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Computing the Frobenius Normal Form of a Sparse Matrix
 CASC 2000 Proc. the Third International Workshop on Computer Algebra in Scientific Computing
, 2000
"... . We probabilistically determine the Frobenius form and thus the characteristic polynomial of a matrix A 2 F nn by O(n log(n)) multiplications of A by vectors and O n 2 log 2 (n) log log(n) arithmetic operations in the eld F. The parameter is the number of distinct invariant factors of A, ..."
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. We probabilistically determine the Frobenius form and thus the characteristic polynomial of a matrix A 2 F nn by O(n log(n)) multiplications of A by vectors and O n 2 log 2 (n) log log(n) arithmetic operations in the eld F. The parameter is the number of distinct invariant factors of A, it is less than 3 p n=2 in the worst case. The method requires O(n) storage space in addition to that needed for the matrix A. 1 Introduction The known complexity estimates of the computation of the characteristic polynomial and a fortiori, of the Frobenius normal form of special { sparse or black box { square matrices A over a eld F, seem to not be satisfactory. We refer to Kaltofen [8, Open Problem 3] and to Pan et al. [16, 15] for discussions on this subject and survey of current solutions. We denote by M(n) the number of operations in F required for nn matrix multiplications. The characteristic polynomial of a general matrix A can be computed at cost of O(n 3 ) or O(M(n) log...
Black Box Frobenius Decompositions over Small Fields (Extended Abstract)
, 2000
"... A new randomized algorithm is presented for computation of the Frobenius form and transition matrix for an n × n matrix over a field. Using standard matrix and polynomial arithmetic, the algorithm has an asymptotic expected complexity that matches the worst case complexity of the best know ..."
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Cited by 6 (0 self)
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A new randomized algorithm is presented for computation of the Frobenius form and transition matrix for an n &times; n matrix over a field. Using standard matrix and polynomial arithmetic, the algorithm has an asymptotic expected complexity that matches the worst case complexity of the best known deterministic algorithmic for this problem, recently given by Storjohann and Villard [16]. The new algorithm is based on the evaluation of Krylov spaces, rather than an elimination technique, and may therefore be superior when applied to sparse or structured matrices with a small number of invariant factors.
Asymptotically efficient algorithms for the Frobenius form
, 2000
"... A new randomized algorithm is presented for computation of the Frobenius form of an nn matrix over a field. A version of the algorithm is presented that uses standard arithmetic whose asymptotic expected complexity matches the worst case complexity of the best known deterministic algorithm for this ..."
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Cited by 4 (0 self)
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A new randomized algorithm is presented for computation of the Frobenius form of an nn matrix over a field. A version of the algorithm is presented that uses standard arithmetic whose asymptotic expected complexity matches the worst case complexity of the best known deterministic algorithm for this problem, recently given by Storjohann and Villard [25], and that seems to be superior when applied to sparse or structured matrices with a small number of invariant factors. A version that uses asymptotically fast matrix multiplication is also presented. This is the first known algorithm for this computation over small fields whose asymptotic complexity matches that of the best algorithm for computations over large fields and that also provides a Frobenius transition matrix over the ground field. As an application, it is shown that a "rational Jordan form" of an nn matrix over a finite field can also be computed asymptotically efficiently.
Rational Normal Form for Dynamical Systems by Carleman Linearization
, 1999
"... We propose in this paper a rational normal form for dynamical systems or vector elds. That is to say if the coecients of the system are in a eld K (which, in practice, is Q;R), so is the computed rational normal form. We give an algorithm for an eective computation of the normal form up to a nite o ..."
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We propose in this paper a rational normal form for dynamical systems or vector elds. That is to say if the coecients of the system are in a eld K (which, in practice, is Q;R), so is the computed rational normal form. We give an algorithm for an eective computation of the normal form up to a nite order. We don't need to compute the Jordan canonical form of the leading matrix neither its eigenvalues. We give also a rational method to test the existence of resonance of a xed order without computing the eigenvalues of the linear part. Our normal form is a renement of the classical normal forms. Our method is applicable for both the nilpotent and the non nilpotent cases.
Fixed Points in Discrete Models for Regulatory Genetic Networks
, 2007
"... It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that ..."
