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ECE 598GG Fall 2007
Prof. George Gross Room 339 Everitt Lab
Homework 2
due: Tuesday, September 18, 2007
- Find the optimal order for the buses shown in the 18 bus network. The optimal ordering is the one that introduces the absolute minimum number of fill ins. In addition, compare the result with the locally optimal (Tinney Scheme 2) ordering algorithm. The network is
with the zero - nonzero pattern of the matrix having the structure
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 • • • • 2 • • • • • 3 • • • 4 • • • • 5 • • • • 6 • • • • • 7 • • • • 8 • • • • 9 • • • • • 10 • • • • • 11 • • • • 12 • • • • 13 • • • • • 14 • • • • 15 • • • • 16 • • • 17 • • • • • 18 • • • •
- Consider the subnetwork consisting of buses {1,2,...,9}. Can you derive the locally optimal ordering from problem 1? What is it?
- Using the derivation of the number of multiply/divide operations for a sparse matrix, show that for a full n x n matrix A****.
N M =
n^3 3 –^
n 3
n 2 2 –^
n 2
n^2 2 +^
n 2
for L U decomposition
for the forward substitution
for the backward substitution
What is N M for the case of A = A T?
1 x x x x x x 2 x x x x 3 x x x 4 x x 5 x x x 6 x x x 7 x x x 8 x x 9 x x 10 x x
- For the matrix having the zero-nonzero pattern shown below determine
( i ) the zero-nonzero pattern after elimination using the given ordering ( ii ) the zero-nonzero pattern after elimination after reordering so that the elimination process introduces exactly two fill-ins ( iii ) draw the associated graph and use the optimal ordering scheme to order the rows of this array. Show the order of eliminating the nodes and the resulting reduced graph at each step. How many fill-ins are introduced?
