

E

YY

YYOR


Optimal Ordering

I

B

I

B

IITBI

BIBB

I

I

E

E

YY

YY Graphical Interpretation of:

Internal

Modified Boundary Bus Injections (I’B)

Note that bus voltages of interest are unscathed by the process.

Fictitious Branches (due to fill) represent equivalent network.

The End


Optimal Ordering

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9

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6

5

4

3

2

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lecture #12 eee 574 dr. dan tylavsky optimal ordering

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