On condition numbers of polynomial eigenvalue problems with nonsingular leading coefficients

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.Comment: 4 figure

Transient growth in Taylor-Couette flow

Transient growth due to non-normality is investigated for the Taylor-Couette problem with counter-rotating cylinders as a function of aspect ratio eta and Reynolds number Re. For all Re < 500, transient growth is enhanced by curvature, i.e. is greater for eta < 1 than for eta = 1, the plane Couette limit. For fixed Re < 130 it is found that the greatest transient growth is achieved for eta between the Taylor-Couette linear stability boundary, if it exists, and one, while for Re > 130 the greatest transient growth is achieved for eta on the linear stability boundary. Transient growth is shown to be approximately 20% higher near the linear stability boundary at Re = 310, eta = 0.986 than at Re = 310, eta = 1, near the threshold observed for transition in plane Couette flow. The energy in the optimal inputs is primarily meridional; that in the optimal outputs is primarily azimuthal. Pseudospectra are calculated for two contrasting cases. For large curvature, eta = 0.5, the pseudospectra adhere more closely to the spectrum than in a narrow gap case, eta = 0.99