Computer algebra handbook
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Probabilistic algorithms based on the Krylov / Wiedemann or the Lanczos method to solve non homogeneous N x N systems Ax = b over a Galois field GF(q), usually require 2N matrix---vector products and O(n2+o(1)) additional arithmetic operations. Only the block Wiedemann algorithm, as given by Kaltofen in [6], has the least number (1+ε)N+O(1) of matrix---vector products of any known algorithm. We extend its analysis to the case of singular matrices A.