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Forward- and Back-solving Steps

These take the factored form , with unit lower-triangular, upper-triangular, and permutation matrices , , and solve Ax = b, using the implicit pivoting approach described in [Skjellum:90d]. Sequentially, the triangular solves each require work proportional to the number of entries in the respective triangular factor, including fill-in. We have yet to find an example of sufficient size for which we actually attain speedup for these operations, at least for the sparse case. At most, we try to prevent these operations from becoming competitive in cost to the B-mode factorization; we detail these efforts in [Skjellum:90d]. In brief, the optimum grid shape for the triangular solves has Q=1, and P somewhat reduced from what we can use in all the other steps. As stated, P small seems better thus far, although for many examples increasing the overhead as a function of increasing P is not unacceptable (see [Skjellum:90d] and the example below).

Guy Robinson
Wed Mar 1 10:19:35 EST 1995