Computational error bounds for multiple or nearly multiple eigenvalues

AbstractIn this paper bounds for clusters of eigenvalues of non-selfadjoint matrices are investigated. We describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues, and for a basis of the corresponding invariant subspaces. The input matrix may be real or complex, dense or sparse. The method is based on a quadratically convergent Newton-like method; it includes the case of defective eigenvalues, uncertain input matrices and the generalized eigenvalue problem. Computational results show that verified bounds are still computed even if other eigenvalues or clusters are nearby the eigenvalues under consideration

Effects of Home-Away Sequencing on the Length of Best-of-Seven Game Playoff Series

We analyze the number of games played in a seven-game playoff series under various homeaway sequences. In doing so, we employ a simple Bernoulli model of home-field advantage in which the outcome of each game in the series depends only on whether it is played at home or away with respect to a designated home team. Considering all such sequences that begin and end at home, we show that, in terms of the number of games played, there are four classes of stochastically different formats, including the popular 2-3 and 2-2 formats both currently used in National Basketball Association (NBA) playoffs. Characterizing the regions in parametric space that give rise to distinct stochastic and expected value orderings of series length among these four format classes, we then investigate where in this parametric space that teams actually play. An extensive analysis of historical 7-game playoff series data from the NBA reveals that this homeaway model is preferable to the simpler, well-studied but ill-fitting binomial model that ignores home-field advantage. The model suggests that switching from the 2-2 series format used for most of the playoffs to the 2-3 format that has been used in the NBA Finals since a switch in 1985 would stochastically lengthen these playoff series, creating an expectation of approximately one extra game per playoff season. Such evidence should encourage television sponsors to lobby for a change of playoff format in order to garner additional television advertising revenues while reducing team and media travel costs

Data Clustering for Fitting Parameters of a Markov Chain Model of Multi-Game Playoff Series

We propose a Markov chain model of a best-of-7 game playoff series that involves game-togame dependence on the current status of the series. To create a relatively parsimonious model, we seek to group transition probabilities of the Markov chain into clusters of similar game-winning frequency. To do so, we formulate a binary optimization problem to minimize several measures of cluster dissimilarity. We apply these techniques on Major League Baseball (MLB) data and test the goodness of fit to historical playoff outcomes. These state-dependent Markov models improve significantly on probability models based solely on home-away game dependence. It turns out that a better two-parameter model ignores where the games are played and instead focuses simply on, for each possible series status, whether or not the team with home-field advantage in the series has been the historical favorite - the more likely winner - in the next game of the series

VARIATIONAL CHARACTERIZATIONS OF THE SIGN-REAL AND THE SIGN-COMPLEX SPECTRAL RADIUS ∗

Key words. Generalized spectral radius, sign-real spectral radius, sign-complex spectral radius, Perron-Frobenius theory. AMS subject classifications. 15A48, 15A18 Abstract. The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one correspondences to classical Perron-Frobenius theory. In this paper the corresponding inf-max characterization as well as variational characterizations of the generalized (real and complex) spectral radius are presented. Again those are almost identical to the corresponding results in classical Perron-Frobenius theory. 1. Introduction. Denote R+: = {x ≥ 0: x ∈ R}, andletK ∈{R+, R, C}. The generalized spectral radius is defined [6] by (1.1) ρ K (A):=max{|λ | : ∃ 0 = x ∈ K n, ∃ λ ∈ K, |Ax | = |λx|} for A ∈ Mn(K)

On relative errors of floating-point operations: optimal bounds and applications

International audienceRounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function fl and barring underflow and overflow, such models bound the relative errors E 1 (t) = |t − fl(t)|/|t| and E 2 (t) = |t − fl(t)|/|fl(t)| by the unit roundoff u. This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 (t) and E 2 (t) are optimally bounded by u/(1 + u) and u, respectively, when t is real or, under mild assumptions on the base and the precision, when t = x ± y or t = xy with x, y two floating-point numbers. We prove that while this remains true for division in base β > 2, smaller, attainable bounds can be derived for both division in base β = 2 and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves

Improved error bounds for inner products in floating-point arithmetic

International audienceGiven two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for inner product returns a floating-point number $\hat r$ such that $|{\hat r}-x^Ty| \le nu|x|^T|y|$ with $u$ the unit roundoff. This result, which holds for any radix and with no restriction on $n$, can be seen as a generalization of a similar bound given in~\cite{Rump12} for recursive summation in radix $2$, namely $|{\hat r}- x^Te| \le (n-1)u|x|^Te$ with $e=[1,1,\ldots,1]^T$. As a direct consequence, the error bound for the floating-point approximation $\hat C$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\hat{C}-AB|\le nu|A||B|$

Experiences of carriers of multidrug-resistant organisms: a systematic review

Contains fulltext : 202240.pdf (publisher's version ) (Open Access)OBJECTIVES: A comprehensive overview of the ways control measures directed at carriers of multidrug-resistant organisms (MDRO) affect daily life of carriers is lacking. In this systematic literature review, we sought to explore how carriers experience being a carrier and how they experience being subjected to control measures by looking at the impact on basic capabilities. METHODS: We searched Medline, Embase and PsychINFO until 26 May 2016 for studies addressing experiences of MDRO carriers. Twenty-seven studies were included, addressing experiences with methicillin-resistant Staphylococcus aureus (n = 21), ESBL (n = 1), multiple MDRO (n = 4) and other (n = 1, not specified). We categorized reported experiences according to Nussbaum's capability approach. RESULTS: Carriage and control measures were found to interfere with quality of care, cause negative emotions, limit interactions with loved ones, cause stigmatization, limit recreational activities and create financial and professional insecurity. Further, carriers have difficulties with full comprehension of the problem of antimicrobial resistance, thus affecting six out of ten basic capabilities. CONCLUSIONS: Applying Nussbaum's capability approach visualizes an array of unintended consequences of control measures. Carriers experience stigmatization, especially in healthcare settings, and have limited understanding of their situation and the complexities of antimicrobial resistance