A modification of the LLL reduction algorithm
Journal of Symbolic Computation
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Asymptotically fast triangularization of matrices over rings
SIAM Journal on Computing
Certifying inconsistency of sparse linear systems
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Fast computation of the Smith form of a sparse integer matrix
Computational Complexity
Modern Computer Algebra
Smith normal form of dense integer matrices fast algorithms into practice
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
On the complexity of computing determinants
Computational Complexity
Computing the rank and a small nullspace basis of a polynomial matrix
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
The shifted number system for fast linear algebra on integer matrices
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Quadratic-time certificates in linear algebra
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Fast matrix rank algorithms and applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Deterministic unimodularity certification
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Fast matrix rank algorithms and applications
Journal of the ACM (JACM)
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Let M = [A B/C D] be a 2n x 2n integer matrix with the principal block A square and nonsingular. An algorithm is presented to determine if the Schur complement D--CA--1 B is equal to the zero matrix in O~(nω log ||M||) bit operations. Here, ω is the exponent of matrix multiplication and ||M|| denotes the largest entry in absolute value. The algorithm is randomized of the Las Vegas type, and either returns the correct answer ("yes" or "no"), or returns fail with probability less than 1/2. This gives a Las Vegas algorithm for computing the rank r of an n x m integer matrix A in an expected number of O~(nmrω--2 log ||A||) bit operations.