1. Solving Scale Linear Systems (Example system continued) Lecture 14 MA/CS 471 Fall 2003

2. Today We will discuss direct methods for a slightly larger loop current problem as introduced last time. Then we will look at a much larger problem and examine the memory/work requirements for direct methods.

3. Note It is important that at least one loop has a resistor that lies only on that loop (i.e. at least one resistor lying on a wire not shared by two loops). Otherwise the entire circuit will short circuit around the boundary of the entire circuit.. For example: 44 33 22 11 66 77 +-+- 11 1 23 45 8 9 44 33 22 11 66 77 +-+- 11 1 2 3 45 8 9

4. Circuit Problem Enlarged Problem: Find the current running through each closed loop 11 44 33 22 11 55 66 77 30V +-+- 11 44 33 22 11 55 66 77 20V +-+- 11 44 33 22 11 55 66 77 10V +-+- 22 11 1 2 3 45 6 7 8 9 10 11 12 14 15 16 17 13

5. Shortcut to Loop Current Matrix To obtain the n’th row of the Matrix: 1)The diagonal entry is equal to the sum of the resistances on the n’th loop 2)There is an off diagonal entry for each neighbor of the loop which is equal to: a) –sum(resistances) on shared wire if loop currents are in opposite direction b) sum(resistances) on shared wire if loop currents are in the same direction 11 44 33 22 11 55 66 77 30V +-+- 11 44 33 22 11 55 66 77 20V +-+- 11 44 33 22 11 55 66 77 10V +-+- 22 11 1 2 3 45 6 7 8 9 10 11 12 14 15 16 17 13

6. 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 22 11 1 2 3 45 6 7 8 9 10 11 12 14 15 16 17 13

7. Using Sparse Matrices in Matlab

8. Run bigcircuit 1)Create the list of non-zeros 2)Convert the list to Matlab’s sparse matrix format 3)Convert the sparse matrix to a full matrix (just for viewing)

9. Counting the Non-Zeros with nnz We can use Matlab’s built in nnz function to find the number of non-zeros: i.e. there are only 58 non-zero entries out of 17x17=289 possible

10. Now We LU Factorize The Sparse Matrix Since there are only a few degrees of freedom we will use a direct method to factorize and solve the system. We can use the built in LU factorization of the matrix… i.e. find two matrices L & U such that A=LU where L is logically lower triangle and U is logically upper triangle..

11. LU Factorization of Loop Current Matrix

12. Solving The System Now we have factorized the system into A=LU we can solve in three stages. 1)Build the source vector, v 2)Solve y = L\v 3)Solve for the currents I = U\y

13. 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 22 11 1 2 3 45 6 7 8 9 10 11 12 14 15 16 17 13 30V 20V 10V Solving for the loop currents using an LU factorization

14. 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 22 11 1 2 3 45 6 7 8 9 10 11 12 14 15 16 17 13 30V 20V 10V Solving for the loop currents using an LU factorization

15. Let’s Renumber Matlab has a built in routine symrcm which takes a symmetric matrix and returns a permutation array so that if we use this to permute the unknowns (by column and row swaps) the bandwidth of the matrix may be reduced…

16. Renumbering with symrcm

17. How Were The Loops Renumbered We can examine the permutation matrix: i.e. the old 13 cell becomes the new 1 cell 6->2, 14->3…

18. 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 22 11 1 2 3 45 6 7 8 9 10 11 12 14 15 16 17 13 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 22 11 12 10 1514 2 11 17 16 7 9 8 3 6 4 5 1

19. Let’s Figure Out the Sequence of Shells in symrcm 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 11 44 33 22 11 55 66 77 +-+- 22 11 13 12 10 1514 2 11 17 16 7 9 8 3 6 4 5 1 Level 1 Level 2 Level 3 Level 4 Level 5 Level 6

20. Let’s Try LU Factorization of the Reordered Matrix

21. Effect of Reordering Before After

22. Notes We could have also used Cholesky factorization (since the loop current matrix is symmetric). Just by reordering unknowns we have changed the amount of fill in the L,U factors of the matrix. Changing the ordering of cells will not change the answer (beyond round off). However, we have reduced the amount of work required in the backsolves.

23. Solving The Reordered System

24. Comparing The Results We can compare the results from solving using the original matrix and using the reordered matrix: No reorderingWith reordering

25. OK – Let’s Get Serious And Look At A Large Circuit Case We can construct a random circuit:

26. The Sparsity Pattern of a Loop Circuit Matrix for a Random Circuit (with 1000 closed loops)

27. Notes For a more realistic circuit (i.e. with less random interconnects) the fill in the L & U matrices will be reduced more after reordering using RCM

28. Lab Task Construct a more realistic “random” circuit for an arbitrary N (i.e. design a process which randomly grows a circuit). Use an interconnect of say 4 per cell on average. Time how long the LU factorization takes for N=10,100,1000,1e4,1e5,1e6,1e7,1e8 and plot a graph of time v. N Calculate the number of non-zeros (i.e. how much memory is taken by the L & U matrices). Plot this as a function of N. Solve with a random source vector for N=10,100,…,1e6 and time how long the backsolves take and plot a graph of time v. N Perform a polynomial fit of the timings (estimate the polynomial growth rate with N)… USE SPARSE MATRICES!. You can use Cholesky if you wish.