A fast parallel algorithm to compute the rank of a matrix over an arbitrary field
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Updating LU factorizations for computing stationary distributions
SIAM Journal on Algebraic and Discrete Methods
Journal of the ACM (JACM)
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Downdating the Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
A Divide-and-Conquer Algorithm for the Bidiagonal SVD
SIAM Journal on Matrix Analysis and Applications
Nearly Optimal Algorithms For Canonical Matrix Forms
SIAM Journal on Computing
Dimensionality reduction for similarity searching in dynamic databases
SIGMOD '98 Proceedings of the 1998 ACM SIGMOD international conference on Management of data
An eigenspace update algorithm for image analysis
Graphical Models and Image Processing
The complexity of the matrix eigenproblem
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Lower bounds for dynamic algebraic problems
Information and Computation
Rational Matrix Functions and Rank-1 Updates
SIAM Journal on Matrix Analysis and Applications
Isomorphism of graphs with bounded eigenvalue multiplicity
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Segmentation Using Eigenvectors: A Unifying View
ICCV '99 Proceedings of the International Conference on Computer Vision-Volume 2 - Volume 2
Spectral partitioning works: planar graphs and finite element meshes
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Deterministic Computation of the Frobenius Form
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Faster algorithms for the characteristic polynomial
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Two uses for updating the partial singular value decomposition in latent semantic indexing
Applied Numerical Mathematics
Theoretical Computer Science
| Hi-index | 5.23 |
We present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix. In the generic symmetric case, our algorithm supports rank-one updates in O(n^2logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n^2klogn) randomized time, where k is the number of invariant factors of the matrix. The algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the Frobenius normal form or the tridiagonal symmetric form. The algorithm can be extended to solve the matrix eigenproblem with relative error 2^-^b in additional O(nlog^2nlogb) time. Furthermore, it can be used to dynamically maintain the singular value decomposition (SVD) of a generic matrix. Together with the algorithm, the hardness of the problem is studied. For the symmetric case, we present an @W(n^2) lower bound for rank-one updates and an @W(n) lower bound for element updates.