Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Complexity of parallel matrix computations
Theoretical Computer Science
Fast parallel computation of hermite and smith forms of polynomial matrices
SIAM Journal on Algebraic and Discrete Methods
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
On computing determinants of matrices without divisions
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm
Mathematics of Computation
A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants
SIAM Journal on Matrix Analysis and Applications
Mathematics of Computation
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
On randomized Lanczos algorithms
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Efficient parallel solution of sparse systems of linear diophantine equations
PASCO '97 Proceedings of the second international symposium on Parallel symbolic computation
Modern computer algebra
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
Approximate algorithms to derive exact solutions to systems of linear equations
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
On computing the determinant and Smith form of an integer matrix
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On the complexity of polynomial matrix computations
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Black box linear algebra with the linbox library
Black box linear algebra with the linbox library
High-order lifting and integrality certification
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Smith normal form of dense integer matrices fast algorithms into practice
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Essentially optimal computation of the inverse of generic polynomial matrices
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
On the complexity of computing determinants
Computational Complexity
A BLAS based C library for exact linear algebra on integer matrices
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Efficient computation of the characteristic polynomial
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Generic matrix multiplication and memory management in linBox
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Computing the rank and a small nullspace basis of a polynomial matrix
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
Computing the smith forms of integer matrices and solving related problems
Computing the smith forms of integer matrices and solving related problems
Algebraic Complexity Theory
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We describe some major recent progress in exact and symbolic linear algebra. These advances concern the improvement of complexity estimates for fundamental problems such as linear system solution, determinant, inversion and computation of canonical forms. The matrices are over a finite field, the integers, or univariate polynomials. We show how selected techniques are key ingredients for the new solutions: randomization and algebraic conditioning, lifting, subspace approach, divide-double and conquer, minimum matrix polynomial, matrix approximants. These algorithmic progress allow the design of new generation high performance libraries such as LinBox, and open various research directions. We refer to [3] for an overview of methods in exact linear algebra, see also [37], [1] (in French), and [7, x2.3]. For fundamentals of computer algebra we refer to [16, 7].