Eberly, Wayne2018-05-032018-05-032018-05-03Eberly, Wayne. “Black Box Linear Algebra.” ACM Communications in Computer Algebra, vol. 50, no. 4, 2017, pp. 164–166., doi:10.1145/3055282.3055291http://hdl.handle.net/1880/106606Wiedemann’s paper, introducing his algorithm for sparse and structured matrix computations over arbitrary fields, also presented a pair of matrix preconditioners for computations over small fields. The analysis of the second of these is extended in order to provide more explicit statements of the expected number of nonzero entries in the matrices obtained as well as bounds on the probability that such matrices have maximal rank. This is part of ongoing work to establish that this matrix preconditioner can also be used to bound the number of nontrivial nilpotent blocks in the Jordan normal form of a preconditioned matrix, in such a way that one can also sample uniformly from the null space of the originally given matrix. If successful this will result in a black box algorithm for the type of matrix computation required when using the number field sieve for integer factorization that is provably reliable and — by a small factor — asymptotically more efficient than alternative techniques that make use of other matrix preconditioners or require computations over field extensions.enUnless otherwise indicated, this material is protected by copyright and has been made available with authorization from the copyright owner. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Matrix ComputationsBlack Box Linear AlgebraConditioningtechnical reportdoi:10.1145/3055282.3055291.10.11575/PRISM/35464