We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex har...

The pseudospectra of banded finite dimensional Toeplitz matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz matrices—not just eigenvalues. What if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite matrices sti...

In recent years, many network perturbation techniques, such as topological perturbations and service perturbations, were employed to study and improve the robustness of complex networks. However, there is no general way to evaluate the network robustness. In this paper, we propose a global measure for a network, the reconstructability coefficient theta , defined as the maximum number of eigenva...

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the ...

Let λ be a nonderogatory eigenvalue of A ∈ C. The sensitivity of λ with respect to matrix perturbations A A + ∆,∆ ∈ ∆, is measured by the structured condition number κ∆(A,λ). Here ∆ denotes the set of admissible perturbations. However, if ∆ is not a vector space over C then κ∆(A, λ) provides only incomplete information about the mobility of λ under small perturbations from ∆. The full informati...

Structured singular values and pseudospectra play an important role in assessing the properties of a linear system under structured perturbations. This paper discusses computational aspects of structured pseudospectra for structures that admit an eigenvalue minimization characterization, including the classes of real, skew-symmetric, Hermitian, and Hamiltonian perturbations. For all these struc...

ARPACK and its Matlab counterpart, eigs, are software packages that calculate some eigenvalues of a large nonsymmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in largescale eigenvalue ...

Let H and H + H be positive deenite matrices. It was shown by Barlow and Demmel, and Demmel and Veseli c that if one takes a component-wise approach one can prove much stronger bounds on i (H)== i (H++H) and the components of the eigenvectors of H and H++H than by using the standard norm-wise perturbation theory. Here a uniied approach is presented that improves on the results of Barlow, Demmel...

The presence of a deformable free surface in thin films driven to spread by body or shear forces gives rise to base states that are spatially nonuniform. This nonuniformity produces linearized disturbance operators that are non-normal and an eigenvalue spectrum that does not necessarily predict stability behavior. The falling film provides a simple example for demonstrating a more generalized, ...

A dynamical system ẋ = Ax is robustly stable when all eigenvalues of complex matrices within a given distance of the square matrix A lie in the left half-plane. The ‘pseudospectral abscissa’, which is the largest real part of such an eigenvalue, measures the robust stability of A. We present an algorithm for computing the pseudospectral abscissa, prove global and local quadratic convergence, an...