The concept of pseudospectrum was introduced by L. N. Trefethen to explain the behavior of nonnormal operators. Many phenomena (for example, hydrodynamic instability and convergence of iterative methods for linear systems) cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra. The straightforward way to compute pseudospectra involves many appl...

The ε-pseudospectral abscissa and radius 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. Existing techniques compute these quantities accurately but the cost is multiple singular value decompositions and eigenvalue decompositions of order n, making them impractical when n is large. We present n...

where B ∈ C, C ∈ C are fixed matrices and ∆ is an element of a subset ∆ of C. It is assumed that ∆ is closed, connected and contains the zero matrix. The size of a perturbation ∆ ∈ ∆ is measured by a norm ‖ · ‖ on C. (a) The structued pseudospectrum (also called spectral value set) of the triple (A,B,C) with respect to the perturbation class ∆, the underlying norm ‖ · ‖ and the perturbation lev...

This paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right-hand sides. A convergence theory is developed based on a representation of the method using matrix-valued polynomials. Relations between the roots of the residual polynomial for block gmres and the matrix "-pseudospectrum are derived, and illustrated with numerical exp...

We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudospectra and boundary conditions in the semi-classical limit.

The -pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance of A. We are interested in two aspects of “optimization and pseudospectra.” The first concerns maximizing the function “real part” over an -pseudospectrum of a fixed matrix: this defines a function known as the -pseudospectral abscissa of a matrix. We pres...

Recent investigations of the pseudospectrum in black hole spacetimes have shown that quasinormal mode frequencies suffer from spectral instabilities. This phenomenon may severely affect gravitational-wave spectroscopy and limit precision tests general relativity. We extend analysis to horizonless exotic compact objects possess a reflective surface arbitrarily close Schwarzschild radius find the...

If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A)−1‖. Five years ago, the author published a...