We consider a variation of the well known problem of computing the unique solution to a nonsingular system Ax = b of n linear equations over a field K. The variation assumes that A has generic rank profile and requires as output not only the single solution vector A −1 b ∈ 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 first 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 field 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 profile computation presented previously by the authors.

Randomized algorithms are given for linear algebra problems on an input matrix A ∈ K n×m over a field K. We give an algorithm that simultaneously computes the row and column rank profiles of A in 2r 3 + (r 2 + n + m + |A|) 1+o(1) field operations from K, where r is the rank of A and |A| denotes the number of nonzero entries of A. Here, the +o(1) in cost estimates captures some missing log n and log m factors. The rank profiles algorithm is randomized of the Monte Carlo type: the correct answer will be returned with probability at least 1/2. Given a b ∈ K n×1 , we give an algorithm that either computes a particular solution vector x ∈ K m×1 to the system Ax = b, or produces an inconsistency certificate vector u ∈ K 1×n such that uA = 0 and ub = 0. The linear solver examines at most r + 1 rows and r columns of A and has running time 2r 3 + (r 2 + n + m + |R| + |C|)1+o (1) field operations from K, where |R| and |C| are the number of nonzero entries in the rows and columns, respectively, that are examined. The solver is randomized of the Las Vegas type: an incorrect result is never returned but the algorithm may report FAIL with probability at most 1/2. These cost estimates are achieved by making use of a novel randomized online data structure for the detection of linearly independent rows and columns.

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