Towards an exact adaptive algorithm for the determinant of a rational matrix

In this paper we propose several strategies for the exact computation of the determinant of a rational matrix. First, we use the Chinese Remaindering Theorem and the rational reconstruction to recover the rational determinant from its modular images. Then we show a preconditioning for the determinant which allows us to skip the rational reconstruction process and reconstruct an integer result. We compare those approaches with matrix preconditioning which allow us to treat integer instead of rational matrices. This allows us to introduce integer determinant algorithms to the rational determinant problem. In particular, we discuss the applicability of the adaptive determinant algorithm of [9] and compare it with the integer Chinese Remaindering scheme. We present an analysis of the complexity of the strategies and evaluate their experimental performance on numerous examples. This experience allows us to develop an adaptive strategy which would choose the best solution at the run time, depending on matrix properties. All strategies have been implemented in LinBox linear algebra library

Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem

We define a notion of network capacity region of networks that generalizes the notion of network capacity defined by Cannons et al. and prove its notable properties such as closedness, boundedness and convexity when the finite field is fixed. We show that the network routing capacity region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. We define the semi-network linear coding capacity region, with respect to a fixed finite field, that inner bounds the corresponding network linear coding capacity region, show that it is a computable rational polytope, and provide exact algorithms and approximation heuristics. We show connections between computing these regions and a polytope reconstruction problem and some combinatorial optimization problems, such as the minimum cost directed Steiner tree problem. We provide an example to illustrate our results. The algorithms are not necessarily polynomial-time.Comment: Appeared in the 2011 IEEE International Symposium on Information Theory, 5 pages, 1 figur

Representation of the Lagrange reconstructing polynomial by combination of substencils

The Lagrange reconstructing polynomial [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] of a function $f(x)$ on a given set of equidistant (\Delta x=\const) points $\bigl\{x_i+\ell\Delta x;\;\ell\in\{-M_-,...,+M_+\}\bigr\}$ is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with $x$) averages on $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to the Lagrange interpolating polynomial of $f(x)$ on the same stencil. We first study the fundamental functions of Lagrange reconstruction, show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) $x_i+n\tfrac{1}{2}\Delta x$ ($n\in\mathbb{Z}$), and obtain several identities. Using these identities, by analogy to the recursive Neville-Aitken-like algorithm applied to the Lagrange interpolating polynomial, we show that there exists a unique representation of the Lagrange reconstructing polynomial on $\{i-M_-,...,i+M_+\}$ as a combination of the Lagrange reconstructing polynomials on the $K_\mathrm{s}+1\leq M:=M_-+M_+>1$ substencils $\{i-M_-+k_\mathrm{s},...,i+M_+-K_\mathrm{s}+k_\mathrm{s}\}$ ($k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$), with weights $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ which are rational functions of $\xi$ ($x=x_i+\xi\Delta x$) [Liu Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503--538], and give an analytical recursive expression of the weight-functions. We then use the analytical expression of the weight-functions $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of $\xi=\tfrac{1}{2}$, under the condition that all of the substencils contain either point $i$ or point $i+1$ (or both).Comment: final corrected version; in print J. Comp. Appl. Mat

Oversampling in shift-invariant spaces with a rational sampling period

8 pages, no figures.It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang-Fix conditions of order r, this formula also provides an approximation scheme of order r valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support.This work has been supported by the Grant MTM2009-08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología

A Comparison of Point Data Selection Schemes for Evolutionary Surface Reconstructions

This article presents a study of the application of Computational Intelligence (CI) methods to the problem of optimal surface reconstruction using triangulations and NURBS (Non-Uniform Rational B-Splines) surface approximations on digitized point data. In mechanical engineering surface reconstructions are used to transform physical objects into mathematical representations for computer aided design purposes. In order to record the geometrical shape of the objects, tactile or optical sensors generate point sets with a huge number of sample points. The number and distribution of these points are decisive for the quality and computational efficiency of the numerical surface representations. Triangulations and NURBS are widely used in CAD/CAM-applications, because they belong to a class of very exible discrete interpolation and approximation methods. In order to verify the suitability of surface model independent point selection schemes and to find model dependent sampling point distributions, optimal surface reconstructions are used

Evolutionary distances in the twilight zone -- a rational kernel approach

Phylogenetic tree reconstruction is traditionally based on multiple sequence alignments (MSAs) and heavily depends on the validity of this information bottleneck. With increasing sequence divergence, the quality of MSAs decays quickly. Alignment-free methods, on the other hand, are based on abstract string comparisons and avoid potential alignment problems. However, in general they are not biologically motivated and ignore our knowledge about the evolution of sequences. Thus, it is still a major open question how to define an evolutionary distance metric between divergent sequences that makes use of indel information and known substitution models without the need for a multiple alignment. Here we propose a new evolutionary distance metric to close this gap. It uses finite-state transducers to create a biologically motivated similarity score which models substitutions and indels, and does not depend on a multiple sequence alignment. The sequence similarity score is defined in analogy to pairwise alignments and additionally has the positive semi-definite property. We describe its derivation and show in simulation studies and real-world examples that it is more accurate in reconstructing phylogenies than competing methods. The result is a new and accurate way of determining evolutionary distances in and beyond the twilight zone of sequence alignments that is suitable for large datasets.Comment: to appear in PLoS ON