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Ordering Sparse Matrices for Iterative Methods

Symmetric sparse matrices can be represented by graphs with elements in equations corresponding to undirected edges in the graph [29]. Ordering a symmetric sparse matrix modifies the order in which rows are solved and is actually little more than changing the labels associated with nodes in an undirected graph. Modifying the ordering of a sparse matrix is simple to perform using a permutation matrix of either zeros or ones that simply generates elementary row and column exchanges. Applying the permutation matrix to the original linear system in equation gif yields the linear system

that is solved using the parallel Gauss-Seidel algorithm. While ordering the matrix greatly simplifies accessing parallelism inherent within the matrix structure, ordering can have an effect on convergence [23]. In section 7.2, we present empirical data to show that in spite of the ordering to yield parallelism, convergence appears to be rapid for positive definite power systems load-flow matrices.



David P. Koester
Sun Oct 22 17:27:14 EDT 1995