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It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that enable the detection of steady states that are modeled by fixed points in discrete finite dynamical systems. We discuss two algebraic models, a univariate model and a multivariate model. We show that these two models are equivalent and that one can be converted to the other by means of a discrete Fourier transform. We give a new, more general definition of a linear finite dynamical system and we give a necessary and sufficient condition for such a system to be a fixed point system, that is, all cycles are of length one. We show how this result for generalized linear systems can be used to determine when certain nonlinear systems (monomial dynamical systems over finite fields) are fixed point systems. We also show how it is possible to determine in polynomial time when an ordinary linear system (defined over a finite field) is a fixed point system. We conclude with a necessary condition for a univariate finite dynamical system to be a fixed point system.
COMPUTING MINIMAL POLYNOMIALS OF MATRICES
"... We present and analyse a MonteCarlo algorithm to compute the minimal polynomial of an n × n matrix over a finite field that requires O(n³) field operations and O(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard a ..."
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We present and analyse a MonteCarlo algorithm to compute the minimal polynomial of an n × n matrix over a finite field that requires O(n³) field operations and O(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard algorithms for polynomial and matrix operations. We compare features of the algorithm with several other algorithms in the literature. In addition we present a deterministic verification procedure which is similarly efficient in most cases but has a worstcase complexity of O(n^4). Finally, we report the results of practical experiments with an implementation of our algorithms in comparison with the current algorithms in the GAP library.
POLYNOMIALTIME ALGORITHMS FOR QUADRATIC ISOMORPHISM OF POLYNOMIALS
, 2013
"... ABSTRACT. Let K be a field, f= ( f1,..., fm) and g=(g1,...,gm) be two sets of m�1 nonlinear polynomials over K[x1,...,xn]. We consider the computational problem of finding – if any – an invertible transformation on the variables mapping f to g. The corresponding equivalence problem is known as Isom ..."
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ABSTRACT. Let K be a field, f= ( f1,..., fm) and g=(g1,...,gm) be two sets of m�1 nonlinear polynomials over K[x1,...,xn]. We consider the computational problem of finding – if any – an invertible transformation on the variables mapping f to g. The corresponding equivalence problem is known as Isomorphism of Polynomials with one Secret (IP1S) and is a fundamental problem in multivariate cryptography; the problem is also calledPolyProj when m=1. Agrawal and Saxena show that Graph Isomorphism (GI) reduces to equivalence of cubic polynomials with respect to an invertible linear change of variables. This strongly suggests that solving equivalence problems efficiently, i.e. in polynomialtime, is a very challenging algorithmic task. Then, following Kayal at SODA’11, we search for large families of polynomials equivalence which can be solved efficiently. The main result is a randomized polynomialtime algorithm for solvingIP1S for quadratic instances, a particular case of importance in cryptography and somewhat justifying a posteriori the fact that GI reduces to only cubic instances of IP1S. To this end, we show that IP1S for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory, which involves to test
Dynamic Load Balancing of Parallel SURF with Vertical Partitioning
"... Abstract—The demand for realtime processing of robust feature detection is one of the major issues in the computer vision field. In order to comply with the requirements, in this paper a parallelization and optimization method to effectively accelerate SURF is proposed. The proposed parallelization ..."
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Abstract—The demand for realtime processing of robust feature detection is one of the major issues in the computer vision field. In order to comply with the requirements, in this paper a parallelization and optimization method to effectively accelerate SURF is proposed. The proposed parallelization method is developed based on a workload analysis of SURF in terms of various aspects, focusing in particular on the load balancing problem. First, the average parallel workload is divided into identical portions using the vertical partitioning method. Then, the load imbalance problem is further resolved using the dynamic partition balancing method. In addition, an optimization method is proposed together with the parallelization method to find and exclude redundant operations in SURF, thus effectively accelerating the feature detection operation when the proposed parallelization method is applied. The proposed method shows a maximum speedup of 19.21 compared to the single threaded performance on a 24core system, achieving a maximum of 83.80 fps in a realmachine experiment, enabling realtime processing. Index Terms—Image processing and computer vision, edge and feature detection, SURF, parallel computing, multithreading F 1