1,720,986 research outputs found
On the exponential of semi-infinite quasi-Toeplitz matrices
Get PDFLet a(z) = ∑ i ∈ Zaizi be a complex valued function defined for | z| = 1 , such that ∑ i ∈ Z| ai| < ∞; define T(a)=(ti,j)i,j∈Z+,ti,j=aj-i for i, j∈ Z+, the semi-infinite Toeplitz matrix associated with the symbol a(z); let E=(ei,j)i,j∈Z+ be a compact operator in lp, with 1 ≤ p≤ ∞. A semi-infinite matrix of the kind A= T(a) + E is said quasi-Toeplitz (QT). The problem of the computation of exp (A) or exp (A) v, with A quasi-Toeplitz and v a vector, arises in many applications. We prove that the exponential of a QT-matrix A is QT, that is, exp (A) = T(exp (a)) + F where F is a compact operator in lp. This property allows the design of an algorithm for computing exp (A) and exp (A) v up to any precision. The case of families of n× n matrices obtained by truncating infinite QT-matrices to finite size is also considered. Numerical experiments show the effectiveness of this approach. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature
Solving Quadratic Matrix Equations Arising in Random Walks in the Quarter Plane
Get PDFQuadratic matrix equations of the kind A_1 X^2 + A_0 X + A_{−1} = X are encountered in the analysis of Quasi–Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution G. In many queueing models, described by random walks in the quarter plane, the coefficients A_1 , A_0 , A_{−1} are infinite tridiagonal matrices with an almost Toeplitz structure. Here, we analyze some fixed point iterations, including Newton’s iteration, for the computation of G and introduce effective algorithms and acceleration strategies which fully exploit the Toeplitz structure of the matrix coefficients and of the current approximation. Moreover, we provide a structured perturbation analysis for the solution G. The results of some numerical experiments which demonstrate the effectiveness of our approach are reported
Numerical Solution of Algebraic Riccati Equations
No full textThis concise and comprehensive treatment of the basic theory of algebraic Riccati equations describes the classical as well as the more advanced algorithms for their solution in a manner that is accessible to both practitioners and scholars. It is the first book in which nonsymmetric algebraic Riccati equations are treated in a clear and systematic way. Some proofs of theoretical results have been simplified and a unified notation has been adopted.
Readers will find
- a unified discussion of doubling algorithms, which are effective in solving algebraic Riccati equations.
- a detailed description of all classical and advanced algorithms for solving algebraic Riccati equations and their MATLAB® codes. This will help the reader gain an understanding of the computational issues and provide ready-to-use implementation of the different solution techniques
A family of fast fixed point iterations for M/G/1-type Markov chains
Get PDFWe consider the problem of computing the minimal non-negative solution G of the nonlinear matrix equation X=∑∞i=−1AiXi+1 where Ai, for i⩾−1, are non-negative square matrices such that ∑∞i=−1Ai is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix G provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of G, which includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension
A note on computing matrix geometric means
No full textA new definition is introduced for the matrix geometric mean of a set of k positive definite n×n matrices together with an iterative method for its computation. The iterative method is locally convergent with cubic convergence and requires O(n 3 k 2) arithmetic operations per step whereas the methods based on the symmetrization technique of Ando et al. (Linear Algebra Appl 385:305–334, 2004) have complexity O(n 3 k!2 k ). The new mean is obtained from the properties of the centroid of a triangle rephrased in terms of geodesics in a suitable Riemannian geometry on the set of positive definite matrices. It satisfies most part of the ten properties stated by Ando, Li and Mathias; a counterexample shows that monotonicity is not fulfilled
Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes
No full textThe class of semi-infinite Analytically Quasi-Toeplitz (AQT) matrices is introduced. This class is formed by matrices which can be written in the form , where is the semi-infinite Toeplitz matrix associated with the symbol , that is , for , is a semi-infinite matrix such that is finite, and is an analytic function over an annulus for . We prove that AQT matrices are closed under multiplication and inversion, moreover we define a matrix norm such that for any pair of AQT matrices. We introduce a finite representation of AQT matrices and algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind , encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Geometric means of structured matrices
No full textThe geometric mean of positive definite matrices is usually identified
with the Karcher mean, which possesses all properties—generalized from the
scalar case— a geometric mean is expected to satisfy. Unfortunately,
the Karcher mean is typically not structure preserving, and destroys,
e.g., Toeplitz and band structures, which emerge in many
applications. For this reason, the Karcher mean is not always
recommended for modeling averages of structured matrices. In this
article a new definition of a geometric mean for structured matrices
is introduced, its properties are outlined, algorithms for its
computation, and numerical experiments are provided. In the Toeplitz case an existing mean based on the Kahler metric is analyzed for comparison.sponsorship: This work was
partially supported by MIUR grant number 2002014121; by the Research
Council KU Leuven, projects OT/11/055 (Spectral Properties of
Perturbed Normal Matrices and their Applications), CoE EF/05/006
Optimization in Engineering (OPTEC); by the Fund for Scientific
Research—Flanders (Belgium) project G034212N (Reestablishing
Smoothness for Matrix Manifold Optimization via Resolution of
Singularities); and by the Interuniversity Attraction Poles
Programme, initiated by the Belgian State, Science Policy Office,
Belgian Network DYSCO (Dynamical Systems, Control, and
Optimization).status: Publishe
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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