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FC '01 Proceedings of the 5th International Conference on Financial Cryptography
An output-sensitive variant of the baby steps/giant steps determinant algorithm
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Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Computer algebra handbook
High-order lifting and integrality certification
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Computing the sign or the value of the determinant of an integer matrix, a complexity survey
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Smith normal form of dense integer matrices fast algorithms into practice
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Improved algorithms for computing determinants and resultants
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
Essentially optimal computation of the inverse of generic polynomial matrices
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
A BLAS based C library for exact linear algebra on integer matrices
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
Solving sparse rational linear systems
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Parallel computation of the rank of large sparse matrices from algebraic K-theory
Proceedings of the 2007 international workshop on Parallel symbolic computation
Schur aggregation for linear systems and determinants
Theoretical Computer Science
Nearly optimal symbolic-numerical algorithms for structured integer matrices and polynomials
Proceedings of the 2009 conference on Symbolic numeric computation
The shifted number system for fast linear algebra on integer matrices
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
Kaltofen's division-free determinant algorithm differentiated for matrix adjoint computation
Journal of Symbolic Computation
Computing hermite forms of polynomial matrices
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Quadratic-time certificates in linear algebra
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Zigzag persistent homology in matrix multiplication time
Proceedings of the twenty-seventh annual symposium on Computational geometry
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CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Fast computation of Smith forms of sparse matrices over local rings
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A/spl isin/Z/sup n/spl times/n/ the algorithm requires O(n/sup 3.5/(log n)/sup 4.5/) bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using asymptotically fast matrix arithmetic, a variant is described which requires O(n/sup 2+/spl theta//2//spl middot/log/sup 2/nloglogn) bit operations, where n/spl times/n matrices can be multiplied with O(n/sup /spl theta//) operations. The determinant is found by computing the Smith form of the integer matrix an extremely useful canonical form in itself. Our algorithm is probabilistic of the Monte Carlo type. That is, it assumes a source of random bits and on any invocation of the algorithm there is a small probability of error.