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Unformatted text preview: CS 290N/219: Sparse matrix algorithms: Homework 2 Assigned October 7, 2009 Due by class time Wednesday, October 14 1. [20 points] Consider the 9-vertex grid graph, numbered by rows, with the edges directed to point from lower to higher numbered vertices. Perform dfs(3,4) , that is, carry out a depth-first search from vertices 3 and then 4. Show the postorder label assigned to each vertex. 2. [20 points] (See Davis problem 6.14.) In the left-looking LU factorization algorithm pre- sented in class on October 7, one can speed up the structure-prediction step by so-called symmetric pruning , which reduces the number of edges in the depth-first-search graph. This speeds up the symbolic step without changing the numerical step. Thus it doesnt change the asymptotic O ( f ) running time, but in practice it typically makes the whole factorization about four times as fast. Pruning can be done in either the no-pivoting ( A = LU ) or partial-pivoting ( PA = LU ) version of the factorization. For this problem, you can just consider the no-pivoting version. Prove that if both l jr and u rj are nonzero for some r < j , then when predicting the structure of any column k > j , the depth-first search in G ( L T ) will still give the correct result if the search...
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- Fall '09