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Ordering

The ordering portion in the preprocessing phase must identify subsections of the matrix that are mutually independent. For symmetric matrices, there is a graph-theoretical interpretation for independent sub-matrices. Independent sub-matrices simply have no shared edges in their undirected graph. A simple example to illustrate the concept of independent subgraphs is illustrated in figure 6. This figure has four independent portions of the graph connected by nodes that form the coupling equations. No subgraph element has edges to any portion of the graph other than within the local subgraph or extending to the coupling equations.

 
Figure 6: Sample Graph with Four Independent Subgraphs/Sub-Matrices  

Few matrices can be readily ordered into block-diagonal-bordered form with equal workload in each block. The exception to this rule are highly regular matrices from the structural analysis community, where the nested dissection ordering technique can produce balanced block-diagonal-bordered matrices on some regular matrices [12]. Recursive spectral bisection can be used to partition irregular matrices [3,17,20], and subsequently, the coupling equations can be extracted. Unfortunately, this technique, as well as nested dissection, relies on dividing the matrix into m equal sized partitions, without considering the coupling equations or considering the number of calculations in each independent block. A third method to order a sparse matrix into block-diagonal-bordered form is referred to as node tearing [5,19], which is a specialized form of diakoptics [11]. This technique attempts to extract the natural structure in the matrix or graph, and generally produces many irregularly sized blocks, while minimizing the number of coupling equations. Load balancing techniques must be used after the node tearing matrix ordering step to uniformly distribute the processing load onto a multi-processor. It is important to note that independent blocks can be assigned to any processor without requirements for interprocessor communications to factor the sub-block.



next up previous
Next: Pseudo Factorization Up: A New Three-step Previous: A New Three-step



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