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
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.