Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Theory of linear and integer programming
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A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
Diophantine linear system solving
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
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ACM Transactions on Mathematical Software (TOMS)
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SIAM Journal on Computing
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EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
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SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
A p-adic algorithm for univariate partial fractions
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
Modern Computer Algebra
On Rational Number Reconstruction and Approximation
SIAM Journal on Computing
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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
Solving sparse rational linear systems
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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ACM Transactions on Mathematical Software (TOMS)
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Journal of Symbolic Computation
Solving Very Sparse Rational Systems of Equations
ACM Transactions on Mathematical Software (TOMS)
Solving Very Sparse Rational Systems of Equations
ACM Transactions on Mathematical Software (TOMS)
Numeric-symbolic exact rational linear system solver
Proceedings of the 36th international symposium on Symbolic and algebraic computation
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Many methods have been developed to symbolically solve systems of linear equations over the rational numbers. A common approach is to use p-adic lifting or iterative refinement to build a modular or approximate solution, then apply rational number reconstruction. An upper bound can be computed on the number of iterations these algorithms must perform before applying rational reconstruction. In practice such bounds can be conservative. Output sensitive lifting is the technique of performing rational reconstruction at intermediate steps of the algorithm and verifying correctness which allows the possibility of early termination when the solution size is small. In this paper we show how using an appropriate output sensitive lifting strategy can improve several algorithms. We show this procedure to be computationally effective and introduce a variant of the iterative-refinement method that incorporates warm starts into the rational reconstruction procedure.