Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The...

We consider the long standing open question on whether one can actually compute spectra and pseudospectra of arbitrary (possibly non-self-adjoint) Schrödinger operators.We conclude that the answer is affirmative for “almost all” such operators, meaning that the operators must satisfy rather weak conditions such as the spectrum cannot be empty nor the whole plane. We include algorithms for the g...

The concept of pseudospectra was introduced by Trefethen during the 1990s and became a popular tool to explain the behavior of non-normal matrices. It is well known that the zeros of a polynomial are equal to the eigenvalues of the associated companion matrix. It is feasible to do the sensitivity analysis of the zeros of polynomials by the tool of pseudospectra of companion matrices. Thus, the ...

The Amoldi iteration, usually viewed as a method for calculating eigenvalues, can also be used to estimate pseudospectra. This possibility may be of practical importance, because in applications involving highly nonnormal matrices or operators, such as hydrodynamic stability, pseudospectra may be physically more significant than spectra. 1. Introduction. Large-scale nonsymmetric matrix eigenval...

The second problem arises for differential operators, particularly in several space dimensions, when the matrix approximations have very high dimensions. Even if A has a sparse matrix, Tt generally has a full matrix, so storing the matrix entries is not feasible. The obvious solution is to find a subspace of relatively small dimension which contains most information of interest. One might try t...

Asymptotic dynamics of ordinary differential equations (ODEs) are commonly understood by looking at eigenvalues of a matrix, and transient dynamics can be bounded above and below by considering the corresponding pseudospectra. While asymptotics for other classes of differential equations have been studied using eigenvalues of a (nonlinear) matrix-valued function, there are no analogous pseudosp...

In this project, I am trying to apply this model to classify networks generated from human brain. Since small-world networks are essentially random graphs whose adjacency matrices are random matrices, I first study the spectrum of them. Next, I assume that a sampled brain network is perturbed version of an averaged network: either that over people with Alzheimer’s disease, or over healthy peopl...

For a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex o...

We prove that non-self-adjoint harmonic and anharmonic oscillator operators have non-trivial pseudospectra. As a consequence, the computation of high energy resonances by the dilation analyticity technique is not numerically stable.

The pseudospectra of banded, nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, related to Hörmander’s commutator condition for partial differential equations, ε-pseudoeigenvectors of...