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(n log n) randomized time and queries in constant time, whereas in the general case the algorithm works in O(nk log n) 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(n log n log b) 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(n) lower bound for rank-one updates and an Ω(n) lower bound for element updates. Introduction. The computation of the characteristic polynomial (CP) of a matrix and the eigenproblem are two important problems in linear algebra and they find an enormous number of applications in mathematics, physics and computer science. Till now almost nothing about the dynamic complexity of these problems has been known. The CP problem is essentially equivalent to the computation of the Frobenius Normal Form (FNF), known also as rational canonical form. All of the efficient algorithms for CP are based on the FNF computation [1–4]. The fastest static algorithms for computing CP are either based on fast matrix multiplication and work in Õ(n) time [4, 1] or they are so called black-box approaches working in Õ(nm) time [2, 3], where m is the number of nonzero entries in the matrix. The latter bound holds only in the generic case, whereas the fastest general algorithm works in O(μnm) [3], where μ is the number of distinct invariant factors of the matrix. All of these results have been obtained very recently. In this paper we are trying to understand the dynamic complexity of these fundamental problems by devising efficient algorithms and by proving matching lower bounds. Note, that in this paper and in all of the papers cited above, we study the arithmetic complexity of the problem, i.e., the notion of time is equivalent to the count of arithmetic operations and discrete control operations. More strictly speaking we work in the real RAM model, for details see [5]. In the first part of the paper we consider the problem of computing the normal form of a real (complex) n× n dynamic matrix A. We assume that the matrix can be changed with use of rank-one updates, i.e., for two n dimensional vectors a and b we allow updates of the form A := A + ab . We want to dynamically compute matrices Q and F such that A = Q−1FQ, where Q is the unitary similarity transformation and F is the normal form in question. The algorithm should support queries to F as well as vector queries to Q, i.e., given vector v it should be able to return Qv or Q−1v. We present the following fully dynamic randomized algorithms for this problem: – for generic symmetric matrices — an algorithm for tridiagonal normal form supporting updates in O(n log n) worst-case time, – for general matrices — an algorithm for Frobenius normal form (for definition see Section 1.1) supporting updates in O(kn log n) worst-case time, where k is the number of invariant factors of the matrix. The queries for F are answered in O(1) time and the queries to Q in O(n log n) worst-case time. After each update the algorithms can compute the characteristic polynomial explicitly and hence support queries for CP in constant time. These are the first known fully dynamic algorithms for the CP and normal form problems. Our results are based on a general result which can be applied to any normal form, under condition of availability of a static algorithm for computing the normal form of a sparse matrix. For the completeness of the presentation we have included the full algorithm for generic symmetric matrices, whereas the included algorithm for Frobenius normal form is the most universal result. This algorithm can be extended to solve the dynamic eigenproblem, i.e., we are asked to maintain with relative error 2−b the eigenvalues λ1, . . . , λn and a matrix Q composed of eigenvectors. In generic case, our algorithm supports updates in O(n log n log b + n log n) worst-case time, queries to λi in constant time and vector queries to Q in O(n log n) worst-case time. You should note that the relative error 2−b is immanent even in the exact arithmetic model, i.e., the eigenvalues can only be computed approximately (for more details please see [6]). Let A be a real (complex) m×n matrix, m ≥ n. The singular value decomposition (SVD) for A consists in two orthogonal (unitary) matrices U and V and a diagonal matrix Σ = diag(σ1, . . . , σn) with nonnegative real entries (the singular values) such that A = UΣV T . Usually the entries of Σ are sorted σ1 ≥ · · · ≥ σn ≥ 0 and in that case Σ is unique. We define Σk to be diag(σ1, . . . , σk, 0, . . . , 0). The dynamic SVD problem considers maintaining the SVD under rank one updates and 2 query operations, one operation returns elements of Σ, another returns the k-rank approximation to A, i.e., given r and v return UΣrV T v. Here, again the results are with a relative error 2−b. Our algorithm for SVD supports updates in O(n log n log b+n log n) worst-case time in the generic case, queries to Σ in constant time and r-rank approximation query in O(n log n) worst-case time. Accompanying the above upper bounds, we provide some lower bounds for the problem of computing the characteristic polynomial. The lower bounds are formulated in the model of history dependent algebraic computation trees [7]. One should note that our algorithms for computing the CP fit into this model. We