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Research
Centre Scientific Computing: Applied Linear Algebra
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2014 - Autumn semester December 26, 15:00, Room #60 December 18, 12:00, Room #60 The only existing algorithms for computing minimal Geršgorin sets are given in the paper of by Varga, Cvetković and Kostić. Here, we present an improvement of this numerical method that is suitable for small and medium size (dense) matrices. New computational technique uses modified Newton's method to find zeros of the parameter dependent left-most eigenvalue of a Z-matrix. For several test examples, behaviour of the new method is compared with the already existing ones. December 4, 12:00, Room #60 We study Maker-Breaker games played on the edge set of the complete graph on n vertices, Kn, for sufficiently large integer n. We look at the (1:b) Perfect matching game and (1:b) Hamiltonicity game and for these two games, we are interested in determining the least number of moves that Maker needs to make in order to win and, on the other hand, for how long can Breaker delay Maker's win. This is a joint work with Asaf Ferber, Dan Hefetz and Miloš Stojaković. November 27, 12:00, Room #60 As nonlinear eigenvalue problems (NLEP) appear in many applications, and the research concerning their proper treatment has drawn a lot of attention in the past few years, there exists a need for developing computationally inexpensive ways to localize eigenvalues of nonlinear matrix valued functions in the complex plane, especially eigenvalues of matrix quadratics. Recently, few variants of Geršgorin localization set for generalized eigenvalue problems were developed and investigated. Here, we will introduce Geršgorin set for quadratic eigenvalues using diagonal dominance, prove some properties of such set and show how this set performs on several test problems. November 13, 12:00, Room #60 Entropy is a measure of uncertainty which is useful in many applications. It quantifies the expected value of information contained in discrete distribution (Shannon entropy) or in continuous distribution (differential entropy). Two different approaches to estimation of the differential entropy are considered in this paper. The first one is nonparametric, based on discretization of a random variable and calculating Shannon entropy. This can be done means of several algorithms, which are included in Entropy package in R. The second one is parametric, based on plugging maximum likelihood estimates of the distribution parameters into the exact differential entropy formula. The main problem in both approaches to estimation of the differential entropy is finite sample bias. The aim of our study is to determine the level of bias for different estimators. For several well known distributions (normal, Student, uniform, lognormal, exponential and Gumbel), we generate a thousand samples for each of the sample sizes 20, 50, 100, 200, 500, and calculate the differential entropy estimates using two different approaches. Using these data, we investigate the distribution of differential entropy estimators. In order to compare the performances of applied procedures for estimation, we compare the distribution of differential entropy estimators with respect to mean value, variance, kurtosis, skewness, bias and mean squared error. October 30, 12:00, Room #60 One of the widest researched matrix nearness problems is how to compute distance to stability of dynamical systems in discrete and continuous sense. These can be viewed as a way to determine robustness of the matrix spectrum localization by the unit disk (discrete) and the open left-half plane (continuous). However, in some applications (population models, vibro-acoustic models, structural mechanics, etc.) other spectrum localization domains in the complex plane determine favorable behavior of the matrix. For example, if one needs to limit the frequency of the sable (or unstable) eigenmodes more complex spectrum inclusion domains need to be used. Motivated by this, we consider Lyapunov-type domains in their general setting and formulate appropriate matrix nearness problems. For this new concept - the distance to delocalization - which generalizes both, the distance to d-stability and the distance to c-stability, we present numerical algorithms for its computation. The cases of medium size dense matrices and large sparse matrices are treated separately, and several implementations that suite certain practical conditions are developed. Finally, for few test problems we present behavior of the algorithms and show how some practical questions arising in applications can be answered. October 23, 12:00, Room #60 September 29, 12:00, Room #59/3 |
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