Abstract
We consider a variation of the well known problem of computing the unique solution to a nonsingular systemAx = b of n linear equations over a eld K. The variation assumes that A has generic rank prole and requires as output not only the single solution vector A 1 b2 K n 1 , but rather the solution to all leading principle subsystems. Most importantly, the rows of the augmented system A b are given one at a time from rst to last, and as soon as the next row is given the solution to the next leading principal subsystem should be produced. We call this problem OnlineSystem. The obvious iterative algorithm for OnlineSystem has a cost in terms of eld operations that is cubic in the dimension of A. In this paper we introduce a relaxed representation for the inverse and show how to obtain an algorithm for OnlineSystem that allows us to incorporate matrix multiplication. As an application we show how to introduce fast matrix multiplication into the inherently iterative algorithm for row rank prole computation presented previously by the authors.
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