Dynamic normal forms and dynamic characteristic polynomial

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(n2logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n2klogn) random...

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Bibliographic Details
Published in:Theoretical computer science Vol. 412; no. 16; pp. 1470 - 1483
Main Authors: Frandsen, Gudmund Skovbjerg, Sankowski, Piotr
Format: Journal Article
Language:English
Published: Elsevier B.V 01.04.2011
Subjects:
Algorithms
Characteristic polynomial
Computation
Decomposition
Dynamic algorithm
Dynamic characteristics
Dynamics
Eigenproblem
Hardness
Invariants
Lower bound
Lower bounds
Matrix normal form
Queries
Lower bound
Matrix normal form
Eigenproblem
Dynamic algorithm
Characteristic polynomial
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary: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(n2logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n2klogn) 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(nlog2nlogb) 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 Ω(n2) lower bound for rank-one updates and an Ω(n) lower bound for element updates.
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2010.11.049