Efficient computation of the characteristic polynomial
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
A block Wiedemann rank algorithm
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Faster inversion and other black box matrix computations using efficient block projections
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Some recent progress in exact linear algebra and related questions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
ACM Communications in Computer Algebra
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages
ACM Transactions on Mathematical Software (TOMS)
Efficient Disjointness Tests for Private Datasets
ACISP '08 Proceedings of the 13th Australasian conference on Information Security and Privacy
Finding the Growth Rate of a Regular of Context-Free Language in Polynomial Time
DLT '08 Proceedings of the 12th international conference on Developments in Language Theory
Unconditionally secure disjointness tests for private datasets
International Journal of Applied Cryptography
On finding multiplicities of characteristic polynomial factors of black-box matrices
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Integer matrix rank certification
Proceedings of the 2009 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
Generating a d-dimensional linear subspace efficiently
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Kaltofen's division-free determinant algorithm differentiated for matrix adjoint computation
Journal of Symbolic Computation
Run-time efficient probabilistic model checking
Proceedings of the 33rd International Conference on Software Engineering
Recent progress in linear algebra and lattice basis reduction
Proceedings of the 36th international symposium on Symbolic and algebraic 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
Supersparse black box rational function interpolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Mathematics of Operations Research
Bayesian learning with mixtures of trees
ECML'06 Proceedings of the 17th European conference on Machine Learning
On regular and logarithmic solutions of ordinary linear differential systems
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
An OPT+1 algorithm for the cutting stock problem with constant number of object lengths
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
An output-sensitive algorithm for computing projections of resultant polytopes
Proceedings of the twenty-eighth annual symposium on Computational geometry
Chip-firing games, potential theory on graphs, and spanning trees
Journal of Combinatorial Theory Series A
Faster geometric algorithms via dynamic determinant computation
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Deterministic unimodularity certification
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Computing the invariant structure of integer matrices: fast algorithms into practice
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
On the matrix berlekamp-massey algorithm
ACM Transactions on Algorithms (TALG)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n 脳 n matrix A with integer entries in $$\left( {n^{3.2} \log \left\| A \right\|} \right)^{1 + o(1)} $$ and $$\left( {n^{2.697263} \log \left\| A \right\|} \right)^{1 + o(1)} $$ bit operations; here $$\left\| A \right\|$$ denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors $$C_1 (\log n)^{C_2 } \left( {\log \log \left\| A \right\|} \right)^{C_3 } $$ for positive real constants C1, C2, C3. The bit complexity $$\left( {n^{3.2} \log \left\| A \right\|} \right)^{1 + o(1)} $$ results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n3.2+o(1) and O(n2.697263) ring additions, subtractions and multiplications.