The ‘Alfvén Paradox’ is that as resistivity decreases, the discrete eigenmodes do not converge to the generalized eigenmodes of the ideal Alfvén continuum. To resolve the paradox, the -pseudospectrum of the RMHD operator is considered. It is proven that for any , the pseudospectrum contains the Alfvén continuum for sufficiently small resistivity. Formal − pseudoeigenmodes are constructed using ...

The ε-pseudospectral abscissa αε and radius ρε of an n× n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048–1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a ...

The pseudospectrum of a linear time-invariant system is the set in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa is defined as the maximum real part of the characteristic roots in the pseudospectrum and, therefore, it is for instance important...

This paper is devoted to the study of the innuence of discretization schemes on the pseudospectrum of the rst derivative operator. Spectral and nite diierence methods are considered. Ample illustrations are provided which show the high sensitivity of the pseudospectrum to the chosen discretization method.

The -pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance of A, measured by the operator 2-norm. Given a nonderogatory matrix A0, for small > 0, we show that the -pseudospectrum of any matrix A near A0 consists of compact convex neighborhoods of the eigenvalues of A0. Furthermore, the dependence of each of these neig...

It has been documented in the literature that the pseudospectrum of a matrix is a powerful concept that broadens our understanding of phenomena based on matrix computations. When the matrix A is non-normal, however, the computation of the pseudospectrum becomes a very expensive computational task. Thus, the use of high performance computing resources becomes key to obtaining useful answers in a...

Pseudospectra of matrix polynomials have been systematically investigated in the last years, since they provide important insights into the sensitivity of polynomial eigenvalue problems. An accurate approximation of the pseudospectrum of a matrix polynomial P (λ) by means of the standard grid method is computationally high demanding. In this paper, we propose an improvement of the grid method, ...

A preliminary example 16 4.2. Quantization of the torus 21 4.3. The complex projective line 26 5. Final Remarks 31 5.1. On the numerical range 31 5.2. The weak Szegö limit theorem 32 Appendix A. Hermite distributions and symbol calculus 33 References 38 1. Introduction The study of pseudospectra of operators is a very active area of research, of importance in applied mathematics and numerical a...

We investigate the behavior of eigenvalues under structured perturbations. We show that for many common structures such as (complex) symmetric, Toeplitz, symmetric Toeplitz, circulant and others the structured condition number is equal to the unstructured condition number for normwise perturbations, and prove similar results for real perturbations. An exception are complex skewsymmetric matrice...