1. Lecture #12 EEE 574 Dr. Dan Tylavsky Optimal Ordering

2. © Copyright 1999 Daniel Tylavsky * L,D,UIndicate positions of native nonzeros. zIndicates positions where fill occurs. –When we discussed fill we observed that depending on the ordering we got different amounts of fill. Table of Factors* 1 2 3 4 5 6 7

3. Optimal Ordering © Copyright 1999 Daniel Tylavsky –Re-label nodes: native node 1 native node 7. 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. 1 2 3 4 5 6 7

4. Optimal Ordering © Copyright 1999 Daniel Tylavsky –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.

5. Optimal Ordering © Copyright 1999 Daniel Tylavsky –Ex: Eliminate node 1. Eliminate node 2. 2 3 4 5 1 2 3 4 5

6. Optimal Ordering © Copyright 1999 Daniel Tylavsky Eliminate node 3. Eliminate node 4. 3 4 5 4 5

7. Optimal Ordering © Copyright 1999 Daniel Tylavsky –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. 123 4 5 6 7 8

8. Optimal Ordering © Copyright 1999 Daniel Tylavsky –Bill Tinney’s sub-optimal ordering methods: Tinney #1: Number nodes in ascending order of valency. 123 4 5 6 7 8

9. Optimal Ordering © Copyright 1999 Daniel Tylavsky –Look at the fill graphically. 4 5 6 7 8 123 4 5 6 7 8

10. Optimal Ordering © Copyright 1999 Daniel Tylavsky 4 5 6 7 8 5 6 7 88 6 7

11. Optimal Ordering © Copyright 1999 Daniel Tylavsky –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.

12. Optimal Ordering © Copyright 1999 Daniel Tylavsky –Bill Tinney’s sub-optimal ordering methods: Tinney #2: Number nodes in ascending order of valency accounting for valency changes due to fill. 4 123 5 6 78

13. Optimal Ordering © Copyright 1999 Daniel Tylavsky 123 4 5 6 7 8 –No fill with Tinney #2 for this simple problem!

14. Optimal Ordering © Copyright 1999 Daniel Tylavsky –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.

15. Optimal Ordering © Copyright 1999 Daniel Tylavsky –Teams (Think Pair Share) - Find the fill due another Tinney #2 ordering. (This yields a form known as border block diagonal form.) 12 3 4 5 6 7 8 9 10 11 12 Is there an ordering scheme consistent with Tinney #2 that has fill?

16. Optimal Ordering © Copyright 1999 Daniel Tylavsky 4 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. Internal External Boundary Buses

17. Optimal Ordering © Copyright 1999 Daniel Tylavsky 4 After reduction of rows associated with the external system: Only submatrix where fill occurs. 4 Premultiplying by the inverse of L. (In practice perform forward substitution.)

18. Optimal Ordering © Copyright 1999 Daniel Tylavsky 4 Writing: 4 Yields: OR

19. Optimal Ordering © Copyright 1999 Daniel Tylavsky 4 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.

20. The End

21. Optimal Ordering © Copyright 1999 Daniel Tylavsky

22. Optimal Ordering © Copyright 1999 Daniel Tylavsky

23. Optimal Ordering © Copyright 1999 Daniel Tylavsky