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# hw3 - CS 290N/219 Sparse matrix algorithms Homework 3...

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CS 290N/219: Sparse matrix algorithms: Homework 3 Assigned October 19, 2009 Due by class Wednesday, October 28 1. [20 points] Let G be the graph of the n -vertex model problem, that is, a k -by- k grid graph with n = k 2 vertices. Prove that there is some constant c > 0 such that for every elimination ordering on G , the filled graph G + contains a complete subgraph with at least c n vertices. (A complete subgraph is a set of vertices such that every pair is joined by an edge.) Hint: Suppose you’re given an ordering for the vertices of G . Think of playing the graph game in the given order, and consider the first time that you’ve either marked all the vertices in any single row of the entire grid or else marked all the vertices in any single column. 2. [40 points] (see Davis problem 6.13). An incomplete LU factorization is an approximate factorization A LU , in which L and U are lower and upper triangular matrices whose product is “approximately” A in some sense, but L and U have fewer nonzeros than the actual LU factors

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hw3 - CS 290N/219 Sparse matrix algorithms Homework 3...

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