Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
The inverses of block Hankel and block Toeplitz matrices
SIAM Journal on Computing
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants
SIAM Journal on Matrix Analysis and Applications
Mathematics of Computation
Efficient parallel solution of sparse systems of linear diophantine equations
PASCO '97 Proceedings of the second international symposium on Parallel symbolic computation
Processor-efficient parallel matrix inversion over abstract fields: two extensions
PASCO '97 Proceedings of the second international symposium on Parallel symbolic computation
Fast rectangular matrix multiplication and applications
Journal of Complexity
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Integer Smith form via the valence: experience with large sparse matrices from homology
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Fast computation of the Smith form of a sparse integer matrix
Computational Complexity
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
On the complexity of polynomial matrix computations
ISSAC '03 Proceedings of the 2003 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
On the complexity of computing determinants
Computational Complexity
Polynomial evaluation and interpolation on special sets of points
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
Solving structured linear systems with large displacement rank
Theoretical Computer Science
On finding multiplicities of characteristic polynomial factors of black-box matrices
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
When e-th roots become easier than factoring
ASIACRYPT'07 Proceedings of the Advances in Crypotology 13th international conference on Theory and application of cryptology and information security
Algorithms for solving linear systems over cyclotomic fields
Journal of Symbolic Computation
Generating a d-dimensional linear subspace efficiently
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
An output-sensitive algorithm for persistent homology
Proceedings of the twenty-seventh annual symposium on Computational geometry
An output-sensitive algorithm for persistent homology
Computational Geometry: Theory and Applications
Fast computation of Smith forms of sparse matrices over local rings
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Block Wiedemann algorithm on multicore architectures
ACM Communications in Computer Algebra
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Efficient block projections of non-singular matrices have recently been used by the authors in [10] to obtain an efficient algorithm to find rational solutions for sparse systems of linear equations. In particular a bound ofO~(n2.5) machine operations is presented for this computation assuming that the input matrix can be multiplied by a vector with constant-sized entries using O~(n) machine operations. Somewhat more general bounds for black-box matrix computations are also derived. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections of non-singular matrices, and this was only conjectured. In this paper we establish the correctness of the algorithm from [10] by proving the existence of efficient block projections for arbitrary non-singular matrices over sufficiently large fields. We further demonstrate the usefulness of these projections by incorporating them into existing black-box matrix algorithms to derive improved bounds for the cost of several matrix problems. We consider, in particular, matrices that can be multiplied by a vector using O~(n) field operations: We show how to compute the inverse of any such non-singular matrix over any field using an expected number of O~(n2.27) operations in that field. A basis for the null space of such a matrix, and a certification of its rank, are obtained at the same cost. An application of this technique to Kaltofen and Villard's Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix is also sketched, yielding algorithms requiring O~(n2.66) machine operations. More general bounds involving the number of black-box matrix operations to be used are also obtained. The derived algorithms are all probabilistic of the Las Vegas type. They are assumed to be able to generate random elements - bits or field elements - at unit cost, and always output the correct answer in the expected time given.