MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Theoretical Computer Science
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We present the first fully dynamic algorithm for computing thecharacteristic polynomial of a matrix. In the generic symmetriccase our algorithm supports rank-one updates inO(n 2 logn) randomized timeand queries in constant time, whereas in the general case thealgorithm works in O(n 2 klogn) randomized time, where k is the number ofinvariant factors of the matrix. The algorithm is based on thefirst dynamic algorithm for computing normal forms of a matrix suchas the Frobenius normal form or the tridiagonal symmetric form. Thealgorithm can be extended to solve the matrix eigenproblem withrelative error 2-b in additionalO(n log2 n logb)time. Furthermore, it can be used to dynamically maintain thesingular value decomposition (SVD) of a generic matrix. Togetherwith the algorithm the hardness of the problem is studied. For thesymmetric case we present an Ω(n2) lower bound for rank-one updates and anΩ(n) lower bound for element updates.