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Nearest common root of a set of polynomials: A structured singular value approach
The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (μ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an “approximate GCD” of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the “approximate GCD” of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations
Graphing of E-Science Data with varying user requirements
Based on our experience in the Swiss Experiment, exploring experimental, scientific data is often done in a visual way. Starting from a global overview the users are zooming in on interesting events. In case of huge data volumes special data structures have to be introduced to provide fast and easy access to the data. Since it is hard to predict on how users will work with the data a generic approach requires self-adaptation of the required special data structures. In this paper we describe the underlying NP-hard problem and present several approaches to address the problem with varying properties. The approaches are illustrated with a small example and are evaluated with a synthetic data set and user queries
On the prime divisors of elements of a quadruple
We show that if {1, b, c, d} is a D(-1) diophantine quadruple with b<c<d and
c=1+s^2, then the cases s=p^k, s=2p^k, c=p and c=2p^k do not occur, where p is
an odd prime and k is a positive integer. For the integer d=1+x^2, we show that
it is not prime and that x is divisible by at least two distinct odd primes.
Furthermore, we present several infinite families of integers b such that the
D(-1) pair {1, b} cannot be extended to a D(-1) quadruple. For instance, we
show that if r=5p where p is an odd prime, then the D(-1) pair {1, r^2+1}
cannot be extended to a D(-1) quadruple
Generalized companion matrix for approximate GCD
We study a variant of the univariate approximate GCD problem, where the
coefficients of one polynomial f(x)are known exactly, whereas the coefficients
of the second polynomial g(x)may be perturbed. Our approach relies on the
properties of the matrix which describes the operator of multiplication by gin
the quotient ring C[x]=(f). In particular, the structure of the null space of
the multiplication matrix contains all the essential information about GCD(f;
g). Moreover, the multiplication matrix exhibits a displacement structure that
allows us to design a fast algorithm for approximate GCD computation with
quadratic complexity w.r.t. polynomial degrees.Comment: Submitted to MEGA 201
The Euclidean Algorithm for Generalized Minimum Distance Decoding of Reed-Solomon Codes
This paper presents a method to merge Generalized Minimum Distance decoding
of Reed-Solomon codes with the extended Euclidean algorithm. By merge, we mean
that the steps taken to perform the Generalized Minimum Distance decoding are
similar to those performed by the extended Euclidean algorithm. The resulting
algorithm has a complexity of O(n^2)
Fast Arithmetics Using Chinese Remaindering
In this paper, some issues concerning the Chinese remaindering representation
are discussed. Some new converting methods, including an efficient
probabilistic algorithm based on a recent result of von zur Gathen and
Shparlinski \cite{Gathen-Shparlinski}, are described. An efficient refinement
of the NC division algorithm of Chiu, Davida and Litow
\cite{Chiu-Davida-Litow} is given, where the number of moduli is reduced by a
factor of
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