Repeated matrix squaring for the parallel solution of linear systems
Given a n×n nonsingular linear system Ax=b, we prove that the solution x can be computed in parallel time ranging from Ω(log n) to O(log2n), provided that the condition number, μ(A), of A is bounded by a polynomial in n. In particular, if μ(A) =O(1), a time bound O(log n) is achieved. To obtain this result, we reduce the computation of x to repeated matrix squaring and prove that a number of steps independent of n is sufficient to approximate x up to a relative error 2−d, d=O(1). This algorithm has both theoretical and practical interest, achieving the same bound of previously published parallel solvers, but being far more simple.
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