The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
Computing GCDs of polynomials over algebraic number fields
Journal of Symbolic Computation
Exact solution of linear equations
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Modern Computer Algebra
Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
A BLAS based C library for exact linear algebra on integer matrices
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Half-GCD and fast rational recovery
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
On exact and approximate interpolation of sparse rational functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
A Computational Introduction to Number Theory and Algebra
A Computational Introduction to Number Theory and Algebra
Factoring univariate polynomials over the rationals
Factoring univariate polynomials over the rationals
An LLL Algorithm with Quadratic Complexity
SIAM Journal on Computing
Solving Very Sparse Rational Systems of Equations
ACM Transactions on Mathematical Software (TOMS)
An LLL-reduction algorithm with quasi-linear time complexity: extended abstract
Proceedings of the forty-third annual ACM symposium on Theory of computing
Gradual sub-lattice reduction and a new complexity for factoring polynomials
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Numeric-symbolic exact rational linear system solver
Proceedings of the 36th international symposium on Symbolic and algebraic computation
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The final step of some algebraic algorithms is to reconstruct the common denominator d of a collection of rational numbers (ni/d)1≤ i≤ n from their images (ai)1≤ i≤ n mod M, subject to a condition such as 0 d ≤ N and Ni}≤ N for a given magnitude bound N. Applying elementwise rational number reconstruction requires that M ∈ Ω(N2). Using the gradual sublattice reduction algorithm of van Hoeij and Novocin, we show how to perform the reconstruction efficiently even when the modulus satisfies a considerably smaller magnitude bound M ∈ Ω(N1+1/c) for c a small constant, for example 2 ≤ c ≤ 5. Assuming c ∈ O(1) the cost of the approach is O(n(log M)3) bit operations using the original LLL lattice reduction algorithm, but is reduced to O(n(log M)2) bit operations by incorporating the L2 variant of Nguyen and Stehle. As an application, we give a robust method for reconstructing the rational solution vector of a linear system from its image, such as obtained by a solver using p-adic lifting.