Mark Embree : Spectra and Pseudospectra

Generalizing Eigenvalue Theorems to Pseudospectra Theorems

Piecewise Continuous Toeplitz Matrices and Operators: Slow Approach to Infinity

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...

How Descriptive Are Gmres Convergence Bounds?

Eigenvalues with the eigenvector condition number, the eld of values, and pseu-dospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coeecient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Reened bounds based on eigenvalues and the eld...

Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

To understand the solution of a linear, time-invariant differential-algebraic equation (DAE), one must analyze a matrix pencil (A,E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left half-plane), the solution can exhibit transient growth before its inevitable decay. When the equation results from the linearization of a nonlinear system, this transient...

Spectra and pseudospectra for pipe Poiseuille flow

Numerically computed spectra and pseudospectra are presented for the linear operator that governs the temporal evolution of infinitesimal perturbations of laminar flow in an infinite circular pipe at Reynolds numbers 1000, 3000 and 10 000. The spectra lie strictly inside the stable complex half-plane, but the pseudospectra protrude significantly into the unstable half-plane, reflecting the larg...