Matrix structure and loss-resilient encoding/decoding

AbstractThe known deterministic algorithms for loss-resilient encoding/decoding involve computations with Cauchy matrices but only weakly exploit the matrix structure. We propose several modifications with more extensive use of the matrix structure to accelerate the computations substantially

Polynomial evaluation over finite fields: new algorithms and complexity bounds

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation in the decoding of Reed-Solomon codes are highlighted.Comment: accepted for publication in Applicable Algebra in Engineering, Communication and Computing. The final publication will be available at springerlink.com. DOI: 10.1007/s00200-011-0160-

New transformations of Cauchy matrices and Trummer's problem

AbstractWe show some new expressions for a Cauchy matrix, which enable us to simplify the solution of Trummer's problem, both in the general case and in the case where the input Cauchy matrix is fixed for the problem whereas the input vector varies

Simultaneous measurements of water optical properties by AC9 transmissometer and ASP-15 Inherent Optical Properties meter in Lake Baikal

Measurements of optical properties in media enclosing Cherenkov neutrino telescopes are important not only at the moment of the selection of an adequate site, but also for the continuous characterization of the medium as a function of time. Over the two last decades, the Baikal collaboration has been measuring the optical properties of the deep water in Lake Baikal (Siberia) where, since April 1998, the neutrino telescope NT-200 is in operation. Measurements have been made with custom devices. The NEMO Collaboration, aiming at the construction of a km3 Cherenkov neutrino detector in the Mediterranean Sea, has developed an experimental setup for the measurement of oceanographic and optical properties of deep sea water. This setup is based on a commercial transmissometer. During a joint campaign of the two collaborations in March and April 2001, light absorption, scattering and attenuation in water have been measured. The results are compatible with previous ones reported by the Baikal Collaboration and show convincing agreement between the two experimental techniques.Comment: 16 pages, submitted to NIM-

Real Polynomial Root-Finding by Means of Matrix and Polynomial Iterations

Recently we proposed to extend the matrix sign classical iteration to the approximation of the real eigenvalues of a companion matrix of a polynomial and consequently to the approximation of its real roots. In our present paper we advance this approach further by combining it with the alternative square root iteration for polynomials and also show a variation using repeated squaring in polynomial algebra

Improved algorithms for computing determinants and resultants

Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via computing the resultant. This is achieved by combining our techniques with the primitive-element method, which leads to an effective implicit representation of the roots. We systematically examine quotient formulae of Sylvester-type resultant matrices, including matrix polynomials and the u-resultant. We reduce the known bit operation complexity bounds by almost an order of magnitude, in terms of the resultant matrix dimension. Our theoretical and practical improvements cover the highly important cases of sparse and degenerate systems. © 2004 Elsevier Inc. All rights reserved