Search results for "Eigenvalue"
showing 10 items of 344 documents
Diagonalization of large matrices: a new parallel algorithm.
2015
On the basis of a dressed matrices formalism, a new algorithm has been devised for obtaining the lowest eigenvalue and the corresponding eigenvector of large real symmetric matrices. Given an N × N matrix, the proposed algorithm consists in the diagonalization of (N - 1)2 × 2 dressed matrices. Both sequential and parallel versions of the proposed algorithm have been implemented. Tests have been performed on a Hilbert matrix, and the results show that this algorithm is up 340 times faster than the corresponding LAPACK routine for N = 10(4) and about 10% faster than the Davidson method. The parallel MPI version has been tested using up to 512 nodes. The speed-up for a N = 10(6) matrix is fair…
Efficient Pruning LMI Conditions for Branch-and-Prune Rank and Chirality-Constrained Estimation of the Dual Absolute Quadric
2014
International audience; We present a new globally optimal algorithm for self- calibrating a moving camera with constant parameters. Our method aims at estimating the Dual Absolute Quadric (DAQ) under the rank-3 and, optionally, camera centers chirality constraints. We employ the Branch-and-Prune paradigm and explore the space of only 5 parameters. Pruning in our method relies on solving Linear Matrix Inequality (LMI) feasibility and Generalized Eigenvalue (GEV) problems that solely depend upon the entries of the DAQ. These LMI and GEV problems are used to rule out branches in the search tree in which a quadric not satisfy- ing the rank and chirality conditions on camera centers is guarantee…
The spectrum of weakly coupled map lattices
1998
We consider weakly coupled analytic expanding circle maps on the lattice Zd (for d 2 l), with small coupling strength c and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a FrCchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure p* previously obtained by Bricmont-Kupiainen (BKl)) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For d = 1 we also construct Banach spaces of densities with respect to pr on which perturbation theory, applied to the diff…
A symmetrization result for Monge–Ampère type equations
2007
In this paper we prove some comparison results for Monge–Ampere type equations in dimension two. We also consider the case of eigenfunctions and we derive a kind of “reverse” inequalities. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Critical points for nondifferentiable functions in presence of splitting
2006
A classical critical point theorem in presence of splitting established by Brézis-Nirenberg is extended to functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. The obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational-hemivariational eigenvalue problems. © 2005 Elsevier Inc. All rights reserved.
On attracting sets in artificial networks: cross activation
2018
Mathematical models of artificial networks can be formulated in terms of dynamical systems describing the behaviour of a network over time. The interrelation between nodes (elements) of a network is encoded in the regulatory matrix. We consider a system of ordinary differential equations that describes in particular also genomic regulatory networks (GRN) and contains a sigmoidal function. The results are presented on attractors of such systems for a particular case of cross activation. The regulatory matrix is then of particular form consisting of unit entries everywhere except the main diagonal. We show that such a system can have not more than three critical points. At least n–1 eigenvalu…
Tunnel effect and symmetries for Kramers–Fokker–Planck type operators
2011
AbstractWe study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.
The ∞-Eigenvalue Problem
1999
. The Euler‐Lagrange equation of the nonlinear Rayleigh quotient \( \left(\int_{\Omega}|\nabla u|^{p}\,dx\right) \bigg/ \left(\int_{\Omega}|u|^{p}\,dx\right)\) is \( -\div\left( |\nabla u|^{p-2}\nabla u \right)= \Lambda_{p}^{p} |u |^{p-2}u,\) where \(\Lambda_{p}^{p}\) is the minimum value of the quotient. The limit as \(p\to\infty\) of these equations is found to be \(\max \left\{ \Lambda_{\infty}-\frac{|\nabla u(x)|}{u(x)},\ \ \Delta_{\infty}u(x)\right\}=0,\) where the constant \(\Lambda_{\infty}=\lim_{p\to\infty}\Lambda_{p}\) is the reciprocal of the maximum of the distance to the boundary of the domain Ω.
Lower bounds for eigenvalues of a quadratic form relative to a positive quadratic form
1968
Abstract : A method is presented for the calculation of lower bounds to eigenvalues of operators that arise from variational problems for one quadratic form relative to a positive definite quadratic form. Eigenvalue problems of this kind occur, for example, in the theory of buckling of continuous linear elastic systems. The technique used is a modification of one introduced earlier, (1) sections II and IVB, for the determination of lower bounds to eigenvalues of semi-bounded self-adjoint operators. Other methods for the latter problem can be carried over without essential changes. The particular difficulty in the case we consider is that some operators which enter the calculation for the lo…
The eigen-structure of the Jacobian in multi-class Lighthill-Whitham-Richards traffic flow models
2007
Characteristic-based High Resolution Shock Capturing schemes for hyperbolic systems of conservation laws require, in their basic design structure, knowledge on the complete eigen-decomposition of the Jacobian matrix of the system. For the Multi-Class Lighthill-Witham-Richards (MCLWR) Traffic flow model considered in [4], there is no explicit formula for the eigenvalues of the Jacobian matrix, which can only be determined numerically. However, once they are determined, the eigen-vectors are easily computed and straightforward formulas can be obtained by exploiting the specific structure of the Jacobian matrix in these models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)