Linearization

Résumé

In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization of a function Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that f(x) is differentiable on [a, b] (or [b, a]) and that a is close to b. In short, linearization approxim

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https://en.wikipedia.org/wiki/Linearization

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Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis

Daniel Kressner

Novel memory-efficient Arnoldi algorithms for solving matrix polynomial eigenvalue problems are presented. More specifically, we consider the case of matrix polynomials expressed in the Chebyshev basis, which is often numerically more appropriate than the standard monomial basis for a larger degree d. The standard way of solving polynomial eigenvalue problems proceeds by linearization, which increases the problem size by a factor d. Consequently, the memory requirements of Krylov subspace methods applied to the linearization grow by this factor. In this paper, we develop two variants of the Arnoldi method that build the Krylov subspace basis implicitly, in a way that only vectors of length equal to the size of the original problem need to be stored. The proposed variants are generalizations of the so-called quadratic Arnoldi method and two-level orthogonal Arnoldi procedure methods, which have been developed for the monomial case. We also show how the typical ingredients of a full implementation of the Arnoldi method, including shift-and-invert and restarting, can be incorporated. Numerical experiments are presented for matrix polynomials up to degree 30 arising from the interpolation of nonlinear eigenvalue problems, which stem from boundary element discretizations of PDE eigenvalue problems. Copyright (C) 2013 John Wiley & Sons, Ltd.

Wiley-Blackwell2014

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Quality factor of Contour Mode Resonators (CMR) are mainly affected by energy losses due to acoustic waves leaving the resonator through the anchors. An engineering of the anchors in order to create a periodic variation in the acoustic impedance of the material, structures known as Phononic Crystals (PnCs), can help improve the Q factor by reflecting part of the acoustic waves. During this project, FEM models have been validated for both 1D and 2D PnCs. The behavior of the band diagram and quantification of transmission has been studied for three different PnC geometries: circle, cross and square unit cell. Parameters such as unit cell size, filling factor, material composition, thickness and number of repetitions have been varied and comparisons between transmission graph and band diagram allowed to understand better the nature of the modes. It has been found that a linear approximation can predict with a good precision the position of the band gap as a function of the unit cell size. These studies have been done in order to select the best configuration for a future integration on the substrate of a CMR whose frequency range of operation is situated between 200MHz and 500 MHz. The results obtained report a better attenuation and broader peak of attenuation for a cross unit cell made out of AlN and an attenuation increasing linearly with the number of repetitions.

2016

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Transactors: Unifying Transactions and Actors

Rachid Guerraoui, Mohsen Lesani, Martin Odersky

Composability and deadlock-freedom are important properties that are stated for transactional memory (TM). Commonly, the Semantics of TM requires linearization of transactions. It turns out that linearization of transactions that have cyclic communication brings incomposability and deadlock. Inspired from TM and Actors, this work proposes Transactors that provide facilities of isolation from TM and communication from Actors. We define the semantics of Transactors including support for cyclic transactional communication. An algorithm implementing this semantics is offered. The soundness of the algorithm is proved.

2009

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