Safe localization of eigenvalues

Abstract : Localizing some eigenvalues of a given large sparse matrix in a domain of the complex plane is a hard task when the matrix is non symmetric, especially when it is highly non normal. For taking into account, possible perturbations of the matrix, the notion of the of \epsilon-spectrum or pseudospectrum of a matrix $A \in \mathbb{R}^{n \times n}$ was separately defined by Godunov and Trefethen. Determining an $\epsilon$-spectrum consists of determining a level curve of the 2-norm of the resolvent $R(z) = (zI-A)^{-1}$. A dual approach can be considered: given some curve $(\Gamma)$ in the complex plane, count the number of eigenvalues of the matrix $A$ that are surrounded by $(\Gamma)$. The number of surrounded eigenvalues is determined by evaluating the integral $\frac{1}{2i\pi} \int_{\Gamma}{\frac{d}{dz}\log \det (zI-A) dz}$. This problem was considered in [Bertrand and Philippe, 2001] where several procedures were proposed and more recently in [Kamgnia and Philippe, to appear] where the stepsize control in the quadrature is deeply studied. The present goal is to combine the two approaches: (i) consider the method PAT [Mezher and Philippe, Numer. Algorithms, 2002, Mezher and Philippe, Parallel Comput, 2002] which is a path following method that determines a level curve of the function $s(z)=\sigma_{\min}(zI-A)$; (ii) apply the method EIGENCNT of [Kamgnia and Philippe, to appear] for computing the number of eigenvalues included. The combined procedure will be based on a computing kernel which provides the two numbers (\sigma_{min}(zI - A), det(zI - A)) for any complex number $z \in \mathbb{C}$. These two numbers are obtained through a common LU factorization of $(zI - A)$. In order to introduce a second level of parallelism, we consider a preprocessing transformation similar to the approach developed in SPIKE [Polizzi and Sameh, 2006].
Type de document :
Communication dans un congrès
Numerical Analysis and Scientific Computation with Applications (NASCA13), 2013, Calais, France. pp.66, 2013
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https://hal.inria.fr/hal-00903739
Contributeur : Géraldine Pichot <>
Soumis le : jeudi 21 novembre 2013 - 10:26:05
Dernière modification le : mardi 16 janvier 2018 - 15:54:11

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  • HAL Id : hal-00903739, version 1

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Bernard Philippe, Emmanuel Kamgnia, Louis Bernard Nguenang. Safe localization of eigenvalues. Numerical Analysis and Scientific Computation with Applications (NASCA13), 2013, Calais, France. pp.66, 2013. 〈hal-00903739〉

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