Differential qd algorithm with shifts for rank-structured matrices

Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable. In particular, in the symmetric tridiagonal case where differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often faster convergence while preserving high relative accuracy (that is not guaranteed in QR algorithm). In eigenvalue computations for rank-structured matrices QR algorithm is also a popular choice since, in the symmetric case, the rank structure is preserved. In the unsymmetric case, however, QR algorithm destroys the rank structure and, hence, LR-type algorithms come to play once again. In the current paper we discover several variants of qd algorithms for quasiseparable matrices. Remarkably, one of them, when applied to Hessenberg matrices becomes a direct generalization of dqds algorithm for tridiagonal matrices. Therefore, it can be applied to such important matrices as companion and confederate, and provides an alternative algorithm for finding roots of a polynomial represented in the basis of orthogonal polynomials. Results of preliminary numerical experiments are presented

A Corpus of Sentence-level Revisions in Academic Writing: A Step towards Understanding Statement Strength in Communication

The strength with which a statement is made can have a significant impact on the audience. For example, international relations can be strained by how the media in one country describes an event in another; and papers can be rejected because they overstate or understate their findings. It is thus important to understand the effects of statement strength. A first step is to be able to distinguish between strong and weak statements. However, even this problem is understudied, partly due to a lack of data. Since strength is inherently relative, revisions of texts that make claims are a natural source of data on strength differences. In this paper, we introduce a corpus of sentence-level revisions from academic writing. We also describe insights gained from our annotation efforts for this task.Comment: 6 pages, to appear in Proceedings of ACL 2014 (short paper

Structured condition numbers for parameterized quasiseparable matrices

Low-rank structured matrices have attracted much attention in the last decades, since they arise in many applications and all share the fundamental property that can be represented by O(n) parameters, where n x n is the size of the matrix. This property has allowed the development of fast algorithms for solving numerically many problems involving low-rank structured matrices by performing operations on the parameters describing the matrices, instead of directly on the matrix entries. Among these problems the solution of linear systems of equations and the computation of the eigenvalues are probably the most basic and relevant ones. Therefore, it is important to measure, via structured computable condition numbers, the relative sensitivity of the solutions of linear systems with low-rank structured coefficient matrices, and of the eigenvalues of those matrices, with respect to relative perturbations of the parameters representing such matrices, since this sensitivity determines the maximum accuracy attainable by fast algorithms and allows us to decide which set of parameters is the most convenient from the point of view of accuracy. In this PhD Thesis we develop and analyze condition numbers for eigenvalues of low-rank matrices and for the solutions of linear systems involving such matrices. To this purpose, general expressions are obtained for the condition numbers of the solution of a linear system of equations whose coefficient matrix is any differentiable function of a vector of parameters with respect to perturbations of such parameters, and also for the eigenvalues of those matrices. Since there are many different classes of low-rank structured matrices and many different types of parameters describing them, it is not possible to cover all of them in this thesis. Therefore, the general expressions of the condition numbers are particularized to the important case of quasiseparable matrices and to the quasiseparable and the Givens-vector representations. In the case of {1,1}-quasiseparable matrices, we provide explicit expressions of the corresponding condition numbers for these two representations that can be estimated in O(n) operations. In addition, detailed theoretical and numerical comparisons of the condition numbers with respect to these two representations between themselves, and with respect to unstructured condition numbers are provided. These comparisons show that there are situations in which the unstructured condition numbers are much larger than the structured ones, but that the opposite never happens (...). The approach presented in this dissertation can be generalized to other classes of low-rank structured matrices and parameterizations, as well as to any class of structured matrices that can be represented by parameters, independently of whether or not they enjoy a “low-rank” structure.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Ana María Urbano Salvador.- Secretario: Fernando de Terán Vergara.- Vocal: J. Javier Martínez Fernández-Hera

Fiedler matrices: numerical and structural properties

The first and second Frobenius companion matrices appear frequently in numerical application, but it is well known that they possess many properties that are undesirable numerically, which limit their use in applications. Fiedler companion matrices, or Fiedler matrices for brevity, introduced in 2003, is a family of matrices which includes the two Frobenius matrices. The main goal of this work is to study whether or not Fiedler companion matrices can be used with more reliability than the Frobenius ones in the numerical applications where Frobenius matrices are used. For this reason, in this work we present a thorough study of Fiedler matrices: their structure and numerical properties, where we mean by numerical properties those properties that are interesting for applying these matrices in numerical computations, and some of their applications in the field on numerical linear algebra. The introduction of Fiedler companion matrices is an example of a simple idea that has been very influential in the development of several lines of research in the numerical linear algebra field. This family of matrices has important connections with a number of topics of current interest, including: polynomial root finding algorithms, linearizations of matrix polynomials, unitary Hessenberg matrices, CMV matrices, Green’s matrices, orthogonal polynomials, rank structured matrices, quasiseparable and semiseparable matrices, etc.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Paul Van Dooren.- Secretario: Juan Bernardo Zaballa Tejada.- Vocal: Françoise Tisseu