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Fillin

It is important to note that in this implementation, L and U can reuse the memory from the initial sparse matrix A, as long as provisions for storage of fillin values are provided. This algorithm assumes no pivoting is required to ensure numerical stability. Fillin results when either and or and are both non-zero and or are initially equal to zero. The amount of fillin in the factorization of a sparse matrix is highly dependent on the order in which calculations are performed. There are many ways to order a sparse matrix to reduce the amount of fillin [5,9,12]. Two noteworthy ordering techniques are minimum degree ordering and nested dissection. The foundation for minimum degree ordering is based in graph theory, which shows that fillin can be minimized by first eliminating rows and columns that have the fewest number of non-zero values, thus minimizing early fillin, which will hopefully minimize fillin later in the factorization. Minimum degree ordering is a greedy algorithm that is far from optimal because no good tie breaking rules exist and numerous rows and columns have equal numbers of elements. Nested dissection recursively breaks a matrix into a block-diagonal-bordered form where fillin is limited by the resulting structure. This technique is marginally effective for irregular networks or sparsity patterns, although it is considered the theoretical benchmark to which the quality of orderings are compared [12]. We have developed other ordering techniques for highly irregular matrices [14] that

This research is introduced later in this paper.



David P. Koester
Sun Oct 22 16:27:33 EDT 1995