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In this paper we present a deterministic algorithm for the computation of a minimal nullspace basis of an m × n input matrix of univariate polynomials over a field K with m ≤ n. This algorithm computes a minimal nullspace basis of a degree d input matrix with a cost of O ∼ (n ω md/n) field operations in K. Here the soft-O notation is Big-O with log factors removed while ω is the exponent of matrix multiplication. The same algorithm also works in the more general situation on computing a shifted minimal nullspace basis, with a given degree shift s ∈ Z n ≥0 whose entries bound the corresponding column degrees of the input matrix. In this case if ρ is the sum of the m largest entries of s, then a s-minimal right nullspace basis can be computed with a cost of O ∼ (n ω ρ/m) field operations.

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Algorithms for solving linear systems of equations over the integers are designed and implemented. The implementations are based on the highly optimized and portable ATLAS/BLAS library for numerical linear algebra and the GNU Multiple Precision library (GMP) for large integer arithmetic.

This paper presents a new algorithm for computing the Hermite normal form H of an A c Z "m of rank m together with a unimodular pre-multiplier matrix U such that UA = H. Our algorithm requires O-(m '-lnM(mlog[lAl\)) bit oper@ions to produce both H and U. Here, IIAII = max,j lAij [, M(t) bit operations are sufficient to multiply two (t]-bit integers, and 0 is the exponent for matrix multiplication over rings: two m x m matrices over a ring R can be multiplied in O(me) ring operations from R, The previously fastest algorithm of Hafner & McCurley requires 0-(m2nM (m log I IAI [)) bit operations to produce H, but does not produce a unimodular matrix U which satisfies UA = H. Previous methods require on the order of 0-(n3M(m log I[Al 1)) bit operations to produce a U-our algorithm improves on this significantly in both a theoretical and practical sense.

To cite this version:Claude-Pierre Jeannerod, Clément Pernet, Arne Storjohann. Rank-profile revealing Gaussian elimination and the CUP matrix decomposition.Journal of Symbolic Computation, Elsevier, 2013, 56, pp.46-68.

The shifted number system is presented: a method for detecting and avoiding error producing carries during approximate computations with truncated expansions of rational numbers. Using the shifted number system the high-order lifting and integrality certification techniques of Storjohann 2003 for polynomial matrices are extended to the integer case. Las Vegas reductions to integer matrix multiplication are given for some problems involving integer matrices: the determinant and a solution of a linear system can be computed with about the same number of bit operations as required to multiply together two matrices having the same dimension and size of entries as the input matrix. The algorithms are space efficient